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Efficient Algebraic Two-Level Schwarz Preconditioner For Sparse Matrices

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Abstract and Figures

Domain decomposition methods are among the most efficient for solving sparse linear systems of equations. Their effectiveness relies on a judiciously chosen coarse space. Originally introduced and theoretically proved to be efficient for self-adjoint operators, spectral coarse spaces have been proposed in the past few years for indefinite and non-self-adjoint operators. This paper presents a new spectral coarse space that can be constructed in a fully-algebraic way unlike most existing spectral coarse spaces. We present theoretical convergence result for Hermitian positive definite diagonally dominant matrices. Numerical experiments and comparisons against state-of-the-art preconditioners in the multigrid community show that the resulting two-level Schwarz preconditioner is efficient especially for non-self-adjoint operators. Furthermore, in this case, our proposed preconditioner outperforms state-of-the-art preconditioners.
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Abstract. Domain decomposition methods are among the most efficient for solving sparse linear
systems of equations. Their effectiveness relies on a judiciously chosen coarse space. Originally
introduced and theoretically proved to be efficient for self-adjoint operators, spectral coarse spaces
have been proposed in the past few years for indefinite and non-self-adjoint operators. This paper
presents a new spectral coarse space that can be constructed in a fully-algebraic way unlike most
existing spectral coarse spaces. We present theoretical convergence result for Hermitian positive
definite diagonally dominant matrices. Numerical experiments and comparisons against state-
of-the-art preconditioners in the multigrid community show that the resulting two-level Schwarz
preconditioner is efficient especially for non-self-adjoint operators. Furthermore, in this case, our
proposed preconditioner outperforms state-of-the-art preconditioners.
Key words. Algebraic domain decomposition, Schwarz preconditioner, sparse linear systems,
diagonally dominant matrices.
1. Introduction. In this paper, we develop an algebraic overlapping Schwarz
preconditioner for the linear system of equations
Ax =b,
for a sparse matrix ACn×nand a given vector bCn. Solving sparse linear
systems of equations is omnipresent in scientific computing. Direct approaches, based
on Gaussian elimination, have proved to be robust and efficient for a wide range of
problems [23]. However, the memory required to apply sparse direct methods often
scales poorly with the problem size, particularly for three-dimensional discretizations
of partial differential equations (PDEs). Furthermore, the algorithms underpinning
sparse direct software are poorly suited to parallel computation, which makes them
difficult to adapt to emerging computing architectures.
Iterative methods for solving linear systems [45] have been an active research topic
since early computers’ days. Their simple structure, at their most basic level requiring
only matrix-vector multiplication and vector-vector operations, makes them attractive
for tackling large-scale problems. However, since the convergence rate depends on the
properties of the linear system, iterative methods are not, in general, robust. For the
class of iterative methods known as Krylov subspace methods, we may alleviate this by
applying a preconditioner, which transforms the problem into one with more favorable
numerical properties. The choice of preconditioner is usually problem-dependent,
and a wide variety of preconditioning techniques have been proposed to improve the
convergence rate of iterative methods, see for example the recent survey [44] and the
references therein.
Multilevel domain decomposition (DD) and multigrid methods are widely used
preconditioners [21,47,52,53,54]. They have proved to be effective on a wide variety
of matrices, but they are especially well suited to sparse Hermitian positive definite
(HPD) matrices arising from the discretization of PDEs. Their efficiency stems from
Submitted to the editors January 6, 2022.
