This paper introduces a new preconditioning technique that is suitable for
matrices arising from the discretization of a system of PDEs on unstructured
grids. The preconditioner satisfies a so-called filtering property, which
ensures that the input matrix is identical with the preconditioner on a given
filtering vector. This vector is chosen to alleviate the effect of low
frequency modes on convergence and so decrease or eliminate the plateau which
is often observed in the convergence of iterative methods. In particular, the
paper presents a general approach that allows to ensure that the filtering
condition is satisfied in a matrix decomposition. The input matrix can have an
arbitrary sparse structure. Hence, it can be reordered using nested dissection,
to allow a parallel computation of the preconditioner and of the iterative
process

Grigori, Laura

Nataf, Frédéric

English

arXiv

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Generalized Filtering Decomposition

HAL - Lille 3

Generalized Filtering DecompositionLaura Grigori, Fre´de´ric NatafTo cite this version:Laura Grigori, Fre´de´ric Nataf. Generalized Filtering Decomposition. International ConferenceOn Preconditioning Techniques For Scientific And Industrial Applications, Preconditioning2011, May 2011, Bordeaux, France. <inria-00581744>HAL Id: inria-00581744https://hal.inria.fr/inria-00581744Submitted on 31 Mar 2011HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Generalized Filtering DecompositionList of authors:L. Grigori 1F. Nataf 2In this talk we discuss a new preconditioning method for solving a large sparse linear systemsof equations arising from the discretization of a partial differential equation (PDE) by a finitedifference or finite volume method on an unstructured grid in two or three dimensions. Thepreconditioner is based on an incomplete block LDU decomposition and is suitable for parallelcomputation.The problem of preconditioning has already been extensively studied, see for example [3], [5],[2] for descriptions of algorithms and comparisons. In particular, one class of methods whichhave proved particularly successful are the algebraic multigrid methods, see for example [4],[6], [7]. They are very successful for scalar PDEs and for a limited number of processors. Thismotivates research on iterative solvers for systems of PDEs and/or larger number of processors.Our work is in the context of filtering preconditioners, that for a given filtering vector t satisfythe relation Mt = At, where A is the input matrix and M is the preconditioner. Previous workon tangential filtering preconditioners [1] has focused on developing preconditioners for matricesarising from the discretization of a partial differential equation (PDE) on a structured grid intwo or three dimensions. The tangential filtering precondioner has been used in combinationwith ILU0.In this talk we present a block preconditioner that satisfies the filtering property and that doesnot impose any particular structure on the input matrix. It can be seen as a generalizationfor unstructured grids of the preconditioner presented in [1] for block tridiagonal matrices. Incontrast to the preconditioner presented in [1] that has been shown to be efficient in combinationwith ILU0, the block preconditioner presented here is efficient as a stand-alone preconditioner.Consider a matrix A of size n × n partitioned into a block matrix of size N × N with squarediagonal blocks (not necessarily of a same size).A =A11 . . . A1N.... . ....AN1 . . . ANN ,1INRIA Saclay - Ile de France, Laboratoire de Recherche en Informatique Universite Paris-Sud 11, France(laura.grigori@inria.fr).2Laboratoire J.L. Lions, CNRS UMR7598, Universite Paris 6, France (nataf@ann.jussieu.fr).1An exact block LDU factorization of A is written asA =D11L21 D22.... . .. . .LN1 . . . LN,N−1 DNND−111D−122. . .D−1NND11 U12 . . . U1N. . .. . ....DN−1,N−1 UN−1,NDNN ,(1)where Dii, i = 1 . . . N are square invertible matrices of size bi × bi with bi < n. In practice,even if the matrix A is very sparse, the factors L, D, U can be much denser. In particular,multiplications that involve the inverse matrices D−1kkcan introduce a lot of fill-in in the factors.In the proposed block filtering preconditioner, the inverse of the diagonal blocks D−1kk, k = 1 . . . nis approximated by a sparse matrix such that the preconditioner M stays sparse. In additionM satisfies the right filtering condition (M −A)t = 0, where t is a filtering vector. The choiceof the vector can be made similar to deflation techniques. However, in practice we find that avector of all ones leads to good convergence results.In our tests we use a reordering of the matrix that is suitable for parallel implementations. Thisreordering, known as nested dissection, allows a parallel implementation of the construction ofthe preconditioner, as well as of the iterative process.We present numerical results for two classes of problems. The first class is composed of matricesarising from the discretization of scalar PDEs on structured grids. The second class is composedof matrices from several real applications as reservoir modelling, nuclear waste disposal, etc.These results show that the preconditioner is very efficient for the matrices in our test set,which are of medium size. In our future work we focus on a parallel implementation of thepreconditioner, that will allow us to test it on larger matrices.References[1] Y. Achdou and F. Nataf. Low frequency tangential filtering decomposition. Numer. LinearAlgebra Appl., 14(2):129–147, 2007.[2] Michele Benzi and Miroslav Tuma. A comparative study of sparse approximate inversepreconditioners. Appl. Num. Math., 30:305–340, 1999.[3] G. Meurant. Computer solution of large linear systems. North-Holland Publishing Co.,Amsterdam, 1999.[4] J. W. Ruge and K. Stu¨ben. Algebraic multigrid. InMultigrid methods, volume 3 of FrontiersAppl. Math., pages 73–130. SIAM, Philadelphia, PA, 1987.[5] Y. Saad. Iterative Methods for Sparse Linear Systems. PWS Publishing Company, 1996.[6] K. Stu¨ben. A review of algebraic multigrid. J. Comput. Appl. Math., 128(1-2):281–309,2001. Numerical analysis 2000, Vol. VII, Partial differential equations.2[7] U. Trottenberg, C. W. Oosterlee, and A. Schu¨ller. Multigrid. Academic Press Inc., SanDiego, CA, 2001. With contributions by A. Brandt, P. Oswald and K. Stu¨ben.3

