Reference documentation for deal.II version 9.5.0
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step-37.h
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1) const
757 *   {
758 *   return 1. / (0.05 + 2. * p.square());
759 *   }
760 *  
761 *  
762 *  
763 *   template <int dim>
764 *   double Coefficient<dim>::value(const Point<dim> & p,
765 *   const unsigned int component) const
766 *   {
767 *   return value<double>(p, component);
768 *   }
769 *  
770 *  
771 * @endcode
772 *
773 *
774 * <a name="Matrixfreeimplementation"></a>
775 * <h3>Matrix-free implementation</h3>
776 *
777
778 *
779 * The following class, called <code>LaplaceOperator</code>, implements the
780 * differential operator. For all practical purposes, it is a matrix, i.e.,
781 * you can ask it for its size (member functions <code>m(), n()</code>) and
782 * you can apply it to a vector (the <code>vmult()</code> function). The
783 * difference to a real matrix of course lies in the fact that this class
784 * does not actually store the <i>elements</i> of the matrix, but only knows
785 * how to compute the action of the operator when applied to a vector.
786 *
787
788 *
789 * The infrastructure describing the matrix size, the initialization from a
790 * MatrixFree object, and the various interfaces to matrix-vector products
791 * through vmult() and Tvmult() methods, is provided by the class
792 * MatrixFreeOperator::Base from which this class derives. The
793 * LaplaceOperator class defined here only has to provide a few interfaces,
794 * namely the actual action of the operator through the apply_add() method
795 * that gets used in the vmult() functions, and a method to compute the
796 * diagonal entries of the underlying matrix. We need the diagonal for the
797 * definition of the multigrid smoother. Since we consider a problem with
798 * variable coefficient, we further implement a method that can fill the
799 * coefficient values.
800 *
801
802 *
803 * Note that the file <code>include/deal.II/matrix_free/operators.h</code>
804 * already contains an implementation of the Laplacian through the class
805 * MatrixFreeOperators::LaplaceOperator. For educational purposes, the
806 * operator is re-implemented in this tutorial program, explaining the
807 * ingredients and concepts used there.
808 *
809
810 *
811 * This program makes use of the data cache for finite element operator
812 * application that is integrated in deal.II. This data cache class is
813 * called MatrixFree. It contains mapping information (Jacobians) and index
814 * relations between local and global degrees of freedom. It also contains
815 * constraints like the ones from hanging nodes or Dirichlet boundary
816 * conditions. Moreover, it can issue a loop over all cells in %parallel,
817 * making sure that only cells are worked on that do not share any degree of
818 * freedom (this makes the loop thread-safe when writing into destination
819 * vectors). This is a more advanced strategy compared to the WorkStream
820 * class described in the @ref threads module. Of course, to not destroy
821 * thread-safety, we have to be careful when writing into class-global
822 * structures.
823 *
824
825 *
826 * The class implementing the Laplace operator has three template arguments,
827 * one for the dimension (as many deal.II classes carry), one for the degree
828 * of the finite element (which we need to enable efficient computations
829 * through the FEEvaluation class), and one for the underlying scalar
830 * type. We want to use <code>double</code> numbers (i.e., double precision,
831 * 64-bit floating point) for the final matrix, but floats (single
832 * precision, 32-bit floating point numbers) for the multigrid level
833 * matrices (as that is only a preconditioner, and floats can be processed
834 * twice as fast). The class FEEvaluation also takes a template argument for
835 * the number of quadrature points in one dimension. In the code below, we
836 * hard-code it to <code>fe_degree+1</code>. If we wanted to change it
837 * independently of the polynomial degree, we would need to add a template
838 * parameter as is done in the MatrixFreeOperators::LaplaceOperator class.
839 *
840
841 *
842 * As a sidenote, if we implemented several different operations on the same
843 * grid and degrees of freedom (like a @ref GlossMassMatrix "mass matrix" and a Laplace matrix), we
844 * would define two classes like the current one for each of the operators
845 * (derived from the MatrixFreeOperators::Base class), and let both of them
846 * refer to the same MatrixFree data cache from the general problem
847 * class. The interface through MatrixFreeOperators::Base requires us to
848 * only provide a minimal set of functions. This concept allows for writing
849 * complex application codes with many matrix-free operations.
850 *
851
852 *
853 * @note Storing values of type <code>VectorizedArray<number></code>
854 * requires care: Here, we use the deal.II table class which is prepared to
855 * hold the data with correct alignment. However, storing e.g. an
856 * <code>std::vector<VectorizedArray<number> ></code> is not possible with
857 * vectorization: A certain alignment of the data with the memory address
858 * boundaries is required (essentially, a VectorizedArray that is 32 bytes
859 * long in case of AVX needs to start at a memory address that is divisible
860 * by 32). The table class (as well as the AlignedVector class it is based
861 * on) makes sure that this alignment is respected, whereas std::vector does
862 * not in general, which may lead to segmentation faults at strange places
863 * for some systems or suboptimal performance for other systems.
864 *
865 * @code
866 *   template <int dim, int fe_degree, typename number>
867 *   class LaplaceOperator
868 *   : public MatrixFreeOperators::
869 *   Base<dim, LinearAlgebra::distributed::Vector<number>>
870 *   {
871 *   public:
872 *   using value_type = number;
873 *  
874 *   LaplaceOperator();
875 *  
876 *   void clear() override;
877 *  
878 *   void evaluate_coefficient(const Coefficient<dim> &coefficient_function);
879 *  
880 *   virtual void compute_diagonal() override;
881 *  
882 *   private:
883 *   virtual void apply_add(
885 *   const LinearAlgebra::distributed::Vector<number> &src) const override;
886 *  
887 *   void
888 *   local_apply(const MatrixFree<dim, number> & data,
891 *   const std::pair<unsigned int, unsigned int> &cell_range) const;
892 *  
893 *   void local_compute_diagonal(
894 *   const MatrixFree<dim, number> & data,
896 *   const unsigned int & dummy,
897 *   const std::pair<unsigned int, unsigned int> &cell_range) const;
898 *  
899 *   Table<2, VectorizedArray<number>> coefficient;
900 *   };
901 *  
902 *  
903 *  
904 * @endcode
905 *
906 * This is the constructor of the @p LaplaceOperator class. All it does is
907 * to call the default constructor of the base class
908 * MatrixFreeOperators::Base, which in turn is based on the Subscriptor
909 * class that asserts that this class is not accessed after going out of scope
910 * e.g. in a preconditioner.
911 *
912 * @code
913 *   template <int dim, int fe_degree, typename number>
914 *   LaplaceOperator<dim, fe_degree, number>::LaplaceOperator()
915 *   : MatrixFreeOperators::Base<dim,
917 *   {}
918 *  
919 *  
920 *  
921 *   template <int dim, int fe_degree, typename number>
922 *   void LaplaceOperator<dim, fe_degree, number>::clear()
923 *   {
924 *   coefficient.reinit(0, 0);
926 *   clear();
927 *   }
928 *  
929 *  
930 *  
931 * @endcode
932 *
933 *
934 * <a name="Computationofcoefficient"></a>
935 * <h4>Computation of coefficient</h4>
936 *
937
938 *
939 * To initialize the coefficient, we directly give it the Coefficient class
940 * defined above and then select the method
941 * <code>coefficient_function.value</code> with vectorized number (which the
942 * compiler can deduce from the point data type). The use of the
943 * FEEvaluation class (and its template arguments) will be explained below.
944 *
945 * @code
946 *   template <int dim, int fe_degree, typename number>
947 *   void LaplaceOperator<dim, fe_degree, number>::evaluate_coefficient(
948 *   const Coefficient<dim> &coefficient_function)
949 *   {
950 *   const unsigned int n_cells = this->data->n_cell_batches();
952 *  
953 *   coefficient.reinit(n_cells, phi.n_q_points);
954 *   for (unsigned int cell = 0; cell < n_cells; ++cell)
955 *   {
956 *   phi.reinit(cell);
957 *   for (unsigned int q = 0; q < phi.n_q_points; ++q)
958 *   coefficient(cell, q) =
959 *   coefficient_function.value(phi.quadrature_point(q));
960 *   }
961 *   }
962 *  
963 *  
964 *  
965 * @endcode
966 *
967 *
968 * <a name="LocalevaluationofLaplaceoperator"></a>
969 * <h4>Local evaluation of Laplace operator</h4>
970 *
971
972 *
973 * Here comes the main function of this class, the evaluation of the
974 * matrix-vector product (or, in general, a finite element operator
975 * evaluation). This is done in a function that takes exactly four
976 * arguments, the MatrixFree object, the destination and source vectors, and
977 * a range of cells that are to be worked on. The method
978 * <code>cell_loop</code> in the MatrixFree class will internally call this
979 * function with some range of cells that is obtained by checking which
980 * cells are possible to work on simultaneously so that write operations do
981 * not cause any race condition. Note that the cell range used in the loop
982 * is not directly the number of (active) cells in the current mesh, but
983 * rather a collection of batches of cells. In other word, "cell" may be
984 * the wrong term to begin with, since FEEvaluation groups data from several
985 * cells together. This means that in the loop over quadrature points we are
986 * actually seeing a group of quadrature points of several cells as one
987 * block. This is done to enable a higher degree of vectorization. The
988 * number of such "cells" or "cell batches" is stored in MatrixFree and can
989 * be queried through MatrixFree::n_cell_batches(). Compared to the deal.II
990 * cell iterators, in this class all cells are laid out in a plain array
991 * with no direct knowledge of level or neighborship relations, which makes
992 * it possible to index the cells by unsigned integers.
993 *
994
995 *
996 * The implementation of the Laplace operator is quite simple: First, we
997 * need to create an object FEEvaluation that contains the computational
998 * kernels and has data fields to store temporary results (e.g. gradients
999 * evaluated on all quadrature points on a collection of a few cells). Note
1000 * that temporary results do not use a lot of memory, and since we specify
1001 * template arguments with the element order, the data is stored on the
1002 * stack (without expensive memory allocation). Usually, one only needs to
1003 * set two template arguments, the dimension as a first argument and the
1004 * degree of the finite element as the second argument (this is equal to the
1005 * number of degrees of freedom per dimension minus one for FE_Q
1006 * elements). However, here we also want to be able to use float numbers for
1007 * the multigrid preconditioner, which is the last (fifth) template
1008 * argument. Therefore, we cannot rely on the default template arguments and
1009 * must also fill the third and fourth field, consequently. The third
1010 * argument specifies the number of quadrature points per direction and has
1011 * a default value equal to the degree of the element plus one. The fourth
1012 * argument sets the number of components (one can also evaluate
1013 * vector-valued functions in systems of PDEs, but the default is a scalar
1014 * element), and finally the last argument sets the number type.