STFC Rutherford Appleton Laboratory, Harwell Campus, Didcot, Oxfordshire, OX11 0QX, UK
CNRS, ENSEEIHT, 2 rue Charles Camichel, 31071 Toulouse Cedex 7, France
a judicious combination of a cheap fine-level solver with a coarse-space correction. In
the last two decades, there has been a great advance in the development of spectral
coarse spaces that yield efficient preconditioners. Spectral coarse spaces were initially
proposed in the multigrid community for elliptic PDEs with self-adjoint operators [15,
19,24,34], and similar ideas were later picked up by the DD community for the same
kind of problems [2,3,4,7,33,32,40,48,49]. The past three years have seen several
approaches to tackle symmetric indefinite systems and non-self-adjoint problems. For
example, spectral coarse spaces for least squares problems and symmetric indefinite
saddle-point systems were proposed in [5,39], where the problem is returned into the
framework of self-adjoint operators. An exciting new development is that a number of
multigrid methods and spectral coarse spaces have been suggested for problems with
indefinite or non-self-adjoint operators [9,10,11,12,13,14,22,36,37]. These coarse
spaces are mainly based on heuristics and show efficiency on several challenging model
problems arising from discretized PDEs.
A variety of mathematical tools such as the fictitious subspace lemma [41] and
local Hermitian positive semi-definite (HPSD) splitting matrices [2] are now available
to analyze and propose effective coarse spaces for self-adjoint operators. However,
these tools may not be directly used for indefinite or non-self-adjoint operators. An
alternative approach to studying the convergence of DD methods in the indefinite or
non-self-adjoint case is to use Elman’s theory [25] of GMRES convergence, see for
example [8,10,18].
In this work, we propose a fully-algebraic spectral coarse space for the two-level
Schwarz preconditioner for general sparse matrices. We review the overlapping DD
framework in section 2, including a summary of the main features of local HPSD
splitting matrices. For each subdomain, we introduce the local block splitting using
lumping in the overlap of a sparse matrix in section 3. The coarse space is then
constructed by solving locally and concurrently a generalized eigenvalue problem
involving the local block splitting matrix and the local subdomain matrix. In the
case where the matrix is HPD diagonally dominant, we prove that the local block
splitting matrices are local HPSD splitting matrices, and in that case we show that one
can bound the condition number of the preconditioned matrix from above by a user-
defined number. Based on this heuristic, we generalize our approach for other cases.
Unlike most existing spectral coarse spaces, especially those suggested for indefinite
or non-self-adjoint operators, we obtain the matrices involved in the local generalized
eigenvalue problem efficiently from the coefficient matrix; the preconditioner is
therefore fully-algebraic. In order to assess the proposed preconditioner, we provide
in section 4 a set of numerical experiments on problems arising from a wide range of
applications including convection-diffusion equation and other linear systems from the
SuiteSparse Collection [20]. Furthermore, we compare our proposed preconditioner
against state-of-the-art preconditioners in the multigrid community. Finally, we give
concluding remarks and future lines of research in section 5.
Notation. Let 1 nmand let MCm×nbe a complex sparse matrix. Let
J1, pKdenote the set of the first ppositive integers, and let S1J1, mKand S2J1, nK.
M(S1,:) is the submatrix of Mformed by the rows whose indices belong to S1and
M(:, S2) is the submatrix of Mformed by the columns whose indices belong to S2.
M(S1, S2) denotes the submatrix formed by taking the rows whose indices belong
to S1and only retaining the columns whose indices belong to S2. [S1, S2] means
the concatenation of any two sets of integers S1and S2, where the order of the
concatenation is important. Inis the identity matrix of size n, the transpose matrix
of Mis denoted M>, and the adjoint of M, denoted MH, is the conjugate transpose
of M, i.e., MH=¯
M>.ker(M) and range(M) denote the null space and the range
of M, respectively.
2. Domain decomposition. Consider G(A), the adjacency graph of the
coefficient matrix in (1), and number its nodes, V, from 1 to n. Using a graph
partitioning algorithm, we split Vinto Nnnonoverlapping subdomains, i.e.,
disjoint subsets ΩIi ,iJ1, N K, of size nIi. Let ΩΓibe the subset, of size nΓi, of
nodes that are distance one in G(A) from the nodes in ΩI i,iJ1, N K. We define
the overlapping subdomain, Ωi, as Ωi= [ΩIi ,Γi], with size ni=nΓi+nIi . The
complement of Ωiin J1, nKis denoted by Ωci.
