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A Class of Efficient Locally Constructed Preconditioners Based on Coarse Spaces

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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.
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A CLASS OF EFFICIENT LOCALLY CONSTRUCTED
PRECONDITIONERS BASED ON COARSE SPACES
HUSSAM AL DAASAND LAURA GRIGORI
Abstract. 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.
Key words. preconditioners, iterative linear solvers, domain decomposition, condition number,
coarse spaces
AMS subject classifications. 65F08, 65F10, 65N55
1. Introduction. The conjugate gradient method CG [8] is a widely known
Krylov iterative method, for solving large linear systems of equations of the form
(1.1) Ax =b,
where ARn×nis a symmetric positive definite (SPD) matrix, bRnis the right-
hand side, and xRnis the vector of unknowns. It finds at iteration jthe approxi-
mate solution xjx0+Kj(A, r0) that minimizes the A-norm of the error kxxjkA,
where x0is the initial guess, r0=bAx0,Kj(A, r0) is the Krylov subspace of dimen-
sion jrelated to Aand r0,xis the exact solution of (1.1), and k.kAis the A-norm.
The convergence of this method is well studied in the literature [16]. The rate of
convergence depends on the condition number of the matrix A. Let κ=λn
λ1be the
spectral condition number of A, where λnand λ1are the largest and the smallest
eigenvalues of Arespectively, the error at iteration jsatisfies the following inequality
(1.2) kxxjkA≤ kxx0kAκ1
κ+ 1j
,
We suppose that the graph of the matrix is partitioned into a number of subdo-
mains by using a k-way partitioning method [10]. To enhance the convergence, it is
common to solve the preconditioned system
(1.3) M1Ax =M1b.
Block Jacobi, additive Schwarz, restricted additive Schwarz, etc., are widely used
preconditioners. These preconditioners are called one-level preconditioners. They
Submitted to the editors June 14, 2018.
ALPINES, INRIA, Paris, France (hussam.al-daas@inria.fr,https://who.rocq.inria.fr/Hussam.
Al-Daas/index.html).
ALPINES, INRIA, Paris, France (laura.grigori@inria.fr,https://who.rocq.inria.fr/Laura.
Grigori/index.html).
1
2H. AL DAAS, L. GRIGORI
correspond to solving subproblems on subdomains. In [2,3] the authors prove that the
largest eigenvalue of the preconditioned system by the additive Schwarz preconditioner
is bounded by a number that is independent of the number of subdomains. However,
no control is guaranteed for the smallest eigenvalue of the preconditioned matrix.
Furthermore, when the number of subdomains increases, the smallest eigenvalue might
become even smaller. Thus, the number of iterations to reach convergence typically
increases. This occurs since this type of preconditioner employs only local information
and does not include global information. For this reason, these preconditioners are
usually combined with a second-level preconditioner, which corresponds to a coarse
space correction or deflation. In principle, it is meant to annihilate the impact of
the smallest eigenvalues of the operator. Different strategies exist in literature to
add this level. In [20], the authors compare different strategies of applying two-level
preconditioners. In [2,21,12,18,3,6,11], the authors propose different methods for
constructing a coarse space correction. Coarse spaces can be categorized in two types,
analytic and algebraic. Analytic coarse spaces depend on the underlying problem
from which the matrix Ais issued. Algebraic coarse spaces depend only on the
coefficient matrix Aand do not require information from the underlying problem
from which Aarises. Based on the underlying partial differential equation (PDE) and
its discretization, several methods that propose analytic coarse spaces are described
in literature [3,2,21,12,18].
In most cases, a generalized (or standard) eigenvalue problem is solved in each
subdomain. Every subdomain then contributes to the construction of the coarse space
by adding certain eigenvectors. These methods are efficient in several applications.
Nevertheless, the dependence on the analytic information makes it impossible to be
made in a pure algebraic way. Algebraic coarse space correction can be found in
literature [6,11]. However, the construction of the coarse space can be even more
costly than solving the linear system (1.1). In this paper we discuss a class of robust
preconditioners that are based on locally constructed coarse spaces. We characterize
the local eigenvalue problems that allow to construct an efficient coarse space related
to the additive Schwarz preconditioner. The paper is organized as follows. In section 2
we review general theory of one- and two-level preconditioners, in section 3 we present
our main result. We introduce the notion of algebraic local SPSD splitting of an SPD
matrix. For a simple case, given the block SPD matrix B=
B11 B12
B21 B22 B23
B32 B33
, the
local symmetric positive semi-definite (SPSD) splitting of Bwith respect to the first
block means finding two SPSD matrices B1,B2of the form B1=
B11 B12
B21
and B2=
B23
B32 B33
, where represents a non-zero block matrix such that
B=B1+B2. We characterize all possible local SPSD splittings. Then we introduce
the τ-filtering subspace. Given two SPSD matrices A, B, a τ-filtering subspace Z
makes the following inequality hold
(uP u)>B(uP u)u>Au, u,
where Pis an orthogonal projection on Z. Based on the local SPSD splitting and the
τ-filtering subspace, we propose in section 4 an efficient coarse space, which bounds
the spectral condition number by a given number defined a priori. Furthermore, we
CELCPCS 3
show how the coarse space can be chosen such that its dimension is minimal. The
resulting spectral condition number depends on three parameters. The first parameter
depends on the sparsity of the matrix, namely, the minimum number of colours kc
needed to colour subdomains such that two subdomains of the same colour are disjoint,
see Lemma 2.7 [2, Theorem 12]. The second parameter kmdepends on the algebraic
local SPSD splitting. It is bounded by the number of subdomains. For a special case
of splitting it can be chosen to be the maximal number of subdomains that share a
degree of freedom. The third parameter is chosen such that the spectral condition
number is bounded by the user-defined upper bound. In all stages of the construction
of this coarse space, no information is necessary but the coefficient matrix Aand
the desired bound on the spectral condition number. We show how the coarse space
constructed analytically by the method GenEO [17,3] corresponds to a special case
of our characterization. We also discuss the extreme cases of the algebraic local SPSD
splitting and the corresponding coarse spaces. We explain how these two choices are
expensive to construct in practice. Afterwards, we propose a practical strategy to
compute efficiently an approximation of the coarse space. In section 5 we present
numerical experiments to illustrate the theoretical and practical impact of our work.