1015 *
1016
1017 *
1018 * Next, we loop over the given cell range and then we continue with the
1019 * actual implementation: <ol> <li>Tell the FEEvaluation object the (macro)
1020 * cell we want to work on. <li>Read in the values of the source vectors
1021 * (@p read_dof_values), including the resolution of constraints. This
1022 * stores @f$u_\mathrm{cell}@f$ as described in the introduction. <li>Compute
1023 * the unit-cell gradient (the evaluation of finite element
1024 * functions). Since FEEvaluation can combine value computations with
1025 * gradient computations, it uses a unified interface to all kinds of
1026 * derivatives of order between zero and two. We only want gradients, no
1027 * values and no second derivatives, so we set the function arguments to
1028 * true in the gradient slot (second slot), and to false in the values slot
1029 * (first slot). There is also a third slot for the Hessian which is
1030 * false by default, so it needs not be given. Note that the FEEvaluation
1031 * class internally evaluates shape functions in an efficient way where one
1032 * dimension is worked on at a time (using the tensor product form of shape
1033 * functions and quadrature points as mentioned in the introduction). This
1034 * gives complexity equal to @f$\mathcal O(d^2 (p+1)^{d+1})@f$ for polynomial
1035 * degree @f$p@f$ in @f$d@f$ dimensions, compared to the naive approach with loops
1036 * over all local degrees of freedom and quadrature points that is used in
1037 * FEValues and costs @f$\mathcal O(d (p+1)^{2d})@f$. <li>Next comes the
1038 * application of the Jacobian transformation, the multiplication by the
1039 * variable coefficient and the quadrature weight. FEEvaluation has an
1040 * access function @p get_gradient that applies the Jacobian and returns the
1041 * gradient in real space. Then, we just need to multiply by the (scalar)
1042 * coefficient, and let the function @p submit_gradient apply the second
1043 * Jacobian (for the test function) and the quadrature weight and Jacobian
1044 * determinant (JxW). Note that the submitted gradient is stored in the same
1045 * data field as where it is read from in @p get_gradient. Therefore, you
1046 * need to make sure to not read from the same quadrature point again after
1047 * having called @p submit_gradient on that particular quadrature point. In
1048 * general, it is a good idea to copy the result of @p get_gradient when it
1049 * is used more often than once. <li>Next follows the summation over
1050 * quadrature points for all test functions that corresponds to the actual
1051 * integration step. For the Laplace operator, we just multiply by the
1052 * gradient, so we call the integrate function with the respective argument
1053 * set. If you have an equation where you test by both the values of the
1054 * test functions and the gradients, both template arguments need to be set
1055 * to true. Calling first the integrate function for values and then
1056 * gradients in a separate call leads to wrong results, since the second
1057 * call will internally overwrite the results from the first call. Note that
1058 * there is no function argument for the second derivative for integrate
1059 * step. <li>Eventually, the local contributions in the vector
1060 * @f$v_\mathrm{cell}@f$ as mentioned in the introduction need to be added into
1061 * the result vector (and constraints are applied). This is done with a call
1062 * to @p distribute_local_to_global, the same name as the corresponding
1063 * function in the AffineConstraints (only that we now store the local vector
1064 * in the FEEvaluation object, as are the indices between local and global
1065 * degrees of freedom). </ol>
1066 *
1067 * @code
1068 *   template <int dim, int fe_degree, typename number>
1069 *   void LaplaceOperator<dim, fe_degree, number>::local_apply(
1070 *   const MatrixFree<dim, number> & data,
1073 *   const std::pair<unsigned int, unsigned int> & cell_range) const
1074 *   {
1076 *  
1077 *   for (unsigned int cell = cell_range.first; cell < cell_range.second; ++cell)
1078 *   {
1079 *   AssertDimension(coefficient.size(0), data.n_cell_batches());
1080 *   AssertDimension(coefficient.size(1), phi.n_q_points);
1081 *  
1082 *   phi.reinit(cell);
1083 *   phi.read_dof_values(src);
1084 *   phi.evaluate(EvaluationFlags::gradients);
1085 *   for (unsigned int q = 0; q < phi.n_q_points; ++q)
1086 *   phi.submit_gradient(coefficient(cell, q) * phi.get_gradient(q), q);
1087 *   phi.integrate(EvaluationFlags::gradients);
1088 *   phi.distribute_local_to_global(dst);
1089 *   }
1090 *   }
1091 *  
1092 *  
1093 *  
1094 * @endcode
1095 *
1096 * This function implements the loop over all cells for the
1097 * Base::apply_add() interface. This is done with the @p cell_loop of the
1098 * MatrixFree class, which takes the operator() of this class with arguments
1099 * MatrixFree, OutVector, InVector, cell_range. When working with MPI
1100 * parallelization (but no threading) as is done in this tutorial program,
1101 * the cell loop corresponds to the following three lines of code:
1102 *
1103
1104 *
1105 * <div class=CodeFragmentInTutorialComment>
1106 * @code
1107 * src.update_ghost_values();
1108 * local_apply(*this->data, dst, src, std::make_pair(0U,
1109 * data.n_cell_batches()));
1110 * dst.compress(VectorOperation::add);
1111 * @endcode
1112 * </div>
1113 *
1114
1115 *
1116 * Here, the two calls update_ghost_values() and compress() perform the data
1117 * exchange on processor boundaries for MPI, once for the source vector
1118 * where we need to read from entries owned by remote processors, and once
1119 * for the destination vector where we have accumulated parts of the
1120 * residuals that need to be added to the respective entry of the owner
1121 * processor. However, MatrixFree::cell_loop does not only abstract away
1122 * those two calls, but also performs some additional optimizations. On the
1123 * one hand, it will split the update_ghost_values() and compress() calls in
1124 * a way to allow for overlapping communication and computation. The
1125 * local_apply function is then called with three cell ranges representing
1126 * partitions of the cell range from 0 to MatrixFree::n_cell_batches(). On
1127 * the other hand, cell_loop also supports thread parallelism in which case
1128 * the cell ranges are split into smaller chunks and scheduled in an
1129 * advanced way that avoids access to the same vector entry by several
1130 * threads. That feature is explained in @ref step_48 "step-48".
1131 *
1132
1133 *
1134 * Note that after the cell loop, the constrained degrees of freedom need to
1135 * be touched once more for sensible vmult() operators: Since the assembly
1136 * loop automatically resolves constraints (just as the
1137 * AffineConstraints::distribute_local_to_global() call does), it does not
1138 * compute any contribution for constrained degrees of freedom, leaving the
1139 * respective entries zero. This would represent a matrix that had empty
1140 * rows and columns for constrained degrees of freedom. However, iterative
1141 * solvers like CG only work for non-singular matrices. The easiest way to
1142 * do that is to set the sub-block of the matrix that corresponds to
1143 * constrained DoFs to an identity matrix, in which case application of the
1144 * matrix would simply copy the elements of the right hand side vector into
1145 * the left hand side. Fortunately, the vmult() implementations
1146 * MatrixFreeOperators::Base do this automatically for us outside the
1147 * apply_add() function, so we do not need to take further action here.
1148 *
1149
1150 *
1151 * When using the combination of MatrixFree and FEEvaluation in parallel
1152 * with MPI, there is one aspect to be careful about &mdash; the indexing
1153 * used for accessing the vector. For performance reasons, MatrixFree and
1154 * FEEvaluation are designed to access vectors in MPI-local index space also
1155 * when working with multiple processors. Working in local index space means
1156 * that no index translation needs to be performed at the place the vector
1157 * access happens, apart from the unavoidable indirect addressing. However,
1158 * local index spaces are ambiguous: While it is standard convention to
1159 * access the locally owned range of a vector with indices between 0 and the
1160 * local size, the numbering is not so clear for the ghosted entries and
1161 * somewhat arbitrary. For the matrix-vector product, only the indices
1162 * appearing on locally owned cells (plus those referenced via hanging node
1163 * constraints) are necessary. However, in deal.II we often set all the
1164 * degrees of freedom on ghosted elements as ghosted vector entries, called
1165 * the
1166 * @ref GlossLocallyRelevantDof "locally relevant DoFs described in the glossary".
1167 * In that case, the MPI-local index of a ghosted vector entry can in
1168 * general be different in the two possible ghost sets, despite referring
1169 * to the same global index. To avoid problems, FEEvaluation checks that
1170 * the partitioning of the vector used for the matrix-vector product does
1171 * indeed match with the partitioning of the indices in MatrixFree by a
1172 * check called
1173 * LinearAlgebra::distributed::Vector::partitioners_are_compatible. To
1174 * facilitate things, the MatrixFreeOperators::Base class includes a
1175 * mechanism to fit the ghost set to the correct layout. This happens in the
1176 * ghost region of the vector, so keep in mind that the ghost region might
1177 * be modified in both the destination and source vector after a call to a
1178 * vmult() method. This is legitimate because the ghost region of a
1179 * distributed deal.II vector is a mutable section and filled on
1180 * demand. Vectors used in matrix-vector products must not be ghosted upon
1181 * entry of vmult() functions, so no information gets lost.
1182 *
1183 * @code
1184 *   template <int dim, int fe_degree, typename number>
1185 *   void LaplaceOperator<dim, fe_degree, number>::apply_add(
1186 *   LinearAlgebra::distributed::Vector<number> & dst,
1187 *   const LinearAlgebra::distributed::Vector<number> &src) const
1188 *   {
1189 *   this->data->cell_loop(&LaplaceOperator::local_apply, this, dst, src);
1190 *   }
1191 *  
1192 *  
1193 *  
1194 * @endcode
1195 *
1196 * The following function implements the computation of the diagonal of the
1197 * operator. Computing matrix entries of a matrix-free operator evaluation
1198 * turns out to be more complicated than evaluating the
1199 * operator. Fundamentally, we could obtain a matrix representation of the
1200 * operator by applying the operator on <i>all</i> unit vectors. Of course,
1201 * that would be very inefficient since we would need to perform <i>n</i>
1202 * operator evaluations to retrieve the whole matrix. Furthermore, this
1203 * approach would completely ignore the matrix sparsity. On an individual
1204 * cell, however, this is the way to go and actually not that inefficient as
1205 * there usually is a coupling between all degrees of freedom inside the
1206 * cell.
1207 *
1208
1209 *
1210 * We first initialize the diagonal vector to the correct parallel
1211 * layout. This vector is encapsulated in a member called
1212 * inverse_diagonal_entries of type DiagonalMatrix in the base class
1213 * MatrixFreeOperators::Base. This member is a shared pointer that we first
1214 * need to initialize and then get the vector representing the diagonal
1215 * entries in the matrix. As to the actual diagonal computation, we again
1216 * use the cell_loop infrastructure of MatrixFree to invoke a local worker
1217 * routine called local_compute_diagonal(). Since we will only write into a
1218 * vector but not have any source vector, we put a dummy argument of type
1219 * <tt>unsigned int</tt> in place of the source vector to confirm with the
1220 * cell_loop interface. After the loop, we need to set the vector entries
1221 * subject to Dirichlet boundary conditions to one (either those on the
1222 * boundary described by the AffineConstraints object inside MatrixFree or
1223 * the indices at the interface between different grid levels in adaptive
1224 * multigrid). This is done through the function
1226 * with the setting in the matrix-vector product provided by the Base
1227 * operator. Finally, we need to invert the diagonal entries which is the
1228 * form required by the Chebyshev smoother based on the Jacobi iteration. In
1229 * the loop, we assert that all entries are non-zero, because they should
1230 * either have obtained a positive contribution from integrals or be
1231 * constrained and treated by @p set_constrained_entries_to_one() following
1232 * cell_loop.
1233 *
1234 * @code
1235 *   template <int dim, int fe_degree, typename number>
1236 *   void LaplaceOperator<dim, fe_degree, number>::compute_diagonal()
1237 *   {
1238 *   this->inverse_diagonal_entries.reset(
1240 *   LinearAlgebra::distributed::Vector<number> &inverse_diagonal =
1241 *   this->inverse_diagonal_entries->get_vector();
1242 *   this->data->initialize_dof_vector(inverse_diagonal);
1243 *   unsigned int dummy = 0;
1244 *   this->data->cell_loop(&LaplaceOperator::local_compute_diagonal,
1245 *   this,
1246 *   inverse_diagonal,
1247 *   dummy);
1248 *  
1249 *   this->set_constrained_entries_to_one(inverse_diagonal);
1250 *  
1251 *   for (unsigned int i = 0; i < inverse_diagonal.locally_owned_size(); ++i)
1252 *   {
1253 *   Assert(inverse_diagonal.local_element(i) > 0.,
1254 *   ExcMessage("No diagonal entry in a positive definite operator "
1255 *   "should be zero"));
1256 *   inverse_diagonal.local_element(i) =
1257 *   1. / inverse_diagonal.local_element(i);
1258 *   }
1259 *   }
1260 *  
1261 *  
1262 *  
1263 * @endcode
1264 *
1265 * In the local compute loop, we compute the diagonal by a loop over all
1266 * columns in the local matrix and putting the entry 1 in the <i>i</i>th
1267 * slot and a zero entry in all other slots, i.e., we apply the cell-wise
1268 * differential operator on one unit vector at a time. The inner part
1269 * invoking FEEvaluation::evaluate, the loop over quadrature points, and
1270 * FEEvalution::integrate, is exactly the same as in the local_apply
1271 * function. Afterwards, we pick out the <i>i</i>th entry of the local
1272 * result and put it to a temporary storage (as we overwrite all entries in
1273 * the array behind FEEvaluation::get_dof_value() with the next loop
1274 * iteration). Finally, the temporary storage is written to the destination
1275 * vector. Note how we use FEEvaluation::get_dof_value() and
1276 * FEEvaluation::submit_dof_value() to read and write to the data field that
1277 * FEEvaluation uses for the integration on the one hand and writes into the
1278 * global vector on the other hand.