Associated with ΩI i is a restriction (or projection) matrix RI i RnIi ×ngiven by
RIi =In(ΩI i,:). RI i maps from the global domain to subdomain ΩI i. Its transpose
Ii is a prolongation matrix that maps from subdomain ΩIi to the global domain.
Similarly, we define Ri=In(Ωi,:) as the restriction operator to the overlapping
subdomain Ωi.
We define the one-level Schwarz preconditioner as
(2.1) M1
ii Ri,
where we assume Aii =RiAR>
iis nonsingular for iJ1, N K.
Applying this preconditioner to a vector involves solving concurrent local
problems in each subdomain. Increasing Nreduces the size of the subdomains, leading
to smaller local problems and, correspondingly, faster computations. However, in
practice, preconditioning by M1
ASM alone is often not be enough for convergence of
the iterative solver to be sufficiently rapid. We can improve convergence, while still
maintaining robustness with respect to N, by applying a suitably chosen coarse space,
or second-level [2,8,21,27].
Let R ⊂ Cnbe a subspace of dimension 0 < n0nand let R0Cn0×n
be a matrix such that the columns of RH
0span the subspace R. Assuming that
A00 =R0ARH
0is nonsingular, we define the two-level Schwarz preconditioner as
(2.2) M1
additive =RH
00 R0+M1
Such preconditioners have been used to solve a large class of systems arising from
a range of engineering applications (see, for example, [3,5,29,35,38,47,51] and
references therein).
We denote by DiRni×ni,iJ1, NK, any non-negative diagonal matrices such
We refer to (Di)1iNas an algebraic partition of unity.
Variants of one- and two-level preconditioners. The so-far presented Schwarz
preconditioners are the additive one-level (2.1) and the additive two-level based on
additive coarse space correction (2.2). It was noticed in [17] that scaling the one-level
Schwarz preconditioner by using the partition of unity yields faster convergence. The
resulting one-level preconditioner is referred to as restricted additive Schwarz and is
defined as:
(2.3) M1
ii Ri.
Furthermore, there is a number of ways of how to combine the coarse space with a
one-level preconditioner such as the additive, deflated, and balanced combinations,
see for example [50]. Given a one-level preconditioner M1
?, where the subscript ?
stands for either ASM or RAS, the two-level preconditioner with additive coarse space
correction is defined as
?,additive =RH
00 R0+M1
The two-level preconditioner based on a deflated coarse space correction is
?,deflated =RH
00 R0+M1
00 R0.
Due to its simple form, the additive two-level Schwarz based on the additive coarse
space correction is the easiest to analyze. However, we observe that the deflated
variant combined with the restricted additive Schwarz preconditioner has better
performance in practice. The theory and presentation in this work employs the
additive two-level Schwarz preconditioner using an additive coarse space correction,
however, all numerical experiments involving the proposed preconditioner employ the
restricted additive two-level Schwarz with deflated coarse space correction so that the
two-level preconditioner used in section 4 reads as
(2.4) M1
deflated =RH
00 R0+M1
00 R0.
Note that the aforementioned variants are agnostic to the choice of the partitioning
and the coarse space. That is, once the restriction operators to the subdomains and
the coarse space are set, all these variants are available.
Local HPSD splitting matrices of sparse HPD matrix. A local HPSD matrix
associated with subdomain iis any HPSD matrix of the form
AΓI,i e
where Pi=In(ΩIi ,Γi,ci,:) is a permutation matrix, AI i =A(ΩIi ,Ii ), AIΓ,i =
ΓI,i =A(ΩI i,Γi), and e
AΓ,i is any HPSD matrix such that the following inequality
Aiuu>Au, u Cn.
First presented and analyzed in [2], local HPSD splitting matrices provide
a framework to construct robust two-level Schwarz preconditioners for sparse
HPD matrices. Recently, this has led to the introduction of robust multilevel
Schwarz preconditioners for finite element SPD matrices [3], sparse normal equations
matrices [5], and sparse general SPD matrices [4].