At the end, we give our conclusion in section 6.
To facilitate the comparison with GenEO we follow the presentation in [3, Chap-
ter 7].
Notation. Let ARn×ndenote a symmetric positive definite matrix. We
use MATLAB notations. Let S1, S2⊂ {1, . . . , n }be two sets of indices. The
concatenation of S1and S2is represented by [S1, S2]. We note that the order of
the concatenation is important. A(S1,:) is the submatrix of Aformed by the rows
whose indices belong to S1.A(:, S1) is the submatrix of Aformed by the columns
whose indices belong to S1.A(S1, S2) := (A(S1,:)) (:, S2). The identity matrix of
size nis denoted In. We suppose that the graph of Ais partitioned into Nnon-
overlapping subdomains, where Nn. The coefficient matrix Ais represented
as (aij )1i,jn. Let N={1, . . . , n}and let Ni,0for i∈ {1, . . . , N }be the sub-
sets of Nsuch that Ni,0stands for the subset of the degrees of freedom, DOF, in
the subdomain i. We refer to Ni,0as the interior DOF in the subdomain i. Let
ifor i∈ {1, . . . , N }be the subset of Nthat represents the neighbors DOF of
the subdomain i, i.e., the DOFs of distance = 1 from the subdomain ithrough the
graph of A. We refer to ∆ias the overlapping DOF in the subdomain i. We denote
Ni= [Ni,0,i],i∈ {1, . . . , N }, the concatenation of the interior and the overlap-
ping DOF of the subdomain i. We denote Ci,i∈ {1, . . . , N }, the complementary of
Niin N, i.e., Ci=N \ Ni. We note ni,0the cardinality of the set Ni,0,δithe cardi-
nality of ∆iand nithe cardinality of the set Ni,i∈ {1, . . . , N }. Let Ri,0Rni,0×n
be defined as Ri,0=In(Ni,0,: ). Let Ri,δ Rδi×nbe defined as Ri,δ =In(∆i,: ).
Let RiRni×nbe defined as Ri=In(Ni,: ). Let Ri,c R(nni)×nbe defined as
Ri,c =In(Ci,: ). Let Pi=In([Ni,0,i,Ci],: ) Rn×n, be a permutation matrix asso-
ciated to the subdomain i, i∈ {1, . . . , N }. We denote DiRni,×ni, i = 1, . . . , N,
any non-negative diagonal matrix such that
(1.4) In=
N
X
i=1
R>
iDiRi.
We refer to (Di)1iNas the algebraic partition of unity. Let n0be a positive integer,
n0n. Let V0Rn×n0be a tall and skinny matrix of full rank. We denote Sthe
4H. AL DAAS, L. GRIGORI
subspace spanned by the columns of V0. This subspace will stand for the coarse space.
We denote R0the projection operator on S. We denote R>
0the interpolation operator
from Sto the global space. Let R1be the operator defined by:
R1:
N
Y
i=1
RniRn
(ui)1iN7→
N
X
i=1
R>
iui.
(1.5)
In the same way we define R2by taking into account the coarse space correction
R2:
N
Y
i=0
RniRn
(ui)0iN7→
N
X
i=0
R>
iui.
(1.6)
We note that the subscripts 1 and 2 in R1and R2refer to one-level and two-level inter-
polation operators respectively. The following example of two-subdomains-partitioned
Aillustrates our notation. Let Abe given as
A=
a11 a12
a21 a22 a23
a32 a33 a34
a43 a44
.
Then, N={1,2,3,4}. The sets of interior DOF of subdomains are N1,0={1,2},
N2,0={3,4}. The sets of overlapping DOF of subdomains are ∆1={3}, ∆2={2}.
The sets of concatenation of the interior DOF and the overlapping DOF of subdomains
are N1={1,2,3},N2={3,4,2}. The restriction operator on the interior DOF of
subdomains is
R1,0=1000
0100, R2,0=0010
0001.