1279 *
1280
1281 *
1282 * Given that we are only interested in the matrix diagonal, we simply throw
1283 * away all other entries of the local matrix that have been computed along
1284 * the way. While it might seem wasteful to compute the complete cell matrix
1285 * and then throw away everything but the diagonal, the integration are so
1286 * efficient that the computation does not take too much time. Note that the
1287 * complexity of operator evaluation per element is @f$\mathcal
1288 * O((p+1)^{d+1})@f$ for polynomial degree @f$k@f$, so computing the whole matrix
1289 * costs us @f$\mathcal O((p+1)^{2d+1})@f$ operations, not too far away from
1290 * @f$\mathcal O((p+1)^{2d})@f$ complexity for computing the diagonal with
1291 * FEValues. Since FEEvaluation is also considerably faster due to
1292 * vectorization and other optimizations, the diagonal computation with this
1293 * function is actually the fastest (simple) variant. (It would be possible
1294 * to compute the diagonal with sum factorization techniques in @f$\mathcal
1295 * O((p+1)^{d+1})@f$ operations involving specifically adapted
1296 * kernels&mdash;but since such kernels are only useful in that particular
1297 * context and the diagonal computation is typically not on the critical
1298 * path, they have not been implemented in deal.II.)
1299 *
1300
1301 *
1302 * Note that the code that calls distribute_local_to_global on the vector to
1303 * accumulate the diagonal entries into the global matrix has some
1304 * limitations. For operators with hanging node constraints that distribute
1305 * an integral contribution of a constrained DoF to several other entries
1306 * inside the distribute_local_to_global call, the vector interface used
1307 * here does not exactly compute the diagonal entries, but lumps some
1308 * contributions located on the diagonal of the local matrix that would end
1309 * up in a off-diagonal position of the global matrix to the diagonal. The
1310 * result is correct up to discretization accuracy as explained in <a
1311 * href="http://dx.doi.org/10.4208/cicp.101214.021015a">Kormann (2016),
1312 * section 5.3</a>, but not mathematically equal. In this tutorial program,
1313 * no harm can happen because the diagonal is only used for the multigrid
1314 * level matrices where no hanging node constraints appear.
1315 *
1316 * @code
1317 *   template <int dim, int fe_degree, typename number>
1318 *   void LaplaceOperator<dim, fe_degree, number>::local_compute_diagonal(
1319 *   const MatrixFree<dim, number> & data,
1321 *   const unsigned int &,
1322 *   const std::pair<unsigned int, unsigned int> &cell_range) const
1323 *   {
1325 *  
1326 *   AlignedVector<VectorizedArray<number>> diagonal(phi.dofs_per_cell);
1327 *  
1328 *   for (unsigned int cell = cell_range.first; cell < cell_range.second; ++cell)
1329 *   {
1330 *   AssertDimension(coefficient.size(0), data.n_cell_batches());
1331 *   AssertDimension(coefficient.size(1), phi.n_q_points);
1332 *  
1333 *   phi.reinit(cell);
1334 *   for (unsigned int i = 0; i < phi.dofs_per_cell; ++i)
1335 *   {
1336 *   for (unsigned int j = 0; j < phi.dofs_per_cell; ++j)
1337 *   phi.submit_dof_value(VectorizedArray<number>(), j);
1338 *   phi.submit_dof_value(make_vectorized_array<number>(1.), i);
1339 *  
1340 *   phi.evaluate(EvaluationFlags::gradients);
1341 *   for (unsigned int q = 0; q < phi.n_q_points; ++q)
1342 *   phi.submit_gradient(coefficient(cell, q) * phi.get_gradient(q),
1343 *   q);
1344 *   phi.integrate(EvaluationFlags::gradients);
1345 *   diagonal[i] = phi.get_dof_value(i);
1346 *   }
1347 *   for (unsigned int i = 0; i < phi.dofs_per_cell; ++i)
1348 *   phi.submit_dof_value(diagonal[i], i);
1349 *   phi.distribute_local_to_global(dst);
1350 *   }
1351 *   }
1352 *  
1353 *  
1354 *  
1355 * @endcode
1356 *
1357 *
1358 * <a name="LaplaceProblemclass"></a>
1359 * <h3>LaplaceProblem class</h3>
1360 *
1361
1362 *
1363 * This class is based on the one in @ref step_16 "step-16". However, we replaced the
1364 * SparseMatrix<double> class by our matrix-free implementation, which means
1365 * that we can also skip the sparsity patterns. Notice that we define the
1366 * LaplaceOperator class with the degree of finite element as template
1367 * argument (the value is defined at the top of the file), and that we use
1368 * float numbers for the multigrid level matrices.
1369 *
1370
1371 *
1372 * The class also has a member variable to keep track of all the detailed
1373 * timings for setting up the entire chain of data before we actually go
1374 * about solving the problem. In addition, there is an output stream (that
1375 * is disabled by default) that can be used to output details for the
1376 * individual setup operations instead of the summary only that is printed
1377 * out by default.
1378 *
1379
1380 *
1381 * Since this program is designed to be used with MPI, we also provide the
1382 * usual @p pcout output stream that only prints the information of the
1383 * processor with MPI rank 0. The grid used for this programs can either be
1384 * a distributed triangulation based on p4est (in case deal.II is configured
1385 * to use p4est), otherwise it is a serial grid that only runs without MPI.
1386 *
1387 * @code
1388 *   template <int dim>
1389 *   class LaplaceProblem
1390 *   {
1391 *   public:
1392 *   LaplaceProblem();
1393 *   void run();
1394 *  
1395 *   private:
1396 *   void setup_system();
1397 *   void assemble_rhs();
1398 *   void solve();
1399 *   void output_results(const unsigned int cycle) const;
1400 *  
1401 *   #ifdef DEAL_II_WITH_P4EST
1403 *   #else
1405 *   #endif
1406 *  
1407 *   FE_Q<dim> fe;
1408 *   DoFHandler<dim> dof_handler;
1409 *  
1410 *   MappingQ1<dim> mapping;
1411 *  
1412 *   AffineConstraints<double> constraints;
1413 *   using SystemMatrixType =
1414 *   LaplaceOperator<dim, degree_finite_element, double>;
1415 *   SystemMatrixType system_matrix;
1416 *  
1417 *   MGConstrainedDoFs mg_constrained_dofs;
1418 *   using LevelMatrixType = LaplaceOperator<dim, degree_finite_element, float>;
1419 *   MGLevelObject<LevelMatrixType> mg_matrices;
1420 *  
1423 *  
1424 *   double setup_time;
1425 *   ConditionalOStream pcout;
1426 *   ConditionalOStream time_details;
1427 *   };
1428 *  
1429 *  
1430 *  
1431 * @endcode
1432 *
1433 * When we initialize the finite element, we of course have to use the
1434 * degree specified at the top of the file as well (otherwise, an exception
1435 * will be thrown at some point, since the computational kernel defined in
1436 * the templated LaplaceOperator class and the information from the finite
1437 * element read out by MatrixFree will not match). The constructor of the
1438 * triangulation needs to set an additional flag that tells the grid to
1439 * conform to the 2:1 cell balance over vertices, which is needed for the
1440 * convergence of the geometric multigrid routines. For the distributed
1441 * grid, we also need to specifically enable the multigrid hierarchy.
1442 *
1443 * @code
1444 *   template <int dim>
1445 *   LaplaceProblem<dim>::LaplaceProblem()
1446 *   #ifdef DEAL_II_WITH_P4EST
1447 *   : triangulation(MPI_COMM_WORLD,
1450 *   dim>::construct_multigrid_hierarchy)
1451 *   #else
1453 *   #endif
1454 *   , fe(degree_finite_element)
1455 *   , dof_handler(triangulation)
1456 *   , setup_time(0.)
1457 *   , pcout(std::cout, Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0)
1458 * @endcode
1459 *
1460 * The LaplaceProblem class holds an additional output stream that
1461 * collects detailed timings about the setup phase. This stream, called
1462 * time_details, is disabled by default through the @p false argument
1463 * specified here. For detailed timings, removing the @p false argument
1464 * prints all the details.
1465 *
1466 * @code
1467 *   , time_details(std::cout,
1468 *   false &&
1469 *   Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0)
1470 *   {}
1471 *  
1472 *  
1473 *  
1474 * @endcode
1475 *
1476 *
1477 * <a name="LaplaceProblemsetup_system"></a>
1478 * <h4>LaplaceProblem::setup_system</h4>
1479 *
1480
1481 *
1482 * The setup stage is in analogy to @ref step_16 "step-16" with relevant changes due to the
1483 * LaplaceOperator class. The first thing to do is to set up the DoFHandler,
1484 * including the degrees of freedom for the multigrid levels, and to
1485 * initialize constraints from hanging nodes and homogeneous Dirichlet
1486 * conditions. Since we intend to use this programs in %parallel with MPI,
1487 * we need to make sure that the constraints get to know the locally
1488 * relevant degrees of freedom, otherwise the storage would explode when
1489 * using more than a few hundred millions of degrees of freedom, see
1490 * @ref step_40 "step-40".
1491 *
1492
1493 *
1494 * Once we have created the multigrid dof_handler and the constraints, we
1495 * can call the reinit function for the global matrix operator as well as
1496 * each level of the multigrid scheme. The main action is to set up the
1497 * <code> MatrixFree </code> instance for the problem. The base class of the
1498 * <code>LaplaceOperator</code> class, MatrixFreeOperators::Base, is
1499 * initialized with a shared pointer to MatrixFree object. This way, we can
1500 * simply create it here and then pass it on to the system matrix and level
1501 * matrices, respectively. For setting up MatrixFree, we need to activate
1502 * the update flag in the AdditionalData field of MatrixFree that enables
1503 * the storage of quadrature point coordinates in real space (by default, it
1504 * only caches data for gradients (inverse transposed Jacobians) and JxW
1505 * values). Note that if we call the reinit function without specifying the
1506 * level (i.e., giving <code>level = numbers::invalid_unsigned_int</code>),
1507 * MatrixFree constructs a loop over the active cells. In this tutorial, we
1508 * do not use threads in addition to MPI, which is why we explicitly disable
1510 * MatrixFree::AdditionalData::none. Finally, the coefficient is evaluated
1511 * and vectors are initialized as explained above.
1512 *
1513 * @code
1514 *   template <int dim>
1515 *   void LaplaceProblem<dim>::setup_system()
1516 *   {
1517 *   Timer time;
1518 *   setup_time = 0;
1519 *  
1520 *   system_matrix.clear();
1521 *   mg_matrices.clear_elements();
1522 *  
1523 *   dof_handler.distribute_dofs(fe);
1524 *   dof_handler.distribute_mg_dofs();
1525 *  
1526 *   pcout << "Number of degrees of freedom: " << dof_handler.n_dofs()
1527 *   << std::endl;
1528 *  
1529 *   const IndexSet locally_relevant_dofs =
1531 *  
1532 *   constraints.clear();
1533 *   constraints.reinit(locally_relevant_dofs);
1534 *   DoFTools::make_hanging_node_constraints(dof_handler, constraints);
1536 *   mapping, dof_handler, 0, Functions::ZeroFunction<dim>(), constraints);
1537 *   constraints.close();
1538 *   setup_time += time.wall_time();
1539 *   time_details << "Distribute DoFs & B.C. (CPU/wall) " << time.cpu_time()
1540 *   << "s/" << time.wall_time() << 's' << std::endl;
1541 *   time.restart();
1542 *  
1543 *   {
1544 *   typename MatrixFree<dim, double>::AdditionalData additional_data;
1545 *   additional_data.tasks_parallel_scheme =
1547 *   additional_data.mapping_update_flags =
1549 *   std::shared_ptr<MatrixFree<dim, double>> system_mf_storage(
1550 *   new MatrixFree<dim, double>());
1551 *   system_mf_storage->reinit(mapping,
1552 *   dof_handler,
1553 *   constraints,
1554 *   QGauss<1>(fe.degree + 1),
1555 *   additional_data);
1556 *   system_matrix.initialize(system_mf_storage);
1557 *   }
1558 *  
1559 *   system_matrix.evaluate_coefficient(Coefficient<dim>());
1560 *  
1561 *   system_matrix.initialize_dof_vector(solution);
1562 *   system_matrix.initialize_dof_vector(system_rhs);
1563 *  
1564 *   setup_time += time.wall_time();
1565 *   time_details << "Setup matrix-free system (CPU/wall) " << time.cpu_time()
1566 *   << "s/" << time.wall_time() << 's' << std::endl;
1567 *   time.restart();
1568 *  
1569 * @endcode
1570 *
1571 * Next, initialize the matrices for the multigrid method on all the
1572 * levels. The data structure MGConstrainedDoFs keeps information about
1573 * the indices subject to boundary conditions as well as the indices on
1574 * edges between different refinement levels as described in the @ref step_16 "step-16"
1575 * tutorial program. We then go through the levels of the mesh and
1576 * construct the constraints and matrices on each level. These follow
1577 * closely the construction of the system matrix on the original mesh,
1578 * except the slight difference in naming when accessing information on
1579 * the levels rather than the active cells.