3. Two-level Schwarz preconditioner for sparse matrices. We present in
this section the construction of a two-level preconditioner for sparse matrices. First,
we introduce a new local splitting matrix associated with subdomain ithat uses local
values of Ato construct a preconditioner which is cheap to setup. In the special chase
where Ais HPD diagonally dominant, we prove that these local matrices are local
HPSD splitting matrices of A. We demonstrate that these matrices, when used to
construct a GenEO-like coarse space, produce a two-level Schwarz preconditioner that
outperforms existing two-level preconditioners in many applications, particularly in
the difficult case where Ais not positive definite.
3.1. Local block splitting matrices of Ausing lumping in the overlap.
Definition 3.1. Local block splitting matrix. Given the overlapping partitioning
of Apresented in section 2, we have for each iJ1, NK
PiAP >
Let sibe the vector whose jth component is the sum of the absolute values of the jth
row of AΓci, and let Si= diag(si). Define e
AΓi=AΓiSi. The local block splitting
matrix of Aassociated with subdomain iis defined to be e
AiiRi, where
Aii =AIi AIΓi
AΓIi e
Note that we only require the sum of the absolute values of each row in the
local matrix AΓcito construct the local block splitting matrix of Aassociated with
subdomain i. Then, each of these values is subtracted from the corresponding diagonal
entry of the local matrix AΓi. We can therefore construct e
Aii cheaply and concurrently
for each subdomain.
The following lemma shows that if Ais HPD diagonally dominant, the local
splitting matrices defined in Definition 3.1 are local HPSD splitting matrices.
Lemma 3.2. Let Abe HPD diagonally dominant. The local block splitting matrix
Aidefined in Definition 3.1 is local HPSD splitting matrix of Awith respect to
subdomain i.
Proof. First, note that the jth diagonal element of e
Ai(j, j) =
A(j, j) if jIi ,
A(j, j)si(j) if jΓi,
0 if jci,
where siis the vector whose jth component is the sum of the absolute values of the jth
row of AΓci. Therefore, by construction, e
Aiis Hermitian diagonally dominant, hence
HPSD. Furthermore, Ae
Aiis Hermitian and diagonally dominant, hence HPSD.
By the local structure of e
Ai, we conclude it is HPSD splitting of Awith respect to
subdomain i.
3.2. Coarse space. In this section we present a coarse space for the two-level
Schwarz preconditioner. For each iJ1, N K, given the local nonsingular matrix
Aii, the local splitting matrix e
Aii =Rie
i, and the partition of unity matrix Di,
let Li=ker (DiAii Di) and Ki=ker e
Aii. Now, define the following generalized
eigenvalue problem:
find (λ, u)C×Cnisuch that
where Πiis the projection on range e
Given a number τ > 0, the coarse space we propose is defined to be the space
generated by the columns of the matrix
1D1Z1· · · R>
where Ziis the matrix whose columns form a basis of the subspace
(3.2) (LiKi)Kispan u|ΠiDiAiiDiΠiu=λe
Aiiu, |λ|>1
where (LiKi)Kiis the complementary subspace of (LiKi) inside Ki.
Note that in the case where Ais sparse HPD diagonally dominant, e
Aiis a local
HPSD splitting matrix of A, and the definition of the coarse space matches the one
defined in [2]. Therefore, the two-level Schwarz preconditioner using the coarse space
defined guarantees an upper bound on the condition number of the preconditioned
ASM,additiveA)(kc+ 1) 2 + (2kc+ 1)km
where kcis the number of colors required to color the graph of Asuch that each
two neighboring subdomains have different colors and kmis the maximum number
of overlapping subdomains sharing a row of A. Therefore, when Ais sparse HPD
diagonally dominant, the upper bound on κ2(M1
ASM,additiveA) is independent of Nand
can be controlled by using the value τ.