The restriction operator on the overlapping DOF of subdomains is
R1=0010, R2=0100.
The restriction operator on the concatenation of the interior DOF and the overlapping
DOF is
R1=
1000
0100
0010
, R2=
0010
0001
0100
.
The permutation matrix associated with each subdomain is
P1=
1000
0100
0010
0001
,P2=
0010
0001
0100
1000
.
CELCPCS 5
The permuted matrix associated with each subdomain is
P1AP>
1=
a11 a12
a21 a22 a23
a32 a33 a34
a43 a44
,P2AP>
2=
a33 a34 a32
a43 a44
a23 a22 a21
a12 a11
.
Finally, the algebraic partition of unity can be defined as
D1=
1 0 0
01
20
0 0 1
2
, D2=
1
20 0
0 1 0
0 0 1
2
.
We note that the reordering of lines in the partition of unity matrices (Di)1iNhas
to be adapted with the lines reordering of (Ri)1iNsuch that (1.4) holds.
2. Background. In this section, we start by presenting three lemmas that help
compare two symmetric positive definite (or semidefinite) matrices. Then, we review
generalities of one- and two-level additive Schwarz preconditioners.
2.1. Auxiliary lemmas. The Lemma 2.1 can be found in [3, Lemma 7.3, p. 164].
This lemma helps prove the effect of the additive Schwarz preconditioner on the largest
eigenvalues of the preconditioned operator.
Lemma 2.1. Let A1, A2Rn×nbe two symmetric positive definite matrices.
Suppose that there is a constant cu>0such that,
(2.1) v>A1vcuv>A2v, vRn.
Then the eigenvalues of A1
2A1are strictly positive and bounded from above by cu.
The Lemma 2.2 is widely known in the community of domain decomposition by
the Fictitious subspace lemma. We announce it following an analog presentation as
in [3, Lemma 7.4 p.164].
Lemma 2.2 (Fictitious subspace lemma). Let ARnA×nA, B RnB×nBbe two
symmetric positive definite matrices. Let Rbe an operator defined as
R:RnBRnA
v7→ Rv,
(2.2)
and let R>be its transpose. Suppose that the following conditions hold:
1. The operator Ris surjective.
2. There exists cu>0such that
(2.3) (Rv)>A(Rv)cuv>Bv, vRnB.
3. There exists cl>0such that vnARnA,vnBRnB|vnA=RvnBand
(2.4) clv>
nBBvnB(RvnB)>A(RvnB) = v>
nAAvnA.
Then, the spectrum of the operator RB1R>Ais contained in the segment [cl, cu].
Proof. We refer the reader to [3, Lemma 7.4 p.164] or [14,13,4] for a detailed
proof.
6H. AL DAAS, L. GRIGORI
We note that there is a general version of Lemma 2.2 for infinite dimensions. This
lemma plays a crucial role in bounding the condition number of our preconditioned
operator. The operator Rwill stand for the interpolation operator. The matrix
Bwill stand for the block diagonal operator of local subdomain problems. It is
important to note that in the finite dimension the existence of the constants cuand
clare guaranteed. This is not the case in the infinite dimension spaces. In the finite
dimension case, the hard part in the fictitious subspace lemma is to find Rsuch that
cu/clis independent of the number of subdomains. When Rand Bare chosen to form
the one- or two-level additive Schwarz operator, the first two conditions are satisfied
for an upper bound cuindependent of the number of subdomains. An algebraic
proof which depends only on the coefficient matrix can be found in [3]. However, the
third condition is still an open question if no information from the underlying PDE
is used. In this paper we address the problem of defining algebraically a surjective
interpolation operator of the two-level additive Schwarz operator such that the third
condition holds for a clindependent of the number of subdomains. This is related to
the stable decomposition property which was introduced in [9]. Later, in [3], the authors
proposed a stable decomposition with the additive Schwarz. This decomposition was
based on the underlying PDE. Thus, when only the coefficient matrix Ais known,
this decomposition is not possible to be computed.
The two following lemmas will be applied to choose the local vectors that con-
tribute to the coarse space. They are based on low rank corrections. In [3], the
authors present two lemmas [3, Lemma 7.6 p.167, Lemma 7.7 p.168] similar to the
following lemmas. The rank correction proposed in their version is not of minimal
rank. We modify these two lemmas to obtain the smallest rank correction.
Lemma 2.3. Let A, B Rm×mbe two symmetric positive semi-definite matrices.
Let ker(A),range(A)denote the nul l space and the range of Arespectively. Let
ker(B)denote the kernel of B. Let L=ker(A)ker(B), we note Lker (A)the
orthogonal complementary of Lin ker(A). Let P0be an orthogonal projection on
range(A). Let τbe a strictly positive real number. Consider the generalized eigenvalue
problem,
P0BP0uk=λkAuk,
ukrange(A),
λkR.
(2.5)
Let Pτbe an orthogonal projection on the subspace
Z=Lker(A)span {uk|λk> τ},
then, the following inequality holds:
(2.6) (uPτu)>B(uPτu)τu>Au, uRm.
Furthermore, Zis the subspace of smallest dimension such that (2.6) holds.