1580 *
1581 * @code
1582 *   const unsigned int nlevels = triangulation.n_global_levels();
1583 *   mg_matrices.resize(0, nlevels - 1);
1584 *  
1585 *   const std::set<types::boundary_id> dirichlet_boundary_ids = {0};
1586 *   mg_constrained_dofs.initialize(dof_handler);
1587 *   mg_constrained_dofs.make_zero_boundary_constraints(dof_handler,
1588 *   dirichlet_boundary_ids);
1589 *  
1590 *   for (unsigned int level = 0; level < nlevels; ++level)
1591 *   {
1592 *   const IndexSet relevant_dofs =
1594 *   AffineConstraints<double> level_constraints;
1595 *   level_constraints.reinit(relevant_dofs);
1596 *   level_constraints.add_lines(
1597 *   mg_constrained_dofs.get_boundary_indices(level));
1598 *   level_constraints.close();
1599 *  
1600 *   typename MatrixFree<dim, float>::AdditionalData additional_data;
1601 *   additional_data.tasks_parallel_scheme =
1603 *   additional_data.mapping_update_flags =
1605 *   additional_data.mg_level = level;
1606 *   std::shared_ptr<MatrixFree<dim, float>> mg_mf_storage_level(
1607 *   new MatrixFree<dim, float>());
1608 *   mg_mf_storage_level->reinit(mapping,
1609 *   dof_handler,
1610 *   level_constraints,
1611 *   QGauss<1>(fe.degree + 1),
1612 *   additional_data);
1613 *  
1614 *   mg_matrices[level].initialize(mg_mf_storage_level,
1615 *   mg_constrained_dofs,
1616 *   level);
1617 *   mg_matrices[level].evaluate_coefficient(Coefficient<dim>());
1618 *   }
1619 *   setup_time += time.wall_time();
1620 *   time_details << "Setup matrix-free levels (CPU/wall) " << time.cpu_time()
1621 *   << "s/" << time.wall_time() << 's' << std::endl;
1622 *   }
1623 *  
1624 *  
1625 *  
1626 * @endcode
1627 *
1628 *
1629 * <a name="LaplaceProblemassemble_rhs"></a>
1630 * <h4>LaplaceProblem::assemble_rhs</h4>
1631 *
1632
1633 *
1634 * The assemble function is very simple since all we have to do is to
1635 * assemble the right hand side. Thanks to FEEvaluation and all the data
1636 * cached in the MatrixFree class, which we query from
1637 * MatrixFreeOperators::Base, this can be done in a few lines. Since this
1638 * call is not wrapped into a MatrixFree::cell_loop (which would be an
1639 * alternative), we must not forget to call compress() at the end of the
1640 * assembly to send all the contributions of the right hand side to the
1641 * owner of the respective degree of freedom.
1642 *
1643 * @code
1644 *   template <int dim>
1645 *   void LaplaceProblem<dim>::assemble_rhs()
1646 *   {
1647 *   Timer time;
1648 *  
1649 *   system_rhs = 0;
1651 *   *system_matrix.get_matrix_free());
1652 *   for (unsigned int cell = 0;
1653 *   cell < system_matrix.get_matrix_free()->n_cell_batches();
1654 *   ++cell)
1655 *   {
1656 *   phi.reinit(cell);
1657 *   for (unsigned int q = 0; q < phi.n_q_points; ++q)
1658 *   phi.submit_value(make_vectorized_array<double>(1.0), q);
1659 *   phi.integrate(EvaluationFlags::values);
1660 *   phi.distribute_local_to_global(system_rhs);
1661 *   }
1662 *   system_rhs.compress(VectorOperation::add);
1663 *  
1664 *   setup_time += time.wall_time();
1665 *   time_details << "Assemble right hand side (CPU/wall) " << time.cpu_time()
1666 *   << "s/" << time.wall_time() << 's' << std::endl;
1667 *   }
1668 *  
1669 *  
1670 *  
1671 * @endcode
1672 *
1673 *
1674 * <a name="LaplaceProblemsolve"></a>
1675 * <h4>LaplaceProblem::solve</h4>
1676 *
1677
1678 *
1679 * The solution process is similar as in @ref step_16 "step-16". We start with the setup of
1680 * the transfer. For LinearAlgebra::distributed::Vector, there is a very
1681 * fast transfer class called MGTransferMatrixFree that does the
1682 * interpolation between the grid levels with the same fast sum
1683 * factorization kernels that get also used in FEEvaluation.
1684 *
1685 * @code
1686 *   template <int dim>
1687 *   void LaplaceProblem<dim>::solve()
1688 *   {
1689 *   Timer time;
1690 *   MGTransferMatrixFree<dim, float> mg_transfer(mg_constrained_dofs);
1691 *   mg_transfer.build(dof_handler);
1692 *   setup_time += time.wall_time();
1693 *   time_details << "MG build transfer time (CPU/wall) " << time.cpu_time()
1694 *   << "s/" << time.wall_time() << "s\n";
1695 *   time.restart();
1696 *  
1697 * @endcode
1698 *
1699 * As a smoother, this tutorial program uses a Chebyshev iteration instead
1700 * of SOR in @ref step_16 "step-16". (SOR would be very difficult to implement because we
1701 * do not have the matrix elements available explicitly, and it is
1702 * difficult to make it work efficiently in %parallel.) The smoother is
1703 * initialized with our level matrices and the mandatory additional data
1704 * for the Chebyshev smoother. We use a relatively high degree here (5),
1705 * since matrix-vector products are comparably cheap. We choose to smooth
1706 * out a range of @f$[1.2 \hat{\lambda}_{\max}/15,1.2 \hat{\lambda}_{\max}]@f$
1707 * in the smoother where @f$\hat{\lambda}_{\max}@f$ is an estimate of the
1708 * largest eigenvalue (the factor 1.2 is applied inside
1709 * PreconditionChebyshev). In order to compute that eigenvalue, the
1710 * Chebyshev initialization performs a few steps of a CG algorithm
1711 * without preconditioner. Since the highest eigenvalue is usually the
1712 * easiest one to find and a rough estimate is enough, we choose 10
1713 * iterations. Finally, we also set the inner preconditioner type in the
1714 * Chebyshev method which is a Jacobi iteration. This is represented by
1715 * the DiagonalMatrix class that gets the inverse diagonal entry provided
1716 * by our LaplaceOperator class.
1717 *
1718
1719 *
1720 * On level zero, we initialize the smoother differently because we want
1721 * to use the Chebyshev iteration as a solver. PreconditionChebyshev
1722 * allows the user to switch to solver mode where the number of iterations
1723 * is internally chosen to the correct value. In the additional data
1724 * object, this setting is activated by choosing the polynomial degree to
1725 * @p numbers::invalid_unsigned_int. The algorithm will then attack all
1726 * eigenvalues between the smallest and largest one in the coarse level
1727 * matrix. The number of steps in the Chebyshev smoother are chosen such
1728 * that the Chebyshev convergence estimates guarantee to reduce the
1729 * residual by the number specified in the variable @p
1730 * smoothing_range. Note that for solving, @p smoothing_range is a
1731 * relative tolerance and chosen smaller than one, in this case, we select
1732 * three orders of magnitude, whereas it is a number larger than 1 when
1733 * only selected eigenvalues are smoothed.
1734 *
1735
1736 *
1737 * From a computational point of view, the Chebyshev iteration is a very
1738 * attractive coarse grid solver as long as the coarse size is
1739 * moderate. This is because the Chebyshev method performs only
1740 * matrix-vector products and vector updates, which typically parallelize
1741 * better to the largest cluster size with more than a few tens of
1742 * thousands of cores than inner product involved in other iterative
1743 * methods. The former involves only local communication between neighbors
1744 * in the (coarse) mesh, whereas the latter requires global communication
1745 * over all processors.
1746 *
1747 * @code
1748 *   using SmootherType =
1749 *   PreconditionChebyshev<LevelMatrixType,
1751 *   mg::SmootherRelaxation<SmootherType,
1753 *   mg_smoother;
1755 *   smoother_data.resize(0, triangulation.n_global_levels() - 1);
1756 *   for (unsigned int level = 0; level < triangulation.n_global_levels();
1757 *   ++level)
1758 *   {
1759 *   if (level > 0)
1760 *   {
1761 *   smoother_data[level].smoothing_range = 15.;
1762 *   smoother_data[level].degree = 5;
1763 *   smoother_data[level].eig_cg_n_iterations = 10;
1764 *   }
1765 *   else
1766 *   {
1767 *   smoother_data[0].smoothing_range = 1e-3;
1768 *   smoother_data[0].degree = numbers::invalid_unsigned_int;
1769 *   smoother_data[0].eig_cg_n_iterations = mg_matrices[0].m();
1770 *   }
1771 *   mg_matrices[level].compute_diagonal();
1772 *   smoother_data[level].preconditioner =
1773 *   mg_matrices[level].get_matrix_diagonal_inverse();
1774 *   }
1775 *   mg_smoother.initialize(mg_matrices, smoother_data);
1776 *  
1778 *   mg_coarse;
1779 *   mg_coarse.initialize(mg_smoother);
1780 *  
1781 * @endcode
1782 *
1783 * The next step is to set up the interface matrices that are needed for the
1784 * case with hanging nodes. The adaptive multigrid realization in deal.II
1785 * implements an approach called local smoothing. This means that the
1786 * smoothing on the finest level only covers the local part of the mesh
1787 * defined by the fixed (finest) grid level and ignores parts of the
1788 * computational domain where the terminal cells are coarser than this
1789 * level. As the method progresses to coarser levels, more and more of the
1790 * global mesh will be covered. At some coarser level, the whole mesh will
1791 * be covered. Since all level matrices in the multigrid method cover a
1792 * single level in the mesh, no hanging nodes appear on the level matrices.
1793 * At the interface between multigrid levels, homogeneous Dirichlet boundary
1794 * conditions are set while smoothing. When the residual is transferred to
1795 * the next coarser level, however, the coupling over the multigrid
1796 * interface needs to be taken into account. This is done by the so-called
1797 * interface (or edge) matrices that compute the part of the residual that
1798 * is missed by the level matrix with
1799 * homogeneous Dirichlet conditions. We refer to the @ref mg_paper
1800 * "Multigrid paper by Janssen and Kanschat" for more details.
1801 *
1802
1803 *
1804 * For the implementation of those interface matrices, there is already a
1805 * pre-defined class MatrixFreeOperators::MGInterfaceOperator that wraps
1807 * MatrixFreeOperators::Base::vmult_interface_up() in a new class with @p
1808 * vmult() and @p Tvmult() operations (that were originally written for
1809 * matrices, hence expecting those names). Note that vmult_interface_down
1810 * is used during the restriction phase of the multigrid V-cycle, whereas
1811 * vmult_interface_up is used during the prolongation phase.
1812 *
1813
1814 *
1815 * Once the interface matrix is created, we set up the remaining Multigrid
1816 * preconditioner infrastructure in complete analogy to @ref step_16 "step-16" to obtain
1817 * a @p preconditioner object that can be applied to a matrix.
1818 *
1819 * @code
1820 *   mg::Matrix<LinearAlgebra::distributed::Vector<float>> mg_matrix(
1821 *   mg_matrices);
1822 *  
1823 *   MGLevelObject<MatrixFreeOperators::MGInterfaceOperator<LevelMatrixType>>
1824 *   mg_interface_matrices;
1825 *   mg_interface_matrices.resize(0, triangulation.n_global_levels() - 1);
1826 *   for (unsigned int level = 0; level < triangulation.n_global_levels();
1827 *   ++level)
1828 *   mg_interface_matrices[level].initialize(mg_matrices[level]);
1829 *   mg::Matrix<LinearAlgebra::distributed::Vector<float>> mg_interface(
1830 *   mg_interface_matrices);
1831 *  
1832 *   Multigrid<LinearAlgebra::distributed::Vector<float>> mg(
1833 *   mg_matrix, mg_coarse, mg_transfer, mg_smoother, mg_smoother);
1834 *   mg.set_edge_matrices(mg_interface, mg_interface);
1835 *  
1836 *   PreconditionMG<dim,
1837 *   LinearAlgebra::distributed::Vector<float>,
1838 *   MGTransferMatrixFree<dim, float>>
1839 *   preconditioner(dof_handler, mg, mg_transfer);
1840 *  
1841 * @endcode
1842 *
1843 * The setup of the multigrid routines is quite easy and one cannot see
1844 * any difference in the solve process as compared to @ref step_16 "step-16". All the
1845 * magic is hidden behind the implementation of the LaplaceOperator::vmult
1846 * operation. Note that we print out the solve time and the accumulated
1847 * setup time through standard out, i.e., in any case, whereas detailed
1848 * times for the setup operations are only printed in case the flag for
1849 * detail_times in the constructor is changed.