4. Numerical experiments. In this section, we validate the effectiveness of
the two-level method when compared to other preconditioners. Table 1 presents
a comparison between four preconditioners: M1
deflated (2.4), BoomerAMG [26],
GAMG [1], and AGMG [42]. The results are for the right-preconditioned GMRES [46]
with a restart parameter of 30 and a relative tolerance set to 108. We highlight
the fact that our proposed preconditioner can handle unstructured systems, not
necessarily stemming from standard PDE discretization schemes, by displaying some
nonzero patterns in Figure 1. For preconditioners used within PETSc [6] (all except
AGMG), the systems are solved using 256 MPI processes. After loading them from
disk, their symmetric part AT+Ais first renumbered by ParMETIS [31]. The resulting
permutation is then applied to Aand the corresponding linear systems are solved using
a random right-hand side. The initial guess is always zero. The code that implements
these steps is given in Appendix A. For our DD method, we leverage the PCHPDDM
framework [30] which is used to assemble spectral coarse spaces using (3.1). The new
option -pc_hpddm_block_splitting, introduced in PETSc 3.17, is used to compute
the local splitting matrices of Afrom subsection 3.1. At most 60 eigenpairs are
computed on each subdomain and the threshold parameter τfrom (3.2) is set to 0.3.
These parameters were found to provide good numerical performance after a very
quick trial-and-error approach on a single problem. We did not want to adjust them
for each problem individually, but it will be shown next that they are fine overall
without additional tuning.
Furthermore, a single subdomain is mapped to each process, i.e., N= 256
in (2.3). Eventually, exact subdomain and second-level operator LU factorizations
are computed.
Table 1
Preconditioner comparison. Iteration counts are reported. M1
deflated is the restricted two-level
overlapping Schwarz preconditioner as in (2.4). No value denotes iteration count exceeds 100.
denotes either a failure in constructing the preconditioner or a breakdown in GMRES. denotes
the problem is complex valued and the preconditioner is unavailable. Matrix identifiers that are
emphasized correspond to symmetric matrices, otherwise matrices are non-self-adjoint.
Identifier nnnz(A) AGMG BoomerAMG GAMG M1
deflated n0
light in tissue 29,282 406,084 15 53 67,230
finan512 74,752 596,992 9 7 8 62,591
consph 83,334 6,010,480 93 31,136
Dubcova3 146,689 3,636,643 72 71 721,047
CO 221,119 7,666,057 25 26 56,135
nxp1 414,604 2,655,880 20 19,707
CoupCons3D 416,800 17,277,420 26 20 28,925
parabolic fem 525,825 3,674,625 12 8 16 524,741
Chevron4 711,450 6,376,412 522,785
apache2 715,176 4,817,870 14 11 35 845,966
tmt sym 726,713 5,080,961 14 10 17 528,253
tmt unsym 917,825 4,584,801 23 13 18 632,947
ecology2 999,999 4,995,991 18 12 18 634,080
thermal2 1,228,045 8,580,313 18 14 20 26 40,098
atmosmodj 1,270,432 8,814,880 8 17 776,368
G3 circuit 1,585,478 7,660,826 25 12 35 871,385
Transport 1,602,111 23,487,281 18 10 98 976,800
memchip 2,707,524 13,343,948 15 36 57,942
circuit5M dc 3,523,317 14,865,409 57 8,629
Fig. 1.Nonzero sparsity pattern of some of the test matrices from Table 1.
(a) nxp1, n= 4.1·105(b) CoupCons3D, n= 4.2·105(c) memchip, n= 2.7·106
A convection-diffusion problem will now be investigated. It reads:
∇ · (V u)ν∇ · (κu) = 0 in
u= 0 in Γ0
u= 1 in Γ1.