Proof. Let mA=dim(range(A)). Let
λ1. . . λmττ < λmτ+1 . . . λmA
be the eigenvalues of the generalized eigenvalue problem (2.5). Let
u1, . . . , umτ, umτ+1, . . . , umA
CELCPCS 7
be the corresponding eigenvectors, A-orthonormalized. Let kB=dim(ker(B)
ker(A)), kA=dim(ker(A)) = mmA. Let v1, . . . , vkBbe an orthogonal basis
of Land let vkB+1, . . . , vkAbe an orthogonal basis of Lker(A)such that v1, . . . , vkA
is an orthogonal basis of ker(A). The symmetry of Aand Bpermits to have
u>
iAuj=δij ,1i, j mA,
u>
iBuj=λiδij ,1i, j mA,
v>
ivj=δij ,1i, j kA,
L= span {v1, . . . , vkB},
Lker(A)= span {vkB+1, . . . , vkA},
where δij stands for the Kronecker symbol. For a vector uRmwe can write:
P0u=
mA
X
k=1
(u>
kAP0u)uk.
Then, we have
Pτu=uP0u
kB
X
k=1
(v>
ku)vk+
mA
X
k=mτ+1
(u>
kAP0u)uk.
Thus,
uPτu=
kB
X
k=1
(v>
ku)vk+
mτ
X
k=1
(u>
kAP0u)uk.
Hence, the left side of (2.6) can be written as:
(uPτu)>B(uPτu) = kB
X
k=1
(v>
ku)vk+
mτ
X
k=1
(u>
kAP0u)uk!>
B kB
X
k=1
(v>
ku)vk+
mτ
X
k=1
(u>
kAP0u)uk!,
= kB
X
k=1
(v>
ku)vk+
mτ
X
k=1
(u>
kAP0u)uk!> mτ
X
k=1
λk(u>
kAP0u)Auk!,
= kB
X
k=1
(v>
ku)Avk+
mτ
X
k=1
(u>
kAP0u)Auk!> mτ
X
k=1
λk(u>
kAP0u)uk!,
= mτ
X
k=1
(u>
kAP0u)Auk!> mτ
X
k=1
λk(u>
kAP0u)uk!,
=
X
k|λkτ
(u>
kAP0u)uk
>
X
k|λkτ
λk(u>
kAP0u)Auk
,
=
X
k|λkτX
j|λjτ
(u>
kAP0u)u>
kλj(u>
jAP0u)Auj
,
=X
k|λkτ
(u>
kAP0u)2λk.
8H. AL DAAS, L. GRIGORI
We obtain (2.6) by remarking that.
X
k|λkτ
(u>
kAP0u)2λkτ
mA
X
k=1
(u>
kAP0u)2,
=τ
mA
X
k=1
(u>
kAP0u)(u>
kAP0u),
=τ(P0u)>AP0u,
=τu>Au.
There remains the minimality of the dimension of Z. First, remark that
u>Bu > τ u>Au, uZ.
To prove the minimality, suppose that there is a subspace Z1of dimension less than
the dimension of Z. By this assumption, there is a non-zero vector w(ZZ1)Z,
where (ZZ1)Zis the orthogonal complementary of (ZZ1) in Z, such that
wZ1. By construction, we have
w>Bw > τ w>Aw.
This contradicts (2.6) and the minimality is proved.
Lemma 2.4. Let ARm×mbe a symmetric positive matrix and BRm×m
be an SPD matrix. Let ker(A), range(A)denote the null space and the range of A
respectively. Let P0be an orthogonal projection on range(A). Let τbe a strictly
positive real number. Consider the following generalized eigenvalue problem,
(2.7) Auk=λkBuk.
Let Pτbe an orthogonal projection on the subspace
Z=span uk|λk<1
τ,
then, the following inequality holds:
(2.8) (uPτu)>B(uPτu)τu>Au uRm.
Zis the subspace of smallest dimension such that (2.8) holds.
Proof. Let u1, . . . , um0be an orthogonal basis vectors of ker(A). Let
0< λm0+1 . . . λmτ<1
τλmτ+1 . . . λm
be the eigenvalues strictly larger than 0 of the generalized eigenvalue problem (2.7).
Let
um0+1, . . . , umτ, umτ+1 , . . . , um
be the corresponding eigenvectors A-orthonormalized. We can suppose that
u>
iAuj=δij , m0+ 1 i, j m,
u>
iBuj=1
λi
δij , m0+ 1 i, j m,
u>
iuj=δij ,1i, j m0,
CELCPCS 9
where δij stands for the Kronecker symbol. We can write
P0u=
m
X
k=m0+1
(u>
kAP0u)uk.
Then, we have
Pτu=uP0u+
mτ
X
k=m0+1
(u>
kAP0u)uk.
Thus,
uPτu=
m
X
k=mτ+1
(u>
kAP0u)uk,
=X
k|λk1
τ
(u>
kAP0u)uk.