1850 *
1851
1852 *
1853 *
1854 * @code
1855 *   SolverControl solver_control(100, 1e-12 * system_rhs.l2_norm());
1856 *   SolverCG<LinearAlgebra::distributed::Vector<double>> cg(solver_control);
1857 *   setup_time += time.wall_time();
1858 *   time_details << "MG build smoother time (CPU/wall) " << time.cpu_time()
1859 *   << "s/" << time.wall_time() << "s\n";
1860 *   pcout << "Total setup time (wall) " << setup_time << "s\n";
1861 *  
1862 *   time.reset();
1863 *   time.start();
1864 *   constraints.set_zero(solution);
1865 *   cg.solve(system_matrix, solution, system_rhs, preconditioner);
1866 *  
1867 *   constraints.distribute(solution);
1868 *  
1869 *   pcout << "Time solve (" << solver_control.last_step() << " iterations)"
1870 *   << (solver_control.last_step() < 10 ? " " : " ") << "(CPU/wall) "
1871 *   << time.cpu_time() << "s/" << time.wall_time() << "s\n";
1872 *   }
1873 *  
1874 *  
1875 *  
1876 * @endcode
1877 *
1878 *
1879 * <a name="LaplaceProblemoutput_results"></a>
1880 * <h4>LaplaceProblem::output_results</h4>
1881 *
1882
1883 *
1884 * Here is the data output, which is a simplified version of @ref step_5 "step-5". We use
1885 * the standard VTU (= compressed VTK) output for each grid produced in the
1886 * refinement process. In addition, we use a compression algorithm that is
1887 * optimized for speed rather than disk usage. The default setting (which
1888 * optimizes for disk usage) makes saving the output take about 4 times as
1889 * long as running the linear solver, while setting
1890 * DataOutBase::CompressionLevel to
1891 * best_speed lowers this to only one fourth the time
1892 * of the linear solve.
1893 *
1894
1895 *
1896 * We disable the output when the mesh gets too large. A variant of this
1897 * program has been run on hundreds of thousands MPI ranks with as many as
1898 * 100 billion grid cells, which is not directly accessible to classical
1899 * visualization tools.
1900 *
1901 * @code
1902 *   template <int dim>
1903 *   void LaplaceProblem<dim>::output_results(const unsigned int cycle) const
1904 *   {
1905 *   Timer time;
1906 *   if (triangulation.n_global_active_cells() > 1000000)
1907 *   return;
1908 *  
1909 *   DataOut<dim> data_out;
1910 *  
1911 *   solution.update_ghost_values();
1912 *   data_out.attach_dof_handler(dof_handler);
1913 *   data_out.add_data_vector(solution, "solution");
1914 *   data_out.build_patches(mapping);
1915 *  
1916 *   DataOutBase::VtkFlags flags;
1918 *   data_out.set_flags(flags);
1919 *   data_out.write_vtu_with_pvtu_record(
1920 *   "./", "solution", cycle, MPI_COMM_WORLD, 3);
1921 *  
1922 *   time_details << "Time write output (CPU/wall) " << time.cpu_time()
1923 *   << "s/" << time.wall_time() << "s\n";
1924 *   }
1925 *  
1926 *  
1927 *  
1928 * @endcode
1929 *
1930 *
1931 * <a name="LaplaceProblemrun"></a>
1932 * <h4>LaplaceProblem::run</h4>
1933 *
1934
1935 *
1936 * The function that runs the program is very similar to the one in
1937 * @ref step_16 "step-16". We do few refinement steps in 3d compared to 2d, but that's
1938 * it.
1939 *
1940
1941 *
1942 * Before we run the program, we output some information about the detected
1943 * vectorization level as discussed in the introduction.
1944 *
1945 * @code
1946 *   template <int dim>
1947 *   void LaplaceProblem<dim>::run()
1948 *   {
1949 *   {
1950 *   const unsigned int n_vect_doubles = VectorizedArray<double>::size();
1951 *   const unsigned int n_vect_bits = 8 * sizeof(double) * n_vect_doubles;
1952 *  
1953 *   pcout << "Vectorization over " << n_vect_doubles
1954 *   << " doubles = " << n_vect_bits << " bits ("
1955 *   << Utilities::System::get_current_vectorization_level() << ')'
1956 *   << std::endl;
1957 *   }
1958 *  
1959 *   for (unsigned int cycle = 0; cycle < 9 - dim; ++cycle)
1960 *   {
1961 *   pcout << "Cycle " << cycle << std::endl;
1962 *  
1963 *   if (cycle == 0)
1964 *   {
1965 *   GridGenerator::hyper_cube(triangulation, 0., 1.);
1966 *   triangulation.refine_global(3 - dim);
1967 *   }
1968 *   triangulation.refine_global(1);
1969 *   setup_system();
1970 *   assemble_rhs();
1971 *   solve();
1972 *   output_results(cycle);
1973 *   pcout << std::endl;
1974 *   };
1975 *   }
1976 *   } // namespace Step37
1977 *  
1978 *  
1979 *  
1980 * @endcode
1981 *
1982 *
1983 * <a name="Thecodemaincodefunction"></a>
1984 * <h3>The <code>main</code> function</h3>
1985 *
1986
1987 *
1988 * Apart from the fact that we set up the MPI framework according to @ref step_40 "step-40",
1989 * there are no surprises in the main function.
1990 *
1991 * @code
1992 *   int main(int argc, char *argv[])
1993 *   {
1994 *   try
1995 *   {
1996 *   using namespace Step37;
1997 *  
1998 *   Utilities::MPI::MPI_InitFinalize mpi_init(argc, argv, 1);
1999 *  
2000 *   LaplaceProblem<dimension> laplace_problem;
2001 *   laplace_problem.run();
2002 *   }
2003 *   catch (std::exception &exc)
2004 *   {
2005 *   std::cerr << std::endl
2006 *   << std::endl
2007 *   << "----------------------------------------------------"
2008 *   << std::endl;
2009 *   std::cerr << "Exception on processing: " << std::endl
2010 *   << exc.what() << std::endl
2011 *   << "Aborting!" << std::endl
2012 *   << "----------------------------------------------------"
2013 *   << std::endl;
2014 *   return 1;
2015 *   }
2016 *   catch (...)
2017 *   {
2018 *   std::cerr << std::endl
2019 *   << std::endl
2020 *   << "----------------------------------------------------"
2021 *   << std::endl;
2022 *   std::cerr << "Unknown exception!" << std::endl
2023 *   << "Aborting!" << std::endl
2024 *   << "----------------------------------------------------"
2025 *   << std::endl;
2026 *   return 1;
2027 *   }
2028 *  
2029 *   return 0;
2030 *   }
2031 * @endcode
2032<a name="Results"></a><h1>Results</h1>
2033
2034
2035<a name="Programoutput"></a><h3>Program output</h3>
2036
2037
2038Since this example solves the same problem as @ref step_5 "step-5" (except for
2039a different coefficient), there is little to say about the
2040solution. We show a picture anyway, illustrating the size of the
2041solution through both isocontours and volume rendering:
2042
2043<img src="https://www.dealii.org/images/steps/developer/step-37.solution.png" alt="">
2044
2045Of more interest is to evaluate some aspects of the multigrid solver.
2046When we run this program in 2D for quadratic (@f$Q_2@f$) elements, we get the
2047following output (when run on one core in release mode):
2048@code
2049Vectorization over 2 doubles = 128 bits (SSE2)
2050Cycle 0
2051Number of degrees of freedom: 81
2052Total setup time (wall) 0.00159788s
2053Time solve (6 iterations) (CPU/wall) 0.000951s/0.000951052s
2054
2055Cycle 1
2056Number of degrees of freedom: 289
2057Total setup time (wall) 0.00114608s
2058Time solve (6 iterations) (CPU/wall) 0.000935s/0.000934839s
2059
2060Cycle 2
2061Number of degrees of freedom: 1089
2062Total setup time (wall) 0.00244665s
2063Time solve (6 iterations) (CPU/wall) 0.00207s/0.002069s
2064
2065Cycle 3
2066Number of degrees of freedom: 4225
2067Total setup time (wall) 0.00678205s
2068Time solve (6 iterations) (CPU/wall) 0.005616s/0.00561595s
2069
2070Cycle 4
2071Number of degrees of freedom: 16641
2072Total setup time (wall) 0.0241671s
2073Time solve (6 iterations) (CPU/wall) 0.019543s/0.0195441s
2074
2075Cycle 5
2076Number of degrees of freedom: 66049
2077Total setup time (wall) 0.0967851s
2078Time solve (6 iterations) (CPU/wall) 0.07457s/0.0745709s
2079
2080Cycle 6
2081Number of degrees of freedom: 263169
2082Total setup time (wall) 0.346374s
2083Time solve (6 iterations) (CPU/wall) 0.260042s/0.265033s
2084@endcode
2085
2086As in @ref step_16 "step-16", we see that the number of CG iterations remains constant with
2087increasing number of degrees of freedom. A constant number of iterations
2088(together with optimal computational properties) means that the computing time
2089approximately quadruples as the problem size quadruples from one cycle to the
2090next. The code is also very efficient in terms of storage. Around 2-4 million
2091degrees of freedom fit into 1 GB of memory, see also the MPI results below. An
2092interesting fact is that solving one linear system is cheaper than the setup,
2093despite not building a matrix (approximately half of which is spent in the
2094DoFHandler::distribute_dofs() and DoFHandler::distribute_mg_dofs()
2095calls). This shows the high efficiency of this approach, but also that the
2096deal.II data structures are quite expensive to set up and the setup cost must
2097be amortized over several system solves.
2098
2099Not much changes if we run the program in three spatial dimensions. Since we
2100use uniform mesh refinement, we get eight times as many elements and
2101approximately eight times as many degrees of freedom with each cycle:
2102
2103@code
2104Vectorization over 2 doubles = 128 bits (SSE2)
2105Cycle 0
2106Number of degrees of freedom: 125
2107Total setup time (wall) 0.00231099s
2108Time solve (6 iterations) (CPU/wall) 0.000692s/0.000922918s
2109
2110Cycle 1
2111Number of degrees of freedom: 729
2112Total setup time (wall) 0.00289083s
2113Time solve (6 iterations) (CPU/wall) 0.001534s/0.0024128s
2114
2115Cycle 2
2116Number of degrees of freedom: 4913
2117Total setup time (wall) 0.0143182s
2118Time solve (6 iterations) (CPU/wall) 0.010785s/0.0107841s
2119
2120Cycle 3
2121Number of degrees of freedom: 35937
2122Total setup time (wall) 0.087064s
2123Time solve (6 iterations) (CPU/wall) 0.063522s/0.06545s
2124
2125Cycle 4
2126Number of degrees of freedom: 274625
2127Total setup time (wall) 0.596306s
2128Time solve (6 iterations) (CPU/wall) 0.427757s/0.431765s
2129
2130Cycle 5
2131Number of degrees of freedom: 2146689
2132Total setup time (wall) 4.96491s
2133Time solve (6 iterations) (CPU/wall) 3.53126s/3.56142s
2134@endcode
2135
2136Since it is so easy, we look at what happens if we increase the polynomial
2137degree. When selecting the degree as four in 3D, i.e., on @f$\mathcal Q_4@f$
2138elements, by changing the line <code>const unsigned int
2139degree_finite_element=4;</code> at the top of the program, we get the
2140following program output:
2141
2142@code
2143Vectorization over 2 doubles = 128 bits (SSE2)
2144Cycle 0
2145Number of degrees of freedom: 729
2146Total setup time (wall) 0.00633097s
2147Time solve (6 iterations) (CPU/wall) 0.002829s/0.00379395s
2148
2149Cycle 1
2150Number of degrees of freedom: 4913
2151Total setup time (wall) 0.0174279s
2152Time solve (6 iterations) (CPU/wall) 0.012255s/0.012254s
2153
2154Cycle 2
2155Number of degrees of freedom: 35937
2156Total setup time (wall) 0.082655s
2157Time solve (6 iterations) (CPU/wall) 0.052362s/0.0523629s
2158
2159Cycle 3
2160Number of degrees of freedom: 274625
2161Total setup time (wall) 0.507943s
2162Time solve (6 iterations) (CPU/wall) 0.341811s/0.345788s
2163
2164Cycle 4
2165Number of degrees of freedom: 2146689
2166Total setup time (wall) 3.46251s
2167Time solve (7 iterations) (CPU/wall) 3.29638s/3.3265s
2168
2169Cycle 5
2170Number of degrees of freedom: 16974593
2171Total setup time (wall) 27.8989s
2172Time solve (7 iterations) (CPU/wall) 26.3705s/27.1077s
2173@endcode
2174
2175Since @f$\mathcal Q_4@f$ elements on a certain mesh correspond to @f$\mathcal Q_2@f$
2176elements on half the mesh size, we can compare the run time at cycle 4 with
2177fourth degree polynomials with cycle 5 using quadratic polynomials, both at
21782.1 million degrees of freedom. The surprising effect is that the solver for
2179@f$\mathcal Q_4@f$ element is actually slightly faster than for the quadratic
2180case, despite using one more linear iteration. The effect that higher-degree
2181polynomials are similarly fast or even faster than lower degree ones is one of
2182the main strengths of matrix-free operator evaluation through sum
2183factorization, see the <a
2184href="http://dx.doi.org/10.1016/j.compfluid.2012.04.012">matrix-free
2185paper</a>. This is fundamentally different to matrix-based methods that get
2186more expensive per unknown as the polynomial degree increases and the coupling
2187gets denser.