The problem is SUPG-stabilized [16] and discretized by FreeFEM [28]. It is important
to keep in mind that the proposed preconditioner is algebraic, thus there is no specific
transfer of information from the discretization kernel to solver backend. The domain
Ω is either the unit square or the unit cube meshed semi-structurally to account for
boundary layers, see an example of such a mesh in Figure 3a. The value of νis
Table 2
Iteration counts of the proposed preconditioner for solving the two- and three-dimensional
convection-diffusion problem from (4.1) with order kLagrange finite element space. The number of
subdomains is Nand the size of the discrete system is n. After each iteration count for each ν, the
size of second-level operator is typeset between parentheses.
Dimension k N n ν
1 101102103104
2 1 1,024 6.3·10623 (52,875) 20 (52,872) 19 (52,759) 20 (47,497) 21 (28,235)
3 2 4,096 8.1·10618 (1.8·105)14 (1.8·105)11 (1.6·105)16 (97,657) 29 (76,853)
Table 3
Iteration counts of GAMG for solving the two- and three-dimensional convection-diffusion
problem from (4.1).denotes either a failure to converge or a breakdown in GMRES.
Dimension nν
1 101102103104
2 6.3·10642 48 88 † †
3 8.1·10640 38 65 † †
constant in Ω. The value of κis given in Figure 3b. The value of the velocity field V
is either:
V(x, y) = x(1 x)(2y1)
y(1 y)(2x1)or V(x, y, z) =
2x(1 x)(2y1)z
y(1 y)(2x1)
z(1 z)(2x1)(2y1)
in 2D and 3D, respectively. These are standard values taken from the literature [43].
The definition of Γ0and Γ1may be inferred by looking at the two- and three-
dimensional solutions in Figures 3c to 3e and Figures 3f to 3h, respectively. The
iteration counts reported in Table 2 show that the proposed preconditioner handles
this problem, even as νtends to zero. In 2D, the operator, resp. grid, complexity is
of at most 1.008, resp. 1.43. In 3D, these figures are 1.02 and 1.7, respectively. For
comparison, GAMG and BoomerAMG iteration counts are also reported in Table 3
and Table 4, respectively.
5. Conclusion. We presented in this work a fully-algebraic two-level Schwarz
preconditioner for large-scale sparse matrices. The proposed preconditioner combines
a classic one-level Schwarz preconditioner with a spectral coarse space. The latter is
constructed efficiently by solving concurrently in each subdomain a local generalized
eigenvalue problem whose pencil matrices are obtained algebraically and cheaply
from the local coefficient matrix. Convergence results were obtained for diagonally
dominant HPD matrices. The proposed preconditioner was compared to state-of-the-
art multigrid preconditioners on a set of challenging matrices arising from a wide
range of applications including a convection-dominant convection-diffusion equation.
The numerical results demonstrated the effectiveness and robustness of the proposed
preconditioner especially for highly non-symmetric matrices.
Acknowledgments. This work was granted access to the GENCI-sponsored
HPC resources of TGCC@CEA under allocation A0110607519.
Fig. 2.(a) Mesh, (b) diffusivity coefficient, and solutions of some of the (c)–(e) two- and
(f)–(h) three-dimensional test cases from Table 2.
(a) Semi-structured mesh
(b) Diffusivity coefficient κ
(c) ν= 1
0 0.2 0.4 0.6 0.8 1
(d) ν= 102(e) ν= 104
(f) ν= 1 (g) ν= 102(h) ν= 104
Table 4
Iteration counts of BoomerAMG for solving the two- and three-dimensional convection-diffusion
problem from (4.1).denotes either a failure to converge or a breakdown in GMRES.
Dimension nν
1 101102103104
2 6.3·10650 49 19 7
3 8.1·10612 9 7 † †
Appendix A. Code reproducibility.