Hence, the left side of (2.8) can be written
(uPτu)>B(uPτu) =
X
k|λk1
τ
(u>
kAP0u)uk
>
B
X
k|λk1
τ
(u>
kAP0u)uk
,
=
X
k|λk1
τ
(u>
kAP0u)uk
>
X
k|λk1
τ
1
λk
(u>
kAP0u)Auk
,
=
X
k|λk1
τX
j|λj1
τ
(u>
kAP0u)u>
k1
λj
(u>
jAP0u)Auj
,
=X
k|λk1
τ
(u>
kAP0u)21
λk
.
We obtain (2.8) by remarking that.
X
k|λk1
τ
(u>
kAP0u)21
λkτ
m
X
k=1
(u>
kAP0u)2,
=τ
m
X
k=m0+1
(u>
kAP0u)(u>
kAP0u),
=τ(P0u)>AP0u,
=τu>Au.
There remains the minimality of Z. First, remark that
u>Bu > τ u>Au, uZ.
To prove the minimality, suppose that there is a subspace Z1of dimension less than
the dimension of Z. By this assumption, there is a non-zero vector w(ZZ1)Z,
10 H. AL DAAS, L. GRIGORI
where (ZZ1)Zis the orthogonal complementary of (ZZ1) in Z, such that
wZ1. By construction, we have
w>Bw > τ w>Aw.
This contradicts the relation (2.8).
The previous lemmas are general and algebraic and not directly related to the pre-
conditioning. In the following section we will review the one- and two-level additive
Schwarz preconditioner.
2.2. One- and two-level additive Schwarz preconditioner. In this section
we review the definition and general properties of one- and two-level additive Schwarz
preconditioners, ASM, ASM2respectively. We review, without proving, several lem-
mas introduced in [2,3]. These lemmas show how the elements of ASM2without any
specific property of the coarse space Sverify the conditions 1 and 2 of the fictitious
subspace Lemma 2.2.
The two-level preconditioner ASM2with coarse space Sis defined as
(2.9) M1
ASM,2=
N
X
i=0
R>
iRiAR>
i1Ri.
If n0= 0, i.e., the subspace Sis trivial, the term
R>
0R0AR>
01R0= 0
by convention.
The following lemma gives the additive Schwarz method a matrix representation as
in [3].
Lemma 2.5. The additive Schwarz operator can be represented as:
(2.10) M1
ASM,2=R2B1R>
2,
where R>
2is the operator adjoint of R2and Bis a block diagonal operator defined as
the following
B:
N
Y
i=0
Rni
N
Y
i=0
Rni
(ui)0iN7→ RiAR>
iui0iN
(2.11)
where RiAR>
ifor 0iNis the ith diagonal block.
Proof. The proof follows directly from the definition of Band R2.
We note that the dimension of the matrix representation of Bis larger than the
dimension of A. More precisely, Bhas the following dimension
nB=
N
X
i=0
ni=n+n0+
N
X
i=1
δi.
The one-level additive Schwarz preconditioner can be defined in the same manner. It
corresponds to the case where the subspace Sis trivial. The following Lemma 2.6,
[3, Lemma 7.10, p. 173] states that the operator R2is surjective without any specific
assumption about the coarse space S.
CELCPCS 11
Lemma 2.6. The operator R2as defined in (1.6) is surjective.
Proof. The proof follows from the definition of R2(1.6) and the definition of the
partition of unity (1.4).
Lemma 2.6 shows that the interpolation operator R2seen as a matrix verifies the
condition 1 in Lemma 2.2. The following Lemma 2.7 guarantees that the matrix
representation of the additive Schwarz verifies condition 2 in Lemma 2.2.
Lemma 2.7. Let kcbe the minimum number of distinct colours so that span{R>
i}1iN
of the same colour are mutually A-orthogonal. Then, we have
(2.12)
(R2uB)>A(R2uB)(kc+ 1)
N
X
i=0
u>
iRiAR>
iui,uB= (ui)0iN
N
Y
i=0
Rni.
Proof. We refer the reader to [2, Theorem 12 p.93] for a detailed proof.
We note that Lemma 2.7 is true for any coarse space S, especially when this sub-
space is trivial. This makes the lemma applicable also for the one-level additive
Schwarz preconditioner (the constant on the right-hand side in Lemma 2.7 becomes
kc). Lemma 2.8 is the first step to obtain a reasonable constant clthat verifies the
third condition in Lemma 2.2
Lemma 2.8. Let uARnAand uB= (ui)0iNQN
i=0 Rnisuch that uA=
R2uB. The additive Schwarz operator without any other restriction on the coarse
space Sverifies the following inequality
(2.13)
N
X
i=0
u>
iRiAR>
iui2u>
AAuA+ (2kc+ 1)
N
X
i=1
u>
iRiAR>
iui,
where kcis defined in Lemma 2.7.
Proof. We refer the reader to [3, Lemma 7.12, p. 175] to view the proof in detail.
In order to apply the fictitious subspace Lemma 2.2, the term PN
i=1 u>
iRiAR>
iuiin
the right-hand side of (2.13) must be bounded by a factor of u>
AAuA. For this aim, the
next section presents an algebraic local decomposition of the matrix A. Combining
this decomposition with the Lemma 2.3 or Lemma 2.4 (depending on the definiteness)
defines a class of local generalized eigenvalue problems. By solving them, we can define
a coarse space S. The additive Schwarz preconditioner combined with Ssatisfy the
three conditions of the fictitious subspace Lemma 2.2. Hence, we can control the
condition number of the preconditioned system.