2188
2189In addition, also the setup gets a bit cheaper for higher order, which is
2190because fewer elements need to be set up.
2191
2192Finally, let us look at the timings with degree 8, which corresponds to
2193another round of mesh refinement in the lower order methods:
2194
2195@code
2196Vectorization over 2 doubles = 128 bits (SSE2)
2197Cycle 0
2198Number of degrees of freedom: 4913
2199Total setup time (wall) 0.0842004s
2200Time solve (8 iterations) (CPU/wall) 0.019296s/0.0192959s
2201
2202Cycle 1
2203Number of degrees of freedom: 35937
2204Total setup time (wall) 0.327048s
2205Time solve (8 iterations) (CPU/wall) 0.07517s/0.075999s
2206
2207Cycle 2
2208Number of degrees of freedom: 274625
2209Total setup time (wall) 2.12335s
2210Time solve (8 iterations) (CPU/wall) 0.448739s/0.453698s
2211
2212Cycle 3
2213Number of degrees of freedom: 2146689
2214Total setup time (wall) 16.1743s
2215Time solve (8 iterations) (CPU/wall) 3.95003s/3.97717s
2216
2217Cycle 4
2218Number of degrees of freedom: 16974593
2219Total setup time (wall) 130.8s
2220Time solve (8 iterations) (CPU/wall) 31.0316s/31.767s
2221@endcode
2222
2223Here, the initialization seems considerably slower than before, which is
2224mainly due to the computation of the diagonal of the matrix, which actually
2225computes a 729 x 729 matrix on each cell and throws away everything but the
2226diagonal. The solver times, however, are again very close to the quartic case,
2227showing that the linear increase with the polynomial degree that is
2228theoretically expected is almost completely offset by better computational
2229characteristics and the fact that higher order methods have a smaller share of
2230degrees of freedom living on several cells that add to the evaluation
2231complexity.
2232
2233<a name="Comparisonwithasparsematrix"></a><h3>Comparison with a sparse matrix</h3>
2234
2235
2236In order to understand the capabilities of the matrix-free implementation, we
2237compare the performance of the 3d example above with a sparse matrix
2238implementation based on TrilinosWrappers::SparseMatrix by measuring both the
2239computation times for the initialization of the problem (distribute DoFs,
2240setup and assemble matrices, setup multigrid structures) and the actual
2241solution for the matrix-free variant and the variant based on sparse
2242matrices. We base the preconditioner on float numbers and the actual matrix
2243and vectors on double numbers, as shown above. Tests are run on an Intel Core
2244i7-5500U notebook processor (two cores and <a
2245href="http://en.wikipedia.org/wiki/Advanced_Vector_Extensions">AVX</a>
2246support, i.e., four operations on doubles can be done with one CPU
2247instruction, which is heavily used in FEEvaluation), optimized mode, and two
2248MPI ranks.
2249
2250<table align="center" class="doxtable">
2251 <tr>
2252 <th>&nbsp;</th>
2253 <th colspan="2">Sparse matrix</th>
2254 <th colspan="2">Matrix-free implementation</th>
2255 </tr>
2256 <tr>
2257 <th>n_dofs</th>
2258 <th>Setup + assemble</th>
2259 <th>&nbsp;Solve&nbsp;</th>
2260 <th>Setup + assemble</th>
2261 <th>&nbsp;Solve&nbsp;</th>
2262 </tr>
2263 <tr>
2264 <td align="right">125</td>
2265 <td align="center">0.0042s</td>
2266 <td align="center">0.0012s</td>
2267 <td align="center">0.0022s</td>
2268 <td align="center">0.00095s</td>
2269 </tr>
2270 <tr>
2271 <td align="right">729</td>
2272 <td align="center">0.012s</td>
2273 <td align="center">0.0040s</td>
2274 <td align="center">0.0027s</td>
2275 <td align="center">0.0021s</td>
2276 </tr>
2277 <tr>
2278 <td align="right">4,913</td>
2279 <td align="center">0.082s</td>
2280 <td align="center">0.012s</td>
2281 <td align="center">0.011s</td>
2282 <td align="center">0.0057s</td>
2283 </tr>
2284 <tr>
2285 <td align="right">35,937</td>
2286 <td align="center">0.73s</td>
2287 <td align="center">0.13s</td>
2288 <td align="center">0.048s</td>
2289 <td align="center">0.040s</td>
2290 </tr>
2291 <tr>
2292 <td align="right">274,625</td>
2293 <td align="center">5.43s</td>
2294 <td align="center">1.01s</td>
2295 <td align="center">0.33s</td>
2296 <td align="center">0.25s</td>
2297 </tr>
2298 <tr>
2299 <td align="right">2,146,689</td>
2300 <td align="center">43.8s</td>
2301 <td align="center">8.24s</td>
2302 <td align="center">2.42s</td>
2303 <td align="center">2.06s</td>
2304 </tr>
2305</table>
2306
2307The table clearly shows that the matrix-free implementation is more than twice
2308as fast for the solver, and more than six times as fast when it comes to
2309initialization costs. As the problem size is made a factor 8 larger, we note
2310that the times usually go up by a factor eight, too (as the solver iterations
2311are constant at six). The main deviation is in the sparse matrix between 5k
2312and 36k degrees of freedom, where the time increases by a factor 12. This is
2313the threshold where the (L3) cache in the processor can no longer hold all
2314data necessary for the matrix-vector products and all matrix elements must be
2315fetched from main memory.
2316
2317Of course, this picture does not necessarily translate to all cases, as there
2318are problems where knowledge of matrix entries enables much better solvers (as
2319happens when the coefficient is varying more strongly than in the above
2320example). Moreover, it also depends on the computer system. The present system
2321has good memory performance, so sparse matrices perform comparably
2322well. Nonetheless, the matrix-free implementation gives a nice speedup already
2323for the <i>Q</i><sub>2</sub> elements used in this example. This becomes
2324particularly apparent for time-dependent or nonlinear problems where sparse
2325matrices would need to be reassembled over and over again, which becomes much
2326easier with this class. And of course, thanks to the better complexity of the
2327products, the method gains increasingly larger advantages when the order of the
2328elements increases (the matrix-free implementation has costs
23294<i>d</i><sup>2</sup><i>p</i> per degree of freedom, compared to
23302<i>p<sup>d</sup></i> for the sparse matrix, so it will win anyway for order 4
2331and higher in 3d).
2332
2333<a name="ResultsforlargescaleparallelcomputationsonSuperMUC"></a><h3> Results for large-scale parallel computations on SuperMUC</h3>
2334
2335
2336As explained in the introduction and the in-code comments, this program can be
2337run in parallel with MPI. It turns out that geometric multigrid schemes work
2338really well and can scale to very large machines. To the authors' knowledge,
2339the geometric multigrid results shown here are the largest computations done
2340with deal.II as of late 2016, run on up to 147,456 cores of the <a
2341href="https://www.lrz.de/services/compute/supermuc/systemdescription/">complete
2342SuperMUC Phase 1</a>. The ingredients for scalability beyond 1000 cores are
2343that no data structure that depends on the global problem size is held in its
2344entirety on a single processor and that the communication is not too frequent
2345in order not to run into latency issues of the network. For PDEs solved with
2346iterative solvers, the communication latency is often the limiting factor,
2347rather than the throughput of the network. For the example of the SuperMUC
2348system, the point-to-point latency between two processors is between 1e-6 and
23491e-5 seconds, depending on the proximity in the MPI network. The matrix-vector
2350products with @p LaplaceOperator from this class involves several
2351point-to-point communication steps, interleaved with computations on each
2352core. The resulting latency of a matrix-vector product is around 1e-4
2353seconds. Global communication, for example an @p MPI_Allreduce operation that
2354accumulates the sum of a single number per rank over all ranks in the MPI
2355network, has a latency of 1e-4 seconds. The multigrid V-cycle used in this
2356program is also a form of global communication. Think about the coarse grid
2357solve that happens on a single processor: It accumulates the contributions
2358from all processors before it starts. When completed, the coarse grid solution
2359is transferred to finer levels, where more and more processors help in
2360smoothing until the fine grid. Essentially, this is a tree-like pattern over
2361the processors in the network and controlled by the mesh. As opposed to the
2362@p MPI_Allreduce operations where the tree in the reduction is optimized to the
2363actual links in the MPI network, the multigrid V-cycle does this according to
2364the partitioning of the mesh. Thus, we cannot expect the same
2365optimality. Furthermore, the multigrid cycle is not simply a walk up and down
2366the refinement tree, but also communication on each level when doing the
2367smoothing. In other words, the global communication in multigrid is more
2368challenging and related to the mesh that provides less optimization
2369opportunities. The measured latency of the V-cycle is between 6e-3 and 2e-2
2370seconds, i.e., the same as 60 to 200 MPI_Allreduce operations.
2371
2372The following figure shows a scaling experiments on @f$\mathcal Q_3@f$
2373elements. Along the lines, the problem size is held constant as the number of
2374cores is increasing. When doubling the number of cores, one expects a halving
2375of the computational time, indicated by the dotted gray lines. The results
2376show that the implementation shows almost ideal behavior until an absolute
2377time of around 0.1 seconds is reached. The solver tolerances have been set
2378such that the solver performs five iterations. This way of plotting data is
2379the <b>strong scaling</b> of the algorithm. As we go to very large core
2380counts, the curves flatten out a bit earlier, which is because of the
2381communication network in SuperMUC where communication between processors
2382farther away is slightly slower.
2383
2384<img src="https://www.dealii.org/images/steps/developer/step-37.scaling_strong.png" alt="">
2385
2386In addition, the plot also contains results for <b>weak scaling</b> that lists
2387how the algorithm behaves as both the number of processor cores and elements
2388is increased at the same pace. In this situation, we expect that the compute
2389time remains constant. Algorithmically, the number of CG iterations is
2390constant at 5, so we are good from that end. The lines in the plot are
2391arranged such that the top left point in each data series represents the same
2392size per processor, namely 131,072 elements (or approximately 3.5 million
2393degrees of freedom per core). The gray lines indicating ideal strong scaling
2394are by the same factor of 8 apart. The results show again that the scaling is
2395almost ideal. The parallel efficiency when going from 288 cores to 147,456
2396cores is at around 75% for a local problem size of 750,000 degrees of freedom
2397per core which takes 1.0s on 288 cores, 1.03s on 2304 cores, 1.19s on 18k
2398cores, and 1.35s on 147k cores. The algorithms also reach a very high
2399utilization of the processor. The largest computation on 147k cores reaches
2400around 1.7 PFLOPs/s on SuperMUC out of an arithmetic peak of 3.2 PFLOPs/s. For
2401an iterative PDE solver, this is a very high number and significantly more is
2402often only reached for dense linear algebra. Sparse linear algebra is limited
2403to a tenth of this value.
2404
2405As mentioned in the introduction, the matrix-free method reduces the memory
2406consumption of the data structures. Besides the higher performance due to less
2407memory transfer, the algorithms also allow for very large problems to fit into
2408memory. The figure below shows the computational time as we increase the
2409problem size until an upper limit where the computation exhausts memory. We do
2410this for 1k cores, 8k cores, and 65k cores and see that the problem size can
2411be varied over almost two orders of magnitude with ideal scaling. The largest
2412computation shown in this picture involves 292 billion (@f$2.92 \cdot 10^{11}@f$)
2413degrees of freedom. On a DG computation of 147k cores, the above algorithms
2414were also run involving up to 549 billion (2^39) DoFs.