1#include < p et s c . h >
st a ti c c h ar h el p [] = " S ol v es a li n ea r s y st e m a fte r h a vi n g r e pa r t it i o ned it s
sy m m et r i c p ar t . \ n \ n ";
in t main( int ar gc ,char * * ar g s )
Ve c b ;
Ma t A , pe r m ;
KS P k sp ;
IS i s , ro w s ;
11 PetscBool fl g ;
PetscViewer viewer;
char name[ PETSC_MAX_PATH_LEN] ;
MatPartitioning mp a rt ;
PetscErrorCode ierr;
ierr = Pe t s cI n it i a li z e (& a rg c , & ar gs ,NU LL , h el p ) ; if (ierr)return ierr;
ierr = Ma t Cr e at e ( PETSC_COMM_WORLD ,& A ) ; CHKERRQ (ierr) ;
ierr = PetscOptionsGetString(N ULL ,N UL L ,"- m a t_ n a me " ,n am e , sizeof ( na m e ) ,& f l g );
CHKERRQ(ierr) ;
if ( ! f lg ) S E TE R RQ ( PETSC_COMM_WORLD ,PETSC_ERR_USER ," M is s in g - m a t_ n a me ") ;
21 ierr = PetscViewerBinaryOpen(P ET SC_ CO MM_ WO RL D ,n am e , FILE_MODE_READ ,& v i ew e r );
CHKERRQ(ierr) ;
ierr = Ma t Lo a d (A , v ie w er ) ; CHKERRQ(ierr ) ;
ierr = P et s cV i ew e rD e s tr o y (& v i ew e r ); CHKERRQ (ierr) ;
ierr = KS P Cr e at e ( PETSC_COMM_WORLD ,& ks p ) ; CHKERRQ(ierr );
ierr = MatPartitioningCreate(P ET SC _C OM M_WO RLD , & m pa r t ); CHKERRQ (ierr) ;
26 {
Ma t B , T ;
ierr = MatTranspose(A, MAT_INITIAL_MATRIX ,& T ) ; CHKERRQ(ierr ) ;
ierr = MatDuplicate(A, MAT_COPY_VALUES ,& B ) ; CHKERRQ(ierr ) ;
31 ierr = M at P a r ti t i on i n g Se t A d ja c e n cy ( mp ar t , B) ; CHKERRQ (ierr) ; / / pa r t it i o n A^T + A
ierr = M at P ar t i ti o n in g Se t F ro m O pt i on s ( m p ar t ) ; CHKERRQ(ierr) ;
ierr = M at P a r ti t i o ni n g A p pl y ( m p ar t ,& i s ) ; CHKERRQ(ierr) ;
ierr = Ma t D e st r o y ( & B) ; CHKERRQ (ierr) ;
ierr = Ma t D e st r o y ( & T) ; CHKERRQ (ierr) ;
36 }
ierr = M at P ar t i ti o ni n g De s tr o y ( & mp a rt ) ; CHKERRQ(ierr );
ierr = IS B u i l dT w o S id e d ( is , NU LL ,& r o ws ) ; CHKERRQ(ierr) ;
ierr = IS D e s tr o y ( & is ) ; CHKERRQ (ierr) ;
ierr = M at C r e at e S u bM a t r ix ( A , ro ws , ro w s , MAT_INITIAL_MATRIX , & p er m ) ; CHKERRQ(ierr) ;
41 ierr = IS D es t r oy ( & r ow s ) ; CHKERRQ(i er r ) ;
ierr = MatHeaderReplace(A,&perm);CHKERRQ(i e rr ) ; // o n ly ke e p th e p e rm u t ed mat r ix
ierr = K SP S e tF r o mO p t io n s ( k sp ) ; CHKERRQ(ierr ) ; // p a rs e c om ma n d - l i ne o p ti o ns
ierr = KS P S e t O pe r a t o r s ( k sp ,A , A ) ; CHKERRQ (ierr) ;
ierr = Ma t C r ea t e V ec s ( A , NULL , & b ) ; CHKERRQ(ierr);
46 ierr = VecSetRandom(b,NULL) ; CHKERRQ(ierr) ; / / r an d om r ig h t - h an d s id e
ierr = KS P S o lv e ( k sp , b , b ) ; CHKERRQ(ierr );
ierr = KS P D es t r oy (& k s p ); CHKERRQ (i er r );
ierr = Ve c D e st r o y ( & b) ; CHKERRQ (ierr) ;
ierr = Ma t D e st r o y ( & A) ; CHKERRQ (ierr) ;
51 ierr = Pe t s c Fi n a l iz e () ;
return ierr;
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Full-text available
We analyze the convergence of the one-level overlapping domain decomposition preconditioner SORAS (Symmetrized Optimized Restricted Additive Schwarz) applied to a generic linear system whose matrix is not necessarily symmetric/self-adjoint nor positive definite. By generalizing the theory for the Helmholtz equation developed in Graham et al. (SIAM J Numer Anal 58(5):2515–2543, 2020., we identify a list of assumptions and estimates that are sufficient to obtain an upper bound on the norm of the preconditioned matrix, and a lower bound on the distance of its field of values from the origin. We stress that our theory is general in the sense that it is not specific to one particular boundary value problem. Moreover, it does not rely on a coarse mesh whose elements are sufficiently small. As an illustration of this framework, we prove new estimates for overlapping domain decomposition methods with Robin-type transmission conditions for the heterogeneous reaction–convection–diffusion equation (to prove the stability assumption for this equation we consider the case of a coercive bilinear form, which is non-symmetric, though).