3. Algebraic local SPSD splitting of an SPD matrix. In this section we
present our main contribution. We introduce the algebraic local SPSD splitting of
an SPD matrix related to a subdomain. Then, we characterize all the algebraic local
SPSD splittings of Athat are related to each subdomain. We give a non-trivial bound
from below for the energy norm of a vector by a locally determined quantity.
We start by defining the algebraic local SPSD splitting of a matrix related to a
subdomain.
Definition 3.1 (Algebraic local SPSD splitting of Arelated to a subdomain).
12 H. AL DAAS, L. GRIGORI
Following the previous notations, let ˜
Aibe the matrix defined as
(3.1) Pi˜
AiP>
i=
Ri,0AR>
i,0Ri,0AR>
i,δ
Ri,δAR>
i,0˜
Ai
δ
0
,
where ˜
Ai
δRδi×δi. We say that ˜
Aiis an algebraic local SPSD splitting of Arelated
to the subdomain iif the following condition holds
(3.2) 0 u>˜
Aiuu>Au, uRn.
For i∈ {1, . . . , N}, the matrix PiAP>
ihas the form of a block tridiagonal matrix
(the permutation matrix Piis defined in the section Notation). The first diagonal
block corresponds to the interior DOF of the subdomain i, the second diagonal block
corresponds to the overlapping DOF in the subdomain i, and the third block diagonal
is associated to the rest of the DOF.
Lemma 3.2. Let m1, m2, m3be strictly positive integers and m=m1+m2+m3,
let BRm×mbe a 3×3block tridiagonal SPD matrix
(3.3) B=
B11 B12
B21 B22 B23
B32 B33
,
where Bii Rmi×mifor i∈ {1,2,3}. Let ˜
B1Rm×mbe
(3.4) ˜
B1=
B11 B12
B21 ˜
B22
0
,
where ˜
B22 Rm2×m2is a symmetric matrix verifying the following inequalities
(3.5) u>B21B1
11 B12uu>˜
B22uu>B22 B23 B1
33 B32u, uRm2,
then, the following inequality holds
(3.6) 0 u>˜
B1uu>Bu, uRm.
Proof. Consider the difference matrix F=B˜
B1. Let F2R(m2+m3)×(m2+m3)
be the lowest 2 ×2 sub-block diagonal matrix of F, i.e.,
F2=B22 ˜
B22 B23
B32 B33.
F2admits the following decomposition,
(3.7) F2=I B23B1
33
IB22 ˜
B22 B23B1
33 B32
B33I
B1
33 B32 I.
Since ˜
B22 satisfies, by assumption, the inequality (3.5),F2satisfies the following
inequality
0u>F2uuR(m2+m3),
CELCPCS 13
This proves the right inequality in (3.6).
Let ER(m1+m2)×(m1+m2)be the upper 2 ×2 sub-block diagonal of ˜
B1.E
admits the following decomposition,
(3.8) E=I
B21B1
11 IB11 ˜
B22 B21B1
11 B12I B1
11 B12
I.
The positivity of ˜
B1follows directly from (3.5).
Lemma 3.3. Using the notations from Lemma 3.2, the following holds
The condition (3.5) in Lemma 3.2 is not trivial, i.e., the set of matrices ˜
B1
that verify the condition (3.5) is not empty
There exist matrices, ˜
B22, that verify the condition (3.5) with strict inequal-
ities
The left inequality in condition (3.5) is optimal, i.e., if there exists a non-zero
vector u2Rm2that verifies
u>
2B21B1
11 B12u2> u>
2˜
B22u2.
Then, there exists a non-zero vector uRmsuch that
u>˜
B1u < 0
The right inequality in condition (3.5) is optimal, i.e., if there exists a non-
zero vector u2Rm2that verifies
u>
2˜
B22u2> u>
2B22 B23B1
33 B32u2.
Then, there exists a non-zero vector uRmsuch that
u>˜
B1u > u>Bu
Proof. First we prove the non-triviality of the set of matrices verifying (3.5).
Indeed, let S(B22) be the Schur complement of B22 in B, namely
S(B22) = B22 B21 B1
11 B12 B23B1
33 B32.
Set ˜
B22 := 1
2S(B22) + B21 B1
11 B12. Then we have,
˜
B22 B21B1
11 B12 =B22 B23B1
33 B32˜
B22 =1
2S(B22),
which is an SPD matrix. Hence, the strict inequalities in (3.5) follow.
Let u2Rm2be a vector such that
u>
2B21B1
11 B12u2> u>
2˜
B22u2,
The block-LDLT factorization (3.8) shows that
u>˜
B1u=u>
2˜
B22 B21B1
11 B12u2<0,
where uis defined as
u=I B1
11 B12
I10
u2.
In the same manner we verify the optimality mentioned in the last point.