This paper introduces a new preconditioning technique that is suitable for matrices arising from the discretization of a system of PDEs on unstructured grids. The preconditioner satisfies a so-called filtering property, which ensures that the input matrix is identical with the preconditioner on a given filtering vector. This vector is chosen to alleviate the effect of low frequency modes on convergence and so decrease or eliminate the plateau which is often observed in the convergence of iterative methods. In particular, the paper presents a general approach that allows to ensure that the filtering condition is satisfied in a matrix decomposition. The input matrix can have an arbitrary sparse structure. Hence, it can be reordered using nested dissection, to allow a parallel computation of the preconditioner and of the iterative process

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HAL Id: inria-00581744https://hal.inria.fr/inria-00581744Submitted on 31 Mar 2011HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Generalized Filtering DecompositionLaura Grigori, Frédéric NatafTo cite this version:Laura Grigori, Frédéric Nataf. Generalized Filtering Decomposition. International Conference OnPreconditioning Techniques For Scientific And Industrial Applications, Preconditioning 2011, May2011, Bordeaux, France. ￿inria-00581744￿Generalized Filtering DecompositionList of authors:L. Grigori 1F. Nataf 2In this talk we discuss a new preconditioning method for solving a large sparse linear systemsof equations arising from the discretization of a partial differential equation (PDE) by a finitedifference or finite volume method on an unstructured grid in two or three dimensions. Thepreconditioner is based on an incomplete block LDU decomposition and is suitable for parallelcomputation.The problem of preconditioning has already been extensively studied, see for example [3], [5],[2] for descriptions of algorithms and comparisons. In particular, one class of methods whichhave proved particularly successful are the algebraic multigrid methods, see for example [4],[6], [7]. They are very successful for scalar PDEs and for a limited number of processors. Thismotivates research on iterative solvers for systems of PDEs and/or larger number of processors.Our work is in the context of filtering preconditioners, that for a given filtering vector t satisfythe relation Mt = At, where A is the input matrix and M is the preconditioner. Previous workon tangential filtering preconditioners [1] has focused on developing preconditioners for matricesarising from the discretization of a partial differential equation (PDE) on a structured grid intwo or three dimensions. The tangential filtering precondioner has been used in combinationwith ILU0.In this talk we present a block preconditioner that satisfies the filtering property and that doesnot impose any particular structure on the input matrix. It can be seen as a generalizationfor unstructured grids of the preconditioner presented in [1] for block tridiagonal matrices. Incontrast to the preconditioner presented in [1] that has been shown to be efficient in combinationwith ILU0, the block preconditioner presented here is efficient as a stand-alone preconditioner.Consider a matrix A of size n × n partitioned into a block matrix of size N × N with squarediagonal blocks (not necessarily of a same size).A =A11 . . . A1N.... . ....AN1 . . . ANN,1INRIA Saclay - Ile de France, Laboratoire de Recherche en Informatique Universite Paris-Sud 11, France(laura.grigori@inria.fr).2Laboratoire J.L. Lions, CNRS UMR7598, Universite Paris 6, France (nataf@ann.jussieu.fr).1An exact block LDU factorization of A is written asA =D11L21 D22.... . .. . .LN1 . . . LN,N−1 DNND−111D−122. . .D−1NND11 U12 . . . U1N. . .. . ....DN−1,N−1 UN−1,NDNN,(1)where Dii, i = 1 . . . N are square invertible matrices of size bi × bi with bi < n. In practice,even if the matrix A is very sparse, the factors L, D, U can be much denser. In particular,multiplications that involve the inverse matrices D−1kkcan introduce a lot of fill-in in the factors.In the proposed block filtering preconditioner, the inverse of the diagonal blocks D−1kk, k = 1 . . . nis approximated by a sparse matrix such that the preconditioner M stays sparse. In additionM satisfies the right filtering condition (M − A)t = 0, where t is a filtering vector. The choiceof the vector can be made similar to deflation techniques. However, in practice we find that avector of all ones leads to good