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Monolithic preconditioners for incompressible fluid flow problems can significantly improve the convergence speed compared to preconditioners based on incomplete block factorizations. However, the computational costs for the setup and the application of monolithic preconditioners are typically higher. In this paper, several techniques to further improve the convergence speed as well as the computing time are applied to monolithic two‐level Generalized Dryja–Smith–Widlund (GDSW) preconditioners. In particular, reduced dimension GDSW (RGDSW) coarse spaces, restricted and scaled versions of the first level, hybrid and parallel coupling of the levels, and recycling strategies are investigated. Using a combination of all these improvements, for a small time‐dependent Navier‐Stokes problem on 240 MPI ranks, a reduction of 86 % of the time‐to‐solution can be obtained. Even without applying recycling strategies, the time‐to‐solution can be reduced by more than 50 % for a larger steady Stokes problem on 4 608 MPI ranks. For the largest problems with 11 979 MPI ranks the scalability deteriorates drastically for the monolithic GDSW coarse space. On the other hand, using the reduced dimension coarse spaces, good scalability up to 11 979 MPI ranks, which corresponds to the largest problem configuration fitting on the employed supercomputer, could be achieved. This article is protected by copyright. All rights reserved.
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In this paper we present a class of robust and fully algebraic two-level preconditioners for SPD matrices. We introduce the notion of algebraic local SPSD splitting of an SPD matrix and we give a characterization of this splitting. This splitting leads to construct algebraically and locally a class of efficient coarse spaces which bound the spectral condition number of the preconditioned system by a number defined a priori. We also introduce the τ-filtering subspace. This concept helps compare the dimension minimality of coarse spaces. Some PDEs-dependant preconditioners correspond to a special case. The examples of the algebraic coarse spaces in this paper are not practical due to expensive construction. We propose a heuristic approximation that is not costly. Numerical experiments illustrate the efficiency of the proposed method.
Solving time-harmonic wave propagation problems in the frequency domain and within heterogeneous media brings many mathematical and computational challenges, especially in the high frequency regime. We will focus here on computational challenges and try to identify the best algorithm and numerical strategy for a few well-known benchmark cases arising in applications. The aim is to cover, through numerical experimentation and consideration of the best implementation strategies, the main two-level domain decomposition methods developed in recent years for the Helmholtz equation. The theory for these methods is either out of reach with standard mathematical tools or does not cover all cases of practical interest. More precisely, we will focus on the comparison of three coarse spaces that yield two-level methods: the grid coarse space, DtN coarse space, and GenEO coarse space. We will show that they display different pros and cons, and properties depending on the problem and particular numerical setting.
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