14 H. AL DAAS, L. GRIGORI
Remark 3.4. We note that the matrix B11 B12
B21 ˜
B22defines a seminorm in Rm1+m2.
Furthermore, if ˜
B22 is set such that the left inequality in (3.5) is strict, then the semi-
norm becomes a norm.
Now, we can apply Lemma 3.2 on PiAP>
ifor each subdomain iby considering
its interior DOF, overlapping DOF, and the rest of the DOF.
Proposition 3.5. For each subdomain i∈ {1, . . . , N }, let ˜
AiRn×nbe defined
as
(3.9) Pi˜
AiP>
i=
Ri,0AR>
i,0Ri,0AR>
i,δ
Ri,δAR>
i,0˜
Ai
δ
0
,
where ˜
Ai
δRδi×δisatisfies the following conditions
uRδi,
u>Ri,δAR>
i,0Ri,0AR>
i,01Ri,0AR>
i,δuu>˜
Ai
δu
u>˜
Ai
δuu>Ri,δAR>
i,δRi,δAR>
i,cRi,c AR>
i,c1Ri,c AR>
i,δu.
Then, i∈ {1, . . . , N }the matrix ˜
Aiis an algebraic local SPSD splitting of Arelated
to the subdomain i. Moreover, the following inequality holds,
(3.10) 0
N
X
i=1
u>˜
Aiukmu>Au uRn,
where kmis a number bounded by N.
Proof. Lemma 3.2 shows that ˜
Aiis an algebraic local SPSD splitting of Arelated
to the subdomain i. The inequality (3.10) holds with the constant Nfor all algebraic
local SPSD splittings of A. Thus, depending on the SPSD splitting related to each
subdomain there exists a number kmNsuch that the inequality holds.
We note that the matrix ˜
Aiis considered local since it has non-zero elements only
in the overlapping subdomain i. More precisely,
j, k ∈ N | j /∈ Nik /∈ Ni,˜
Ai(j, k)=0.
Proposition 3.5 shows that the A-norm of a vector vRncan be bounded from below
by a sum of local seminorms, Remark 3.4.
4. Algebraic stable decomposition with R2.In the previous section we
introduced the algebraic local SPSD splitting of A. In this section we present the
τ-filtering subspace that is associated with each SPSD splitting. In each subdomain a
τ-filtering subspace will contribute to the coarse space. We show how this leads to a
class of stable decomposition with R2. We note that the previous results of section 2
hold for any coarse space S. Those results are sufficient to determine the constant cu
in the second condition of the fictitious subspace lemma, Lemma 2.2. However, they
do not allow to control the constant clof the third condition of the same lemma.
As we will see, the GenEO coarse space [17,3] corresponds to a special SPSD
splitting of A. Therefore, we follow the presentation in [3] in the construction of the
coarse space. We note that the proof of Theorem 4.4 is similar to the proof of [3,
Theorem 7.17, p.177]. We present it for the sake of completeness.
CELCPCS 15
Definition 4.1. Let ˜
Aibe an algebraic local SPSD splitting of Arelated to the
subdomain i, for i= 1, . . . , N. Let τ > 0. Let ˜
ZiRnibe a subspace and let ˜
Pibe
an orthogonal projection on ˜
Zi. We say that ˜
Ziis a τ-filtering subspace if
u>
iRiAR>
iuiτ(Riu)>Ri˜
AiR>
i(Riu),uRn,
where ui=DiIni˜
PiRiuand Diis the partition of unity, for i= 1, . . . , N.
After the characterization of the local SPSD splitting of Arelated to each subdomain,
we characterize the associated smallest τ-filtering subspace.
Lemma 4.2. Let ˜
Aibe an algebraic local SPSD splitting of Arelated to the sub-
domain i, for i= 1, . . . , N . Let τ > 0. For all subdomains 1iN, let
˜
Gi=DiRiAR>
iDi,
where Diis the partition of unity. Let ˜
P0,i be the projection on range(Ri˜
AiR>
i)
parallel to ker(Ri˜
AiR>
i). Let K=ker(Ri˜
AiR>
i),L=ker(˜
Gi)K, and LKthe
orthogonal complementary of Lin K.
If ˜
Giis indefinite, consider the following generalized eigenvalue problem
Find (ui,k, λi,k )range(Ri˜
AiR>
i)×R
such that ˜
P0,i ˜
Gi˜
P0,iui,k =λi,k Ri˜
AiR>
iui,k.
Set
(4.1) ˜
Zτ,i =LKspan {ui,k |λi,k > τ}.
If ˜
Giis definite, consider the following generalized eigenvalue problem
Find (ui,k, λi,k )Rni×R
such that Ri˜
AiR>
iui,k =λi,k ˜
Giui,k.
Set
(4.2) ˜
Zτ,i =span ui,k |λi,k <1
τ.
Then, ˜
Zτ,i is the smallest dimension τ-filtering subspace and the following inequality
holds
u>
iRiAR>
iuiτ(Riu)>Ri˜
AiR>
i(Riu),
where ui=DiIni˜
Pτ,i Riu, and ˜
Pτ,i is the orhtogonal projection on ˜
Zτ,i .