2415
2416<img src="https://www.dealii.org/images/steps/developer/step-37.scaling_size.png" alt="">
2417
2418Finally, we note that while performing the tests on the large-scale system
2419shown above, improvements of the multigrid algorithms in deal.II have been
2420developed. The original version contained the sub-optimal code based on
2421MGSmootherPrecondition where some MPI_Allreduce commands (checking whether
2422all vector entries are zero) were done on each smoothing
2423operation on each level, which only became apparent on 65k cores and
2424more. However, the following picture shows that the improvement already pay
2425off on a smaller scale, here shown on computations on up to 14,336 cores for
2426@f$\mathcal Q_5@f$ elements:
2427
2428<img src="https://www.dealii.org/images/steps/developer/step-37.scaling_oldnew.png" alt="">
2429
2430
2431<a name="Adaptivity"></a><h3> Adaptivity</h3>
2432
2433
2434As explained in the code, the algorithm presented here is prepared to run on
2435adaptively refined meshes. If only part of the mesh is refined, the multigrid
2436cycle will run with local smoothing and impose Dirichlet conditions along the
2437interfaces which differ in refinement level for smoothing through the
2438MatrixFreeOperators::Base class. Due to the way the degrees of freedom are
2439distributed over levels, relating the owner of the level cells to the owner of
2440the first descendant active cell, there can be an imbalance between different
2441processors in MPI, which limits scalability by a factor of around two to five.
2442
2443<a name="Possibilitiesforextensions"></a><h3> Possibilities for extensions</h3>
2444
2445
2446<a name="Kellyerrorestimator"></a><h4> Kelly error estimator </h4>
2447
2448
2449As mentioned above the code is ready for locally adaptive h-refinement.
2450For the Poisson equation one can employ the Kelly error indicator,
2451implemented in the KellyErrorEstimator class. However one needs to be careful
2452with the ghost indices of parallel vectors.
2453In order to evaluate the jump terms in the error indicator, each MPI process
2454needs to know locally relevant DoFs.
2455However MatrixFree::initialize_dof_vector() function initializes the vector only with
2456some locally relevant DoFs.
2457The ghost indices made available in the vector are a tight set of only those indices
2458that are touched in the cell integrals (including constraint resolution).
2459This choice has performance reasons, because sending all locally relevant degrees
2460of freedom would be too expensive compared to the matrix-vector product.
2461Consequently the solution vector as-is is
2462not suitable for the KellyErrorEstimator class.
2463The trick is to change the ghost part of the partition, for example using a
2464temporary vector and LinearAlgebra::distributed::Vector::copy_locally_owned_data_from()
2465as shown below.
2466
2467@code
2468const IndexSet locally_relevant_dofs = DoFTools::extract_locally_relevant_dofs(dof_handler);
2469LinearAlgebra::distributed::Vector<double> copy_vec(solution);
2470solution.reinit(dof_handler.locally_owned_dofs(),
2471 locally_relevant_dofs,
2472 triangulation.get_communicator());
2473solution.copy_locally_owned_data_from(copy_vec);
2474constraints.distribute(solution);
2475solution.update_ghost_values();
2476@endcode
2477
2478<a name="Sharedmemoryparallelization"></a><h4> Shared-memory parallelization</h4>
2479
2480
2481This program is parallelized with MPI only. As an alternative, the MatrixFree
2482loop can also be issued in hybrid mode, for example by using MPI parallelizing
2483over the nodes of a cluster and with threads through Intel TBB within the
2484shared memory region of one node. To use this, one would need to both set the
2485number of threads in the MPI_InitFinalize data structure in the main function,
2486and set the MatrixFree::AdditionalData::tasks_parallel_scheme to
2487partition_color to actually do the loop in parallel. This use case is
2488discussed in @ref step_48 "step-48".
2489
2490<a name="InhomogeneousDirichletboundaryconditions"></a><h4> Inhomogeneous Dirichlet boundary conditions </h4>
2491
2492
2493The presented program assumes homogeneous Dirichlet boundary conditions. When
2494going to non-homogeneous conditions, the situation is a bit more intricate. To
2495understand how to implement such a setting, let us first recall how these
2496arise in the mathematical formulation and how they are implemented in a
2497matrix-based variant. In essence, an inhomogeneous Dirichlet condition sets
2498some of the nodal values in the solution to given values rather than
2499determining them through the variational principles,
2500@f{eqnarray*}
2501u_h(\mathbf{x}) = \sum_{i\in \mathcal N} \varphi_i(\mathbf{x}) u_i =
2502\sum_{i\in \mathcal N \setminus \mathcal N_D} \varphi_i(\mathbf{x}) u_i +
2503\sum_{i\in \mathcal N_D} \varphi_i(\mathbf{x}) g_i,
2504@f}
2505where @f$u_i@f$ denotes the nodal values of the solution and @f$\mathcal N@f$ denotes
2506the set of all nodes. The set @f$\mathcal N_D\subset \mathcal N@f$ is the subset
2507of the nodes that are subject to Dirichlet boundary conditions where the
2508solution is forced to equal @f$u_i = g_i = g(\mathbf{x}_i)@f$ as the interpolation
2509of boundary values on the Dirichlet-constrained node points @f$i\in \mathcal
2510N_D@f$. We then insert this solution
2511representation into the weak form, e.g. the Laplacian shown above, and move
2512the known quantities to the right hand side:
2513@f{eqnarray*}
2514(\nabla \varphi_i, \nabla u_h)_\Omega &=& (\varphi_i, f)_\Omega \quad \Rightarrow \\
2515\sum_{j\in \mathcal N \setminus \mathcal N_D}(\nabla \varphi_i,\nabla \varphi_j)_\Omega \, u_j &=&
2516(\varphi_i, f)_\Omega
2517-\sum_{j\in \mathcal N_D} (\nabla \varphi_i,\nabla\varphi_j)_\Omega\, g_j.
2518@f}
2519In this formula, the equations are tested for all basis functions @f$\varphi_i@f$
2520with @f$i\in N \setminus \mathcal N_D@f$ that are not related to the nodes
2521constrained by Dirichlet conditions.
2522
2523In the implementation in deal.II, the integrals @f$(\nabla \varphi_i,\nabla \varphi_j)_\Omega@f$
2524on the right hand side are already contained in the local matrix contributions
2525we assemble on each cell. When using
2526AffineConstraints::distribute_local_to_global() as first described in the
2527@ref step_6 "step-6" and @ref step_7 "step-7" tutorial programs, we can account for the contribution of
2528inhomogeneous constraints <i>j</i> by multiplying the columns <i>j</i> and
2529rows <i>i</i> of the local matrix according to the integrals @f$(\varphi_i,
2530\varphi_j)_\Omega@f$ by the inhomogeneities and subtracting the resulting from
2531the position <i>i</i> in the global right-hand-side vector, see also the @ref
2532constraints module. In essence, we use some of the integrals that get
2533eliminated from the left hand side of the equation to finalize the right hand
2534side contribution. Similar mathematics are also involved when first writing
2535all entries into a left hand side matrix and then eliminating matrix rows and
2537
2538In principle, the components that belong to the constrained degrees of freedom
2539could be eliminated from the linear system because they do not carry any
2540information. In practice, in deal.II we always keep the size of the linear
2541system the same to avoid handling two different numbering systems and avoid
2542confusion about the two different index sets. In order to ensure that the
2543linear system does not get singular when not adding anything to constrained
2544rows, we then add dummy entries to the matrix diagonal that are otherwise
2545unrelated to the real entries.
2546
2547In a matrix-free method, we need to take a different approach, since the @p
2548LaplaceOperator class represents the matrix-vector product of a
2549<b>homogeneous</b> operator (the left-hand side of the last formula). It does
2550not matter whether the AffineConstraints object passed to the
2551MatrixFree::reinit() contains inhomogeneous constraints or not, the
2552MatrixFree::cell_loop() call will only resolve the homogeneous part of the
2553constraints as long as it represents a <b>linear</b> operator.
2554
2555In our matrix-free code, the right hand side computation where the
2556contribution of inhomogeneous conditions ends up is completely decoupled from
2557the matrix operator and handled by a different function above. Thus, we need
2558to explicitly generate the data that enters the right hand side rather than
2559using a byproduct of the matrix assembly. Since we already know how to apply
2560the operator on a vector, we could try to use those facilities for a vector
2561where we only set the Dirichlet values:
2562@code
2563 // interpolate boundary values on vector solution
2564 std::map<types::global_dof_index, double> boundary_values;
2566 dof_handler,
2567 0,
2568 BoundaryValueFunction<dim>(),
2569 boundary_values);
2570 for (const std::pair<const types::global_dof_index, double> &pair : boundary_values)
2571 if (solution.locally_owned_elements().is_element(pair.first))
2572 solution(pair.first) = pair.second;
2573@endcode
2574or, equivalently, if we already had filled the inhomogeneous constraints into
2575an AffineConstraints object,
2576@code
2577 solution = 0;
2578 constraints.distribute(solution);
2579@endcode
2580
2581We could then pass the vector @p solution to the @p
2582LaplaceOperator::vmult_add() function and add the new contribution to the @p
2583system_rhs vector that gets filled in the @p LaplaceProblem::assemble_rhs()
2584function. However, this idea does not work because the
2585FEEvaluation::read_dof_values() call used inside the vmult() functions assumes
2586homogeneous values on all constraints (otherwise the operator would not be a
2587linear operator but an affine one). To also retrieve the values of the
2588inhomogeneities, we could select one of two following strategies.
2589
2590<a name="UseFEEvaluationread_dof_values_plaintoavoidresolvingconstraints"></a><h5> Use FEEvaluation::read_dof_values_plain() to avoid resolving constraints </h5>
2591
2592
2593The class FEEvaluation has a facility that addresses precisely this
2594requirement: For non-homogeneous Dirichlet values, we do want to skip the
2595implicit imposition of homogeneous (Dirichlet) constraints upon reading the
2596data from the vector @p solution. For example, we could extend the @p
2597LaplaceProblem::assemble_rhs() function to deal with inhomogeneous Dirichlet
2598values as follows, assuming the Dirichlet values have been interpolated into
2599the object @p constraints:
2600@code
2601template <int dim>
2602void LaplaceProblem<dim>::assemble_rhs()
2603{
2604 solution = 0;
2605 constraints.distribute(solution);
2606 solution.update_ghost_values();
2607 system_rhs = 0;
2608
2609 const Table<2, VectorizedArray<double>> &coefficient = system_matrix.get_coefficient();
2610 FEEvaluation<dim, degree_finite_element> phi(*system_matrix.get_matrix_free());
2611 for (unsigned int cell = 0;
2612 cell < system_matrix.get_matrix_free()->n_cell_batches();
2613 ++cell)
2614 {
2615 phi.reinit(cell);
2616 phi.read_dof_values_plain(solution);
2617 phi.evaluate(EvaluationFlags::gradients);
2618 for (unsigned int q = 0; q < phi.n_q_points; ++q)
2619 {
2620 phi.submit_gradient(-coefficient(cell, q) * phi.get_gradient(q), q);
2621 phi.submit_value(make_vectorized_array<double>(1.0), q);
2622 }
2624 phi.distribute_local_to_global(system_rhs);
2625 }
2626 system_rhs.compress(VectorOperation::add);
2627}
2628@endcode
2629
2630In this code, we replaced the FEEvaluation::read_dof_values() function for the
2631tentative solution vector by FEEvaluation::read_dof_values_plain() that
2632ignores all constraints. Due to this setup, we must make sure that other
2633constraints, e.g. by hanging nodes, are correctly distributed to the input
2634vector already as they are not resolved as in
2635FEEvaluation::read_dof_values_plain(). Inside the loop, we then evaluate the
2636Laplacian and repeat the second derivative call with
2637FEEvaluation::submit_gradient() from the @p LaplaceOperator class, but with the
2638sign switched since we wanted to subtract the contribution of Dirichlet
2639conditions on the right hand side vector according to the formula above. When
2640we invoke the FEEvaluation::integrate() call, we then set both arguments
2641regarding the value slot and first derivative slot to true to account for both
2642terms added in the loop over quadrature points. Once the right hand side is
2643assembled, we then go on to solving the linear system for the homogeneous
2644problem, say involving a variable @p solution_update. After solving, we can
2645add @p solution_update to the @p solution vector that contains the final
2646(inhomogeneous) solution.