convergence results.In our tests we use a reordering of the matrix that is suitable for parallel implementations. Thisreordering, known as nested dissection, allows a parallel implementation of the construction ofthe preconditioner, as well as of the iterative process.We present numerical results for two classes of problems. The first class is composed of matricesarising from the discretization of scalar PDEs on structured grids. The second class is composedof matrices from several real applications as reservoir modelling, nuclear waste disposal, etc.These results show that the preconditioner is very efficient for the matrices in our test set,which are of medium size. In our future work we focus on a parallel implementation of thepreconditioner, that will allow us to test it on larger matrices.References[1] Y. Achdou and F. Nataf. Low frequency tangential filtering decomposition. Numer. LinearAlgebra Appl., 14(2):129–147, 2007.[2] Michele Benzi and Miroslav Tuma. A comparative study of sparse approximate inversepreconditioners. Appl. Num. Math., 30:305–340, 1999.[3] G. Meurant. Computer solution of large linear systems. North-Holland Publishing Co.,Amsterdam, 1999.[4] J. W. Ruge and K. Stüben. Algebraic multigrid. In Multigrid methods, volume 3 of FrontiersAppl. Math., pages 73–130. SIAM, Philadelphia, PA, 1987.[5] Y. Saad. Iterative Methods for Sparse Linear Systems. PWS Publishing Company, 1996.[6] K. Stüben. A review of algebraic multigrid. J. Comput. Appl. Math., 128(1-2):281–309,2001. Numerical analysis 2000, Vol. VII, Partial differential equations.2[7] U. Trottenberg, C. W. Oosterlee, and A. Schüller. Multigrid. Academic Press Inc., SanDiego, CA, 2001. With contributions by A. Brandt, P. Oswald and K. Stüben.3

Hal-Diderot

Generalized Filtering Decomposition
Laura Grigori, Fre´de´ric Nataf
To cite this version:
Laura Grigori, Fre´de´ric Nataf. Generalized Filtering Decomposition. International Conference
On Preconditioning Techniques For Scientific And Industrial Applications, Preconditioning
2011, May 2011, Bordeaux, France. <inria-00581744>
HAL Id: inria-00581744
https://hal.inria.fr/inria-00581744
Submitted on 31 Mar 2011
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destine´e au de´poˆt et a` la diffusion de documents
scientifiques de niveau recherche, publie´s ou non,
e´manant des e´tablissements d’enseignement et de
recherche franc¸ais ou e´trangers, des laboratoires
publics ou prive´s.
Generalized Filtering Decomposition
List of authors:
L. Grigori 1
F. Nataf 2
In this talk we discuss a new preconditioning method for solving a large sparse linear systems
of equations arising from the discretization of a partial differential equation (PDE) by a finite
difference or finite volume method on an unstructured grid in two or three dimensions. The
preconditioner is based on an incomplete block LDU decomposition and is suitable for parallel
computation.
The problem of preconditioning has already been extensively studied, see for example [3], [5],
[2] for descriptions of algorithms and comparisons. In particular, one class of methods which
have proved particularly successful are the algebraic multigrid methods, see for example [4],
[6], [7]. They are very successful for scalar PDEs and for a limited number of processors. This
motivates research on iterative solvers for systems of PDEs and/or larger number of processors.
Our work is in the context of filtering preconditioners, that for a given filtering vector t satisfy
the relation Mt = At, where A is the input matrix and M is the preconditioner. Previous work
on tangential filtering preconditioners [1] has focused on developing preconditioners for matrices
arising from the discretization of a partial differential equation (PDE) on a structured grid in
two or three dimensions. The tangential filtering precondioner has been used in combination
with ILU0.
In this talk we present a block preconditioner that satisfies the filtering property and that does
not impose any particular structure on the input matrix. It can be seen as a generalization
for unstructured grids of the preconditioner presented in [1] for block tridiagonal matrices. In
contrast to the preconditioner presented in [1] that has been shown to be efficient in combination
with ILU0, the block preconditioner presented here is efficient as a stand-alone preconditioner.
Consider a matrix A of size n × n partitioned into a block matrix of size N × N with square
diagonal blocks (not necessarily of a same size).
A =