Proof. Direct application of Lemma 2.3 and Lemma 2.4.
We will refer to the smallest dimension τ-filtering subspace as ˜
Zτ,i and to the projec-
tion on it as ˜
Pτ,i . Note that for each algebraic local SPSD splitting of Arelated to
a subdomain i, the τ-filtering subspace ˜
Zτ,i defined in Definition 4.1 changes. Thus,
the projection ˜
Pτ,i depends on the algebraic local SPSD splitting of Arelated to the
subdomain i.
In the rest of the paper, the notations ˜
Zτ,i and ˜
Pτ,i will be used according to the
algebraic local SPSD splitting of Athat we deal with and following Lemma 4.2.
16 H. AL DAAS, L. GRIGORI
Definition 4.1 leads us to bound the sum in (2.13) by a sum of scalar products
associated to algebraic SPSD splittings of A. Therefore, a factor, which depends on
the value of τ, of the scalar product associated to Awill bound the inequality in
(2.13).
Definition 4.3 (Coarse space based on algebraic local SPSD splitting of A,
ALS). Let ˜
Aibe an algebraic local SPSD splitting of Arelated to the subdomain
i, for i= 1, . . . , N . Let ˜
Zτ,i be the subspace associated to ˜
Aias defined in Lemma 4.2.
We define Sthe coarse space based on the algebraic local splitting of Arelated to each
subdomain, as the sum of expanded weighted τ-filtering subspaces associated to the
algebraic local splitting of Arelated to each subdomain,
(4.3) S=
N
M
i=1
R>
iDi˜
Zτ,i .
Let ˜
Z0be a matrix whose columns form a basis of S. We denote its transpose by
R0=˜
Z>
0.
As mentioned previously, the key point to apply the fictitious subspace lemma, Lemma 2.2,
is to find a coarse space that induces a relatively large clin the third condition of the
lemma. The following theorem proves that ALS satisfies this.
Theorem 4.4. Let ˜
Aibe an algebraic local SPSD splitting of Arelated to the
subdomain i, for i= 1, . . . , N. Let ˜
Zτ,i be the τ-filtering subspace associated to ˜
Ai,
and ˜
Pτ,i be the projection on ˜
Zτ,i as defined in Lemma 4.2. Let uRnand let
ui=DiIni˜
Pτ,i Riufor i= 1, . . . , N . Let u0be defined as,
u0=R0R>
01R0 N
X
i=1
R>
iDi˜
Pτ,i Riu!.
Let cl= (2 + (2kc+ 1)kmτ)1. Then,
u=
N
X
i=0
R>
iui,
and
cl
N
X
i=0
u>
iRiAR>
iuiu>Au.
Proof. Since y∈ S, y =R>
0R0R>
01R0y, the relation
u=
N
X
i=0
R>
iui=R2(ui)0iN,
follows directly. Lemma 2.8 shows that
N
X
i=0
u>
iRiAR>
iui2u>Au + (2kc+ 1)
N
X
i=1
u>
iRiAR>
iui.
CELCPCS 17
By using Lemma 4.2 we can write
N
X
i=0
u>
iRiAR>
iui2u>Au + (2kc+ 1)τ
N
X
i=1
(Riu)>Ri˜
AiR>
i(Riu).
Since ˜
Aiis local, we can write
N
X
i=0
u>
iRiAR>
iui2u>Au + (2kc+ 1)τ
N
X
i=1
u>˜
Aiu.
Then, by applying Proposition 3.5, we can write
N
X
i=0
u>
iRiAR>
iui2u>Au + (2kc+ 1)kmτ u>Au,
N
X
i=0
u>
iRiAR>
iui(2 + (2kc+ 1)kmτ)u>Au.
Theorem 4.5. Let MALS be the two-level ASM preconditioner combined with
ALS. The following inequality holds,
κM1
ALS A(kc+ 1) (2 + (2kc+ 1)kmτ)
Proof. Lemma 2.6,Lemma 2.7, and Theorem 4.4 show that the two-level precon-
ditioner associated with ALS verifies the conditions of the fictitious subspace lemma,
Lemma 2.2. Hence, the eigenvalues of M1
ALS Averify the following inequality,
1
2 + (2kc+ 1)kmτλM1
ALS A(kc+ 1),
and the result follows.
Remark 4.6. Since any τ-filtering subspace ˜
Zican replace ˜
Zτ,i in Theorem 4.4, the
Theorem 4.5 applies for coarse spaces of the form S=LN
i=1 R>
iDi˜
Zi. The difference
is that the dimension of the coarse space is minimal by choosing ˜
Zτ,i , see Lemma 4.2.
We note that the previous theorem, Theorem 4.5, shows that the spectral condition
number of the preconditioned system does not depend on the number of subdomains.
It depends only on kc, km,and τ.kcis bounded by the maximum number of neighbors
of a subdomain. kmis a number bounded by the number of subdomains. It depends
on the algebraic local SPSD splitting of each subdomain. Partitioned graphs of sparse
matrices have structures such that kcis small. The parameter τcan be chosen small
enough such that ALS has a relatively small dimension.
4.1. GenEO coarse space. In [3