2647
2648Note that the negative sign for the Laplacian alongside with a positive sign
2649for the forcing that we needed to build the right hand side is a more general
2650concept: We have implemented nothing else than Newton's method for nonlinear
2651equations, but applied to a linear system. We have used an initial guess for
2652the variable @p solution in terms of the Dirichlet boundary conditions and
2653computed a residual @f$r = f - Au_0@f$. The linear system was then solved as
2654@f$\Delta u = A^{-1} (f-Au)@f$ and we finally computed @f$u = u_0 + \Delta u@f$. For a
2655linear system, we obviously reach the exact solution after a single
2656iteration. If we wanted to extend the code to a nonlinear problem, we would
2657rename the @p assemble_rhs() function into a more descriptive name like @p
2658assemble_residual() that computes the (weak) form of the residual, whereas the
2659@p LaplaceOperator::apply_add() function would get the linearization of the
2660residual with respect to the solution variable.
2661
2662<a name="UseLaplaceOperatorwithasecondAffineConstraintsobjectwithoutDirichletconditions"></a><h5> Use LaplaceOperator with a second AffineConstraints object without Dirichlet conditions </h5>
2663
2664
2665A second alternative to get the right hand side that re-uses the @p
2666LaplaceOperator::apply_add() function is to instead add a second LaplaceOperator
2667that skips Dirichlet constraints. To do this, we initialize a second MatrixFree
2668object which does not have any boundary value constraints. This @p matrix_free
2669object is then passed to a @p LaplaceOperator class instance @p
2670inhomogeneous_operator that is only used to create the right hand side:
2671@code
2672template <int dim>
2673void LaplaceProblem<dim>::assemble_rhs()
2674{
2675 system_rhs = 0;
2676 AffineConstraints<double> no_constraints;
2677 no_constraints.close();
2678 LaplaceOperator<dim, degree_finite_element, double> inhomogeneous_operator;
2679
2680 typename MatrixFree<dim, double>::AdditionalData additional_data;
2681 additional_data.mapping_update_flags =
2683 std::shared_ptr<MatrixFree<dim, double>> matrix_free(
2685 matrix_free->reinit(dof_handler,
2686 no_constraints,
2687 QGauss<1>(fe.degree + 1),
2688 additional_data);
2689 inhomogeneous_operator.initialize(matrix_free);
2690
2691 solution = 0.0;
2692 constraints.distribute(solution);
2693 inhomogeneous_operator.evaluate_coefficient(Coefficient<dim>());
2694 inhomogeneous_operator.vmult(system_rhs, solution);
2695 system_rhs *= -1.0;
2696
2698 *inhomogeneous_operator.get_matrix_free());
2699 for (unsigned int cell = 0;
2700 cell < inhomogeneous_operator.get_matrix_free()->n_cell_batches();
2701 ++cell)
2702 {
2703 phi.reinit(cell);
2704 for (unsigned int q = 0; q < phi.n_q_points; ++q)
2705 phi.submit_value(make_vectorized_array<double>(1.0), q);
2706 phi.integrate(EvaluationFlags::values);
2707 phi.distribute_local_to_global(system_rhs);
2708 }
2709 system_rhs.compress(VectorOperation::add);
2710}
2711@endcode
2712
2713A more sophisticated implementation of this technique could reuse the original
2714MatrixFree object. This can be done by initializing the MatrixFree object with
2715multiple blocks, where each block corresponds to a different AffineConstraints
2716object. Doing this would require making substantial modifications to the
2717LaplaceOperator class, but the MatrixFreeOperators::LaplaceOperator class that
2718comes with the library can do this. See the discussion on blocks in
2719MatrixFreeOperators::Base for more information on how to set up blocks.
2720
2721<a name="Furtherperformanceimprovements"></a><h4> Further performance improvements </h4>
2722
2723
2724While the performance achieved in this tutorial program is already very good,
2725there is functionality in deal.II to further improve the performance. On the
2726one hand, increasing the polynomial degree to three or four will further
2727improve the time per unknown. (Even higher degrees typically get slower again,
2728because the multigrid iteration counts increase slightly with the chosen
2729simple smoother. One could then use hybrid multigrid algorithms to use
2730polynomial coarsening through MGTransferGlobalCoarsening, to reduce the impact
2731of the coarser level on the communication latency.) A more significant
2732improvement can be obtained by data-locality optimizations. The class
2733PreconditionChebyshev, when combined with a `DiagonalMatrix` inner
2734preconditioner as in the present class, can overlap the vector operations with
2735the matrix-vector product. As the former are typically constrained by memory
2736bandwidth, reducing the number of loads helps to achieve this goal. The two
2737ingredients to achieve this are
2738<ol>
2739<li> to provide LaplaceOperator class of this tutorial program with a `vmult`
2740function that takes two `std::function` objects, which can be passed on to
2741MatrixFree::cell_loop with the respective signature (PreconditionChebyshev
2742will then pick up this interface and schedule its vector operations), and </li>
2743<li> to compute a numbering that optimizes for data locality, as provided by
2745</ol>
2746 *
2747 *
2748<a name="PlainProg"></a>
2749<h1> The plain program</h1>
2750@include "step-37.cc"
2751*/
void reinit(const IndexSet &local_constraints=IndexSet())
void distribute(VectorType &vec) const
void attach_dof_handler(const DoFHandler< dim, spacedim > &)
value_type get_dof_value(const unsigned int dof) const
void read_dof_values(const VectorType &src, const unsigned int first_index=0, const std::bitset< VectorizedArrayType::size()> &mask=std::bitset< VectorizedArrayType::size()>().flip())
void evaluate(const EvaluationFlags::EvaluationFlags evaluation_flag)
Definition fe_q.h:551
void reinit(const size_type size, const bool omit_zeroing_entries=false)
void initialize(const MGSmootherBase< VectorType > &coarse_smooth)
void resize(const unsigned int new_minlevel, const unsigned int new_maxlevel, Args &&...args)
void set_constrained_entries_to_one(VectorType &dst) const
Definition operators.h:1444
void vmult_add(VectorType &dst, const VectorType &src) const
Definition operators.h:1476
void vmult_interface_down(VectorType &dst, const VectorType &src) const
Definition operators.h:1624
std::shared_ptr< DiagonalMatrix< VectorType > > inverse_diagonal_entries
Definition operators.h:448
unsigned int n_cell_batches() const
void initialize_dof_vector(VectorType &vec, const unsigned int dof_handler_index=0) const
void cell_loop(const std::function< void(const MatrixFree< dim, Number, VectorizedArrayType > &, OutVector &, const InVector &, const std::pair< unsigned int, unsigned int > &)> &cell_operation, OutVector &dst, const InVector &src, const bool zero_dst_vector=false) const
Definition point.h:112
Definition timer.h:118
Point< 3 > vertices[4]
Point< 2 > second
Definition grid_out.cc:4616
Point< 2 > first
Definition grid_out.cc:4615
unsigned int level
Definition grid_out.cc:4618
__global__ void reduction(Number *result, const Number *v, const size_type N)
__global__ void set(Number *val, const Number s, const size_type N)
#define Assert(cond, exc)
#define AssertDimension(dim1, dim2)
void loop(ITERATOR begin, std_cxx20::type_identity_t< ITERATOR > end, DOFINFO &dinfo, INFOBOX &info, const std::function< void(DOFINFO &, typename INFOBOX::CellInfo &)> &cell_worker, const std::function< void(DOFINFO &, typename INFOBOX::CellInfo &)> &boundary_worker, const std::function< void(DOFINFO &, DOFINFO &, typename INFOBOX::CellInfo &, typename INFOBOX::CellInfo &)> &face_worker, ASSEMBLER &assembler, const LoopControl &lctrl=LoopControl())
Definition loop.h:439
void make_hanging_node_constraints(const DoFHandler< dim, spacedim > &dof_handler, AffineConstraints< number > &constraints)
@ update_JxW_values
Transformed quadrature weights.
@ update_gradients
Shape function gradients.
@ update_quadrature_points
Transformed quadrature points.
void apply(const Kokkos::TeamPolicy< MemorySpace::Default::kokkos_space::execution_space >::member_type &team_member, const Kokkos::View< Number *, MemorySpace::Default::kokkos_space > shape_data, const ViewTypeIn in, ViewTypeOut out)
void matrix_free_data_locality(DoFHandler< dim, spacedim > &dof_handler, const MatrixFree< dim, Number, VectorizedArrayType > &matrix_free)
IndexSet extract_locally_relevant_dofs(const DoFHandler< dim, spacedim > &dof_handler)
IndexSet extract_locally_relevant_level_dofs(const DoFHandler< dim, spacedim > &dof_handler, const unsigned int level)
void scale(const double scaling_factor, Triangulation< dim, spacedim > &triangulation)
@ matrix
Contents is actually a matrix.
@ diagonal
Matrix is diagonal.
@ general
No special properties.
void compute_diagonal(const MatrixFree< dim, Number, VectorizedArrayType > &matrix_free, VectorType &diagonal_global, const std::function< void(FEEvaluation< dim, fe_degree, n_q_points_1d, n_components, Number, VectorizedArrayType > &)> &local_vmult, const unsigned int dof_no=0, const unsigned int quad_no=0, const unsigned int first_selected_component=0)
void apply_boundary_values(const std::map< types::global_dof_index, number > &boundary_values, SparseMatrix< number > &matrix, Vector< number > &solution, Vector< number > &right_hand_side, const bool eliminate_columns=true)
Point< spacedim > point(const gp_Pnt &p, const double tolerance=1e-10)
Definition utilities.cc:189
SymmetricTensor< 2, dim, Number > e(const Tensor< 2, dim, Number > &F)
SymmetricTensor< 2, dim, Number > b(const Tensor< 2, dim, Number > &F)
SymmetricTensor< 2, dim, Number > d(const Tensor< 2, dim, Number > &F, const Tensor< 2, dim, Number > &dF_dt)
void call(const std::function< RT()> &function, internal::return_value< RT > &ret_val)
VectorType::value_type * end(VectorType &V)
VectorType::value_type * begin(VectorType &V)
void free(T *&pointer)
Definition cuda.h:97
std::vector< unsigned int > serial(const std::vector< unsigned int > &targets, const std::function< RequestType(const unsigned int)> &create_request, const std::function< AnswerType(const unsigned int, const RequestType &)> &answer_request, const std::function< void(const unsigned int, const AnswerType &)> &process_answer, const MPI_Comm comm)
T sum(const T &t, const MPI_Comm mpi_communicator)
unsigned int this_mpi_process(const MPI_Comm mpi_communicator)
Definition mpi.cc:161
T reduce(const T &local_value, const MPI_Comm comm, const std::function< T(const T &, const T &)> &combiner, const unsigned int root_process=0)
std::string compress(const std::string &input)
Definition utilities.cc:390
void interpolate_boundary_values(const Mapping< dim, spacedim > &mapping, const DoFHandler< dim, spacedim > &dof, const std::map< types::boundary_id, const Function< spacedim, number > * > &function_map, std::map< types::global_dof_index, number > &boundary_values, const ComponentMask &component_mask=ComponentMask())
void run(const Iterator &begin, const std_cxx20::type_identity_t< Iterator > &end, Worker worker, Copier copier, const ScratchData &sample_scratch_data, const CopyData &sample_copy_data, const unsigned int queue_length, const unsigned int chunk_size)
unsigned int n_cells(const internal::TriangulationImplementation::NumberCache< 1 > &c)
Definition tria.cc:13826
void copy(const T *begin, const T *end, U *dest)
int(&) functions(const void *v1, const void *v2)
void assemble(const MeshWorker::DoFInfoBox< dim, DOFINFO > &dinfo, A *assembler)
Definition loop.h:71
void reinit(MatrixBlock< MatrixType > &v, const BlockSparsityPattern &p)
Definition mg.h:82
static const unsigned int invalid_unsigned_int
Definition types.h:213
STL namespace.
Definition types.h:33
unsigned int global_dof_index
Definition types.h:82
const ::parallel::distributed::Triangulation< dim, spacedim > * triangulation
DataOutBase::CompressionLevel compression_level
TasksParallelScheme tasks_parallel_scheme
DEAL_II_HOST constexpr Number determinant(const SymmetricTensor< 2, dim, Number > &)
DEAL_II_HOST constexpr SymmetricTensor< 2, dim, Number > invert(const SymmetricTensor< 2, dim, Number > &)
std::array< Number, 1 > eigenvalues(const SymmetricTensor< 2, 1, Number > &T)