A11 . . . A1N
...
. . .
...
AN1 . . . ANN

 ,
1INRIA Saclay - Ile de France, Laboratoire de Recherche en Informatique Universite Paris-Sud 11, France
(laura.grigori@inria.fr).
2Laboratoire J.L. Lions, CNRS UMR7598, Universite Paris 6, France (nataf@ann.jussieu.fr).
1
An exact block LDU factorization of A is written as
A =


D11
L21 D22
...
. . .
. . .
LN1 . . . LN,N−1 DNN




D−1
11
D−1
22
. . .
D−1NN




D11 U12 . . . U1N
. . .
. . .
...
DN−1,N−1 UN−1,N
DNN

 ,
(1)
where Dii, i = 1 . . . N are square invertible matrices of size bi × bi with bi < n. In practice,
even if the matrix A is very sparse, the factors L, D, U can be much denser. In particular,
multiplications that involve the inverse matrices D−1
kk
can introduce a lot of fill-in in the factors.
In the proposed block filtering preconditioner, the inverse of the diagonal blocks D−1
kk
, k = 1 . . . n
is approximated by a sparse matrix such that the preconditioner M stays sparse. In addition
M satisfies the right filtering condition (M −A)t = 0, where t is a filtering vector. The choice
of the vector can be made similar to deflation techniques. However, in practice we find that a
vector of all ones leads to good convergence results.
In our tests we use a reordering of the matrix that is suitable for parallel implementations. This
reordering, known as nested dissection, allows a parallel implementation of the construction of
the preconditioner, as well as of the iterative process.
We present numerical results for two classes of problems. The first class is composed of matrices
arising from the discretization of scalar PDEs on structured grids. The second class is composed
of matrices from several real applications as reservoir modelling, nuclear waste disposal, etc.
These results show that the preconditioner is very efficient for the matrices in our test set,
which are of medium size. In our future work we focus on a parallel implementation of the
preconditioner, that will allow us to test it on larger matrices.
References
[1] Y. Achdou and F. Nataf. Low frequency tangential filtering decomposition. Numer. Linear
Algebra Appl., 14(2):129–147, 2007.
[2] Michele Benzi and Miroslav Tuma. A comparative study of sparse approximate inverse
preconditioners. Appl. Num. Math., 30:305–340, 1999.
[3] G. Meurant. Computer solution of large linear systems. North-Holland Publishing Co.,
Amsterdam, 1999.
[4] J. W. Ruge and K. Stu¨ben. Algebraic multigrid. InMultigrid methods, volume 3 of Frontiers
Appl. Math., pages 73–130. SIAM, Philadelphia, PA, 1987.
[5] Y. Saad. Iterative Methods for Sparse Linear Systems. PWS Publishing Company, 1996.
[6] K. Stu¨ben. A review of algebraic multigrid. J. Comput. Appl. Math., 128(1-2):281–309,
2001. Numerical analysis 2000, Vol. VII, Partial differential equations.
2
[7] U. Trottenberg, C. W. Oosterlee, and A. Schu¨ller. Multigrid. Academic Press Inc., San
Diego, CA, 2001. With contributions by A. Brandt, P. Oswald and K. Stu¨ben.
3


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