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// $Id: schur_matrix.h 20934 2010-04-02 13:09:20Z bangerth $
// Version: $Name$
//
// Copyright (C) 2001, 2002, 2003, 2004, 2005, 2006, 2009, 2010 by the deal.II authors
//
// This file is subject to QPL and may not be distributed
// without copyright and license information. Please refer
// to the file deal.II/doc/license.html for the text and
// further information on this license.
//
//---------------------------------------------------------------------------
#ifndef __deal2__schur_matrix_h
#define __deal2__schur_matrix_h
#include <base/config.h>
#include <base/subscriptor.h>
#include <base/smartpointer.h>
#include <base/logstream.h>
#include <lac/vector_memory.h>
#include <lac/block_vector.h>
#include <vector>
DEAL_II_NAMESPACE_OPEN
/*! @addtogroup Matrix2
*@{
*/
/**
* Schur complement of a block matrix.
*
* Given a non-singular matrix @p A (often positive definite) and a
* positive semi-definite matrix @p C as well as matrices @p B and
* @p Dt of full rank, this class implements a new matrix, the Schur
* complement a the system of equations of the structure
*
* @verbatim
* / \ / \ / \
* | A Dt | | u | - | f |
* | -B C | | p | - | g |
* \ / \ / \ /
* @endverbatim
*
* Multiplication with the Schur matrix @p S is the operation
* @verbatim
* S p = C p + B A-inverse Dt-transpose p,
* @endverbatim
* which is an operation within the space for @p p.
*
* The data handed to the Schur matrix are as follows:
*
* @p A: the inverse of @p A is stored, instead of @p A. This
* allows the application to use the most efficient form of inversion,
* iterative or direct.
*
* @p B, @p C: these matrices are stored "as is".
*
* @p Dt: the computation of the Schur complement involves the
* function @p Tvmult of the matrix @p Dt, not @p vmult! This way,
* it is sufficient to build only one matrix @p B for the symmetric
* Schur complement and use it twice.
*
* All matrices involved are of arbitrary type and vectors are
* BlockVectors. This way, @p SchurMatrix can be coupled with
* any matrix classes providing @p vmult and @p Tvmult and can be
* even nested. Since SmartPointers of matrices are stored, the
* matrix blocks should be derived from Subscriptor.
*
* Since the Schur complement of a matrix corresponds to a Gaussian
* block elimination, the right hand side of the condensed system must
* be preprocessed. Furthermore, the eliminated variable must be
* reconstructed after solving.
*
* @verbatim
* g = g + B A-inverse f
* u = A-inverse (f - D-transpose p)
* @endverbatim
*
* Applying these transformations, the solution of the system above by a
* @p SchurMatrix @p schur is coded as follows:
*
* @verbatim
* schur.prepare_rhs (g, f);
* solver.solve (schur, p, g, precondition);
* schur.postprocess (u, p);
* @endverbatim
*
* @see @ref GlossBlockLA "Block (linear algebra)"
* @author Guido Kanschat, 2000, 2001, 2002
*/
template <class MA_inverse, class MB, class MDt, class MC>
class SchurMatrix : public Subscriptor
{
public:
/**
* Constructor. This constructor
* receives all the matrices
* needed. Furthermore, it gets a
* reference to a memory structure
* for obtaining block vectors.
*
* Optionally, the length of the
* @p u-vector can be provided.
*
* For the meaning of the matrices
* see the class documentation.
*/
SchurMatrix(const MA_inverse& Ainv,
const MB& B,
const MDt& Dt,
const MC& C,
VectorMemory<BlockVector<double> >& mem,
const std::vector<unsigned int>& signature = std::vector<unsigned int>(0));
/**
* Do block elimination of the
* right hand side. Given right
* hand sides for both components
* of the block system, this
* function provides the right hand
* side for the Schur complement.
*
* The result is stored in the
* first argument, which is also
* part of the input data. If it is
* necessary to conserve the data,
* @p dst must be copied before
* calling this function. This is
* reasonable, since in many cases,
* only the pre-processed right
* hand side is needed.
*/
void prepare_rhs (BlockVector<double>& dst,
const BlockVector<double>& src) const;
/**
* Multiplication with the Schur
* complement.
*/
void vmult (BlockVector<double>& dst,
const BlockVector<double>& src) const;
// void Tmult(BlockVector<double>& dst, const BlockVector<double>& src) const;
/**
* Computation of the residual of
* the Schur complement.
*/
double residual (BlockVector<double>& dst,
const BlockVector<double>& src,
const BlockVector<double>& rhs) const;
/**
* Compute the eliminated variable
* from the solution of the Schur
* complement problem.
*/
void postprocess (BlockVector<double>& dst,
const BlockVector<double>& src,
const BlockVector<double>& rhs) const;
/**
* Select debugging information for
* log-file. Debug level 1 is
* defined and writes the norm of
* every vector before and after
* each operation. Debug level 0
* turns off debugging information.
*/
void debug_level(unsigned int l);
private:
/**
* No copy constructor.
*/
SchurMatrix (const SchurMatrix<MA_inverse, MB, MDt, MC>&);
/**
* No assignment.
*/
SchurMatrix& operator = (const SchurMatrix<MA_inverse, MB, MDt, MC>&);
/**
* Pointer to inverse of upper left block.
*/
const SmartPointer<const MA_inverse,SchurMatrix<MA_inverse,MB,MDt,MC> > Ainv;
/**
* Pointer to lower left block.
