Reference documentation for deal.II version Git d902c8c1aa 2019-12-15 00:36:45 -0500
FilteredMatrix< VectorType > Class Template Reference

#include <deal.II/lac/filtered_matrix.h>

Inheritance diagram for FilteredMatrix< VectorType >:
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## Classes

class  Accessor

class  const_iterator

struct  PairComparison

## Public Types

using size_type = types::global_dof_index

using IndexValuePair = std::pair< size_type, double >

## Public Member Functions

std::size_t memory_consumption () const

Constructors and initialization
FilteredMatrix ()

FilteredMatrix (const FilteredMatrix &fm)

template<typename MatrixType >
FilteredMatrix (const MatrixType &matrix, const bool expect_constrained_source=false)

FilteredMatrixoperator= (const FilteredMatrix &fm)

template<typename MatrixType >
void initialize (const MatrixType &m, const bool expect_constrained_source=false)

void clear ()

Managing constraints
void add_constraint (const size_type i, const double v)

template<class ConstraintList >

void clear_constraints ()

void apply_constraints (VectorType &v, const bool matrix_is_symmetric) const

void apply_constraints (VectorType &v) const

void vmult (VectorType &dst, const VectorType &src) const

void Tvmult (VectorType &dst, const VectorType &src) const

void vmult_add (VectorType &dst, const VectorType &src) const

void Tvmult_add (VectorType &dst, const VectorType &src) const

Iterators
const_iterator begin () const

const_iterator end () const

Public Member Functions inherited from Subscriptor
Subscriptor ()

Subscriptor (const Subscriptor &)

Subscriptor (Subscriptor &&) noexcept

virtual ~Subscriptor ()

Subscriptoroperator= (const Subscriptor &)

Subscriptoroperator= (Subscriptor &&) noexcept

void subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

void unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

unsigned int n_subscriptions () const

template<typename StreamType >
void list_subscribers (StreamType &stream) const

void list_subscribers () const

template<class Archive >
void serialize (Archive &ar, const unsigned int version)

## Private Types

using const_index_value_iterator = typename std::vector< IndexValuePair >::const_iterator

## Private Member Functions

void pre_filter (VectorType &v) const

void post_filter (const VectorType &in, VectorType &out) const

## Private Attributes

bool expect_constrained_source

std::shared_ptr< PointerMatrixBase< VectorType > > matrix

std::vector< IndexValuePairconstraints

Static Public Member Functions inherited from Subscriptor
static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)

## Detailed Description

### template<typename VectorType> class FilteredMatrix< VectorType >

This class is a wrapper for linear systems of equations with simple equality constraints fixing individual degrees of freedom to a certain value such as when using Dirichlet boundary values.

In order to accomplish this, the vmult(), Tvmult(), vmult_add() and Tvmult_add functions modify the same function of the original matrix such as if all constrained entries of the source vector were zero. Additionally, all constrained entries of the destination vector are set to zero.

### Usage

Usage is simple: create an object of this type, point it to a matrix that shall be used for $$A$$ above (either through the constructor, the copy constructor, or the set_referenced_matrix() function), specify the list of boundary values or other constraints (through the add_constraints() function), and then for each required solution modify the right hand side vector (through apply_constraints()) and use this object as matrix object in a linear solver. As linear solvers should only use vmult() and residual() functions of a matrix class, this class should be as good a matrix as any other for that purpose.

Furthermore, also the precondition_Jacobi() function is provided (since the computation of diagonal elements of the filtered matrix $$A_X$$ is simple), so you can use this as a preconditioner. Some other functions useful for matrices are also available.

A typical code snippet showing the above steps is as follows:

// set up sparse matrix A and right hand side b somehow
...
// initialize filtered matrix with matrix and boundary value constraints
FilteredMatrix<Vector<double> > filtered_A (A);
// set up a linear solver
SolverControl control (1000, 1.e-10, false, false);
SolverCG<Vector<double> > solver (control, mem);
// set up a preconditioner object
prec.initialize (A, 1.2);
FilteredMatrix<Vector<double> > filtered_prec (prec);
// compute modification of right hand side
filtered_A.apply_constraints (b, true);
// solve for solution vector x
solver.solve (filtered_A, x, b, filtered_prec);

### Connection to other classes

The function MatrixTools::apply_boundary_values() does exactly the same that this class does, except for the fact that that function actually modifies the matrix. Consequently, it is only possible to solve with a matrix to which MatrixTools::apply_boundary_values() was applied for one right hand side and one set of boundary values since the modification of the right hand side depends on the original matrix.