*/
const SmartPointer<const MB,SchurMatrix<MA_inverse,MB,MDt,MC> > B;
/**
* Pointer to transpose of upper right block.
*/
const SmartPointer<const MDt,SchurMatrix<MA_inverse,MB,MDt,MC> > Dt;
/**
* Pointer to lower right block.
*/
const SmartPointer<const MC,SchurMatrix<MA_inverse,MB,MDt,MC> > C;
/**
* Auxiliary memory for vectors.
*/
VectorMemory<BlockVector<double> >& mem;
/**
* Optional signature of the @p u-vector.
*/
std::vector<unsigned int> signature;
/**
* Switch for debugging information.
*/
unsigned int debug;
};
/*@}*/
//---------------------------------------------------------------------------
template <class MA_inverse, class MB, class MDt, class MC>
SchurMatrix<MA_inverse, MB, MDt, MC>
::SchurMatrix(const MA_inverse& Ainv,
const MB& B,
const MDt& Dt,
const MC& C,
VectorMemory<BlockVector<double> >& mem,
const std::vector<unsigned int>& signature)
: Ainv(&Ainv), B(&B), Dt(&Dt), C(&C),
mem(mem),
signature(signature),
debug(0)
{
}
template <class MA_inverse, class MB, class MDt, class MC>
void
SchurMatrix<MA_inverse, MB, MDt, MC>
::debug_level(unsigned int l)
{
debug = l;
}
template <class MA_inverse, class MB, class MDt, class MC>
void SchurMatrix<MA_inverse, MB, MDt, MC>
::vmult(BlockVector<double>& dst,
const BlockVector<double>& src) const
{
deallog.push("Schur");
if (debug > 0)
deallog << "src:" << src.l2_norm() << std::endl;
C->vmult(dst, src);
if (debug > 0)
deallog << "C:" << dst.l2_norm() << std::endl;
BlockVector<double>* h1 = mem.alloc();
if (signature.size()>0)
h1->reinit(signature);
else
h1->reinit(B->n_block_cols(), src.block(0).size());
Dt->Tvmult(*h1,src);
if (debug > 0)
deallog << "Dt:" << h1->l2_norm() << std::endl;
BlockVector<double>* h2 = mem.alloc();
h2->reinit(*h1);
Ainv->vmult(*h2, *h1);
if (debug > 0)
deallog << "Ainverse:" << h2->l2_norm() << std::endl;
mem.free(h1);
B->vmult_add(dst, *h2);
if (debug > 0)
deallog << "dst:" << dst.l2_norm() << std::endl;
mem.free(h2);
deallog.pop();
}
template <class MA_inverse, class MB, class MDt, class MC>
double SchurMatrix<MA_inverse, MB, MDt, MC>
::residual(BlockVector<double>& dst,
const BlockVector<double>& src,
const BlockVector<double>& rhs) const
{
vmult(dst, src);
dst.scale(-1.);
dst += rhs;
return dst.l2_norm();
}
template <class MA_inverse, class MB, class MDt, class MC>
void SchurMatrix<MA_inverse, MB, MDt, MC>
::prepare_rhs(BlockVector<double>& dst,
const BlockVector<double>& src) const
{
Assert (src.n_blocks() == B->n_block_cols(),
ExcDimensionMismatch(src.n_blocks(), B->n_block_cols()));
Assert (dst.n_blocks() == B->n_block_rows(),
ExcDimensionMismatch(dst.n_blocks(), B->n_block_rows()));
deallog.push("Schur-prepare");
if (debug > 0)
deallog << "src:" << src.l2_norm() << std::endl;
BlockVector<double>* h1 = mem.alloc();
if (signature.size()>0)
h1->reinit(signature);
else
h1->reinit(B->n_block_cols(), src.block(0).size());
Ainv->vmult(*h1, src);
if (debug > 0)
deallog << "Ainverse:" << h1->l2_norm() << std::endl;
B->vmult_add(dst, *h1);
if (debug > 0)
deallog << "dst:" << dst.l2_norm() << std::endl;
mem.free(h1);
deallog.pop();
}
template <class MA_inverse, class MB, class MDt, class MC>
void SchurMatrix<MA_inverse, MB, MDt, MC>
::postprocess(BlockVector<double>& dst,
const BlockVector<double>& src,
const BlockVector<double>& rhs) const
{
Assert (dst.n_blocks() == B->n_block_cols(),
ExcDimensionMismatch(dst.n_blocks(), B->n_block_cols()));
Assert (rhs.n_blocks() == B->n_block_cols(),
ExcDimensionMismatch(rhs.n_blocks(), B->n_block_cols()));
Assert (src.n_blocks() == B->n_block_rows(),
ExcDimensionMismatch(src.n_blocks(), B->n_block_rows()));
deallog.push("Schur-post");
if (debug > 0)
deallog << "src:" << src.l2_norm() << std::endl;
BlockVector<double>* h1 = mem.alloc();
if (signature.size()>0)
h1->reinit(signature);
else
h1->reinit(B->n_block_cols(), src.block(0).size());
Dt->Tvmult(*h1, src);
if (debug > 0)
deallog << "Dt:" << h1->l2_norm() << std::endl;
h1->sadd(-1.,rhs);
Ainv->vmult(dst,*h1);
if (debug > 0)
deallog << "dst:" << dst.l2_norm() << std::endl;
mem.free(h1);
deallog.pop();
}
DEAL_II_NAMESPACE_CLOSE
#endif
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