While this is a feasible method in cases where only one solution of the linear system is required, for example in solving linear stationary systems, one would often like to have the ability to solve multiple times with the same matrix in nonlinear problems (where one often does not want to update the Hessian between Newton steps, despite having different right hand sides in subsequent steps) or time dependent problems, without having to re-assemble the matrix or copy it to temporary matrices with which one then can work. For these cases, this class is meant.

### Some background

Mathematically speaking, it is used to represent a system of linear equations $$Ax=b$$ with the constraint that $$B_D x = g_D$$, where $$B_D$$ is a rectangular matrix with exactly one $$1$$ in each row, and these $$1$$s in those columns representing constrained degrees of freedom (e.g. for Dirichlet boundary nodes, thus the index $$D$$) and zeroes for all other diagonal entries, and $$g_D$$ having the requested nodal values for these constrained nodes. Thus, the underdetermined equation $$B_D x = g_D$$ fixes only the constrained nodes and does not impose any condition on the others. We note that $$B_D B_D^T = 1_D$$, where $$1_D$$ is the identity matrix with dimension as large as the number of constrained degrees of freedom. Likewise, $$B_D^T B_D$$ is the diagonal matrix with diagonal entries $$0$$ or $$1$$ that, when applied to a vector, leaves all constrained nodes untouched and deletes all unconstrained ones.

For solving such a system of equations, we first write down the Lagrangian $$L=1/2 x^T A x - x^T b + l^T B_D x$$, where $$l$$ is a Lagrange multiplier for the constraints. The stationarity condition then reads

[ A B_D^T ] [x] = [b ]
[ B_D 0 ] [l] = [g_D]

The first equation then reads $$B_D^T l = b-Ax$$. On the other hand, if we left-multiply the first equation by $$B_D^T B_D$$, we obtain $$B_D^T B_D A x + B_D^T l = B_D^T B_D b$$ after equating $$B_D B_D^T$$ to the identity matrix. Inserting the previous equality, this yields $$(A - B_D^T B_D A) x = (1 - B_D^T B_D)b$$. Since $$x=(1 - B_D^T B_D) x + B_D^T B_D x = (1 - B_D^T B_D) x + B_D^T g_D$$, we can restate the linear system: $$A_D x = (1 - B_D^T B_D)b - (1 - B_D^T B_D) A B^T g_D$$, where $$A_D = (1 - B_D^T B_D) A (1 - B_D^T B_D)$$ is the matrix where all rows and columns corresponding to constrained nodes have been deleted.

The last system of equation only defines the value of the unconstrained nodes, while the constrained ones are determined by the equation $$B_D x = g_D$$. We can combine these two linear systems by using the zeroed out rows of $$A_D$$: if we set the diagonal to $$1$$ and the corresponding zeroed out element of the right hand side to that of $$g_D$$, then this fixes the constrained elements as well. We can write this as follows: $$A_X x = (1 - B_D^T B_D)b - (1 - B_D^T B_D) A B^T g_D + B_D^T g_D$$, where $$A_X = A_D + B_D^T B_D$$. Note that the two parts of the latter matrix operate on disjoint subspaces (the first on the unconstrained nodes, the latter on the constrained ones).

In iterative solvers, it is not actually necessary to compute $$A_X$$ explicitly, since only matrix-vector operations need to be performed. This can be done in a three-step procedure that first clears all elements in the incoming vector that belong to constrained nodes, then performs the product with the matrix $$A$$, then clears again. This class is a wrapper to this procedure, it takes a pointer to a matrix with which to perform matrix- vector products, and does the cleaning of constrained elements itself. This class therefore implements an overloaded vmult function that does the matrix-vector product, as well as Tvmult for transpose matrix-vector multiplication and residual for residual computation, and can thus be used as a matrix replacement in linear solvers.

It also has the ability to generate the modification of the right hand side, through the apply_constraints() function.

### Template arguments

This class takes as template arguments a matrix and a vector class. The former must provide vmult, vmult_add, Tvmult, and residual member function that operate on the vector type (the second template argument). The latter template parameter must provide access to individual elements through operator(), assignment through operator=.

The functions that operate as a matrix and do not change the internal state of this object are synchronized and thus threadsafe. Consequently, you do not need to serialize calls to vmult or residual .

Deprecated:
Use LinearOperator instead. See the documentation of constrained_linear_operator().

Definition at line 202 of file filtered_matrix.h.

## ◆ size_type

template<typename VectorType>
 using FilteredMatrix< VectorType >::size_type = types::global_dof_index

Declare the type of container size.

Definition at line 210 of file filtered_matrix.h.

## ◆ IndexValuePair

template<typename VectorType>
 using FilteredMatrix< VectorType >::IndexValuePair = std::pair

Typedef defining a type that represents a pair of degree of freedom index and the value it shall have.

Definition at line 333 of file filtered_matrix.h.

## ◆ const_index_value_iterator

template<typename VectorType>
 using FilteredMatrix< VectorType >::const_index_value_iterator = typename std::vector::const_iterator
private

Declare an abbreviation for an iterator into the array constraint pairs, since that data type is so often used and is rather awkward to write out each time.

Definition at line 525 of file filtered_matrix.h.

## ◆ FilteredMatrix() [1/3]

template<typename VectorType >
 FilteredMatrix< VectorType >::FilteredMatrix ( )
inline

Default constructor. You will have to set the matrix to be used later using initialize().

Definition at line 722 of file filtered_matrix.h.

## ◆ FilteredMatrix() [2/3]

template<typename VectorType >
 FilteredMatrix< VectorType >::FilteredMatrix ( const FilteredMatrix< VectorType > & fm )
inline

Copy constructor. Use the matrix and the constraints set in the given object for the present one as well.

Definition at line 729 of file filtered_matrix.h.

## ◆ FilteredMatrix() [3/3]

template<typename VectorType >
template<typename MatrixType >
 FilteredMatrix< VectorType >::FilteredMatrix ( const MatrixType & matrix, const bool expect_constrained_source = false )
inline

Constructor. Use the given matrix for future operations.

• m: The matrix being used in multiplications.

Definition at line 740 of file filtered_matrix.h.

## ◆ operator=()

template<typename VectorType >
 FilteredMatrix< VectorType > & FilteredMatrix< VectorType >::operator= ( const FilteredMatrix< VectorType > & fm )
inline

Copy operator. Take over matrix and constraints from the other object.

Definition at line 751 of file filtered_matrix.h.

## ◆ initialize()

template<typename VectorType >
template<typename MatrixType >
 void FilteredMatrix< VectorType >::initialize ( const MatrixType & m, const bool expect_constrained_source = false )
inline

Set the matrix to be used further on. You will probably also want to call the clear_constraints() function if constraints were previously added.

• m: The matrix being used in multiplications.

Definition at line 712 of file filtered_matrix.h.

## ◆ clear()

template<typename VectorType >
 void FilteredMatrix< VectorType >::clear ( )
inline

Delete all constraints and the matrix pointer.

Definition at line 807 of file filtered_matrix.h.

template<typename VectorType >
 void FilteredMatrix< VectorType >::add_constraint ( const size_type i, const double v )
inline

Add the constraint that the value with index i should have the value v.

Definition at line 763 of file filtered_matrix.h.

template<typename VectorType >
template<class ConstraintList >
 void FilteredMatrix< VectorType >::add_constraints ( const ConstraintList & new_constraints )
inline

Add a list of constraints to the ones already managed by this object. The actual data type of this list must be so that dereferenced iterators are pairs of indices and the corresponding values to be enforced on the respective solution vector's entry. Thus, the data type might be, for example, a std::list or std::vector of IndexValuePair objects, but also a std::map<unsigned, double>.

The second component of these pairs will only be used in apply_constraints(). The first is used to set values to zero in matrix vector multiplications.

It is an error if the argument contains an entry for a degree of freedom that has already been constrained previously.

Definition at line 775 of file filtered_matrix.h.

## ◆ clear_constraints()

template<typename VectorType >
 void FilteredMatrix< VectorType >::clear_constraints ( )
inline

Delete the list of constraints presently in use.

Definition at line 796 of file filtered_matrix.h.

## ◆ apply_constraints() [1/2]

template<typename VectorType >
 void FilteredMatrix< VectorType >::apply_constraints ( VectorType & v, const bool matrix_is_symmetric ) const
inline

Vector operations Apply the constraints to a right hand side vector. This needs to be done before starting to solve with the filtered matrix. If the matrix is symmetric (i.e. the matrix itself, not only its sparsity pattern), set the second parameter to true to use a faster algorithm. Note: This method is deprecated as matrix_is_symmetric parameter is no longer used.

Definition at line 817 of file filtered_matrix.h.

## ◆ apply_constraints() [2/2]

template<typename VectorType >
 void FilteredMatrix< VectorType >::apply_constraints ( VectorType & v ) const
inline

Apply the constraints to a right hand side vector. This needs to be done before starting to solve with the filtered matrix.

Definition at line 827 of file filtered_matrix.h.

## ◆ vmult()

template<typename VectorType >
 void FilteredMatrix< VectorType >::vmult ( VectorType & dst, const VectorType & src ) const
inline

Matrix-vector multiplication: this operation performs pre_filter(), multiplication with the stored matrix and post_filter() in that order.

Definition at line 888 of file filtered_matrix.h.

## ◆ Tvmult()

template<typename VectorType >
 void FilteredMatrix< VectorType >::Tvmult ( VectorType & dst, const VectorType & src ) const
inline

Matrix-vector multiplication: this operation performs pre_filter(), transposed multiplication with the stored matrix and post_filter() in that order.

Definition at line 915 of file filtered_matrix.h.

template<typename VectorType >
 void FilteredMatrix< VectorType >::vmult_add ( VectorType & dst, const VectorType & src ) const
inline

Note
The result vector of this multiplication will have the constraint entries set to zero, independent of the previous value of dst. We expect that in most cases this is the required behavior.

Definition at line 942 of file filtered_matrix.h.

template<typename VectorType >
 void FilteredMatrix< VectorType >::Tvmult_add ( VectorType & dst, const VectorType & src ) const
inline

Note
The result vector of this multiplication will have the constraint entries set to zero, independent of the previous value of dst. We expect that in most cases this is the required behavior.

Definition at line 970 of file filtered_matrix.h.

## ◆ begin()

template<typename number >
 FilteredMatrix< number >::const_iterator FilteredMatrix< number >::begin ( ) const
inline

Iterator to the first constraint.

Definition at line 685 of file filtered_matrix.h.

## ◆ end()

template<typename number >
 FilteredMatrix< number >::const_iterator FilteredMatrix< number >::end ( ) const
inline

Final iterator.

Definition at line 693 of file filtered_matrix.h.

## ◆ memory_consumption()

template<typename VectorType >
 std::size_t FilteredMatrix< VectorType >::memory_consumption ( ) const
inline

Determine an estimate for the memory consumption (in bytes) of this object. Since we are not the owner of the matrix referenced, its memory consumption is not included.

Definition at line 998 of file filtered_matrix.h.

## ◆ pre_filter()

template<typename VectorType >
 void FilteredMatrix< VectorType >::pre_filter ( VectorType & v ) const
inlineprivate

Do the pre-filtering step, i.e. zero out those components that belong to constrained degrees of freedom.

Definition at line 856 of file filtered_matrix.h.

## ◆ post_filter()

template<typename VectorType >
 void FilteredMatrix< VectorType >::post_filter ( const VectorType & in, VectorType & out ) const
inlineprivate

Do the postfiltering step, i.e. set constrained degrees of freedom to the value of the input vector, as the matrix contains only ones on the diagonal for these degrees of freedom.

Definition at line 870 of file filtered_matrix.h.

## ◆ expect_constrained_source

template<typename VectorType>
 bool FilteredMatrix< VectorType >::expect_constrained_source
private

Determine, whether multiplications can expect that the source vector has all constrained entries set to zero.

If so, the auxiliary vector can be avoided and memory as well as time can be saved.

We expect this for instance in Newton's method, where the residual already should be zero on constrained nodes. This is, because there is no test function in these nodes.

Definition at line 517 of file filtered_matrix.h.

## ◆ matrix

template<typename VectorType>
 std::shared_ptr > FilteredMatrix< VectorType >::matrix
private

Pointer to the sparsity pattern used for this matrix.

Definition at line 543 of file filtered_matrix.h.

## ◆ constraints

template<typename VectorType>
 std::vector FilteredMatrix< VectorType >::constraints
private

Sorted list of pairs denoting the index of the variable and the value to which it shall be fixed.

Definition at line 549 of file filtered_matrix.h.

The documentation for this class was generated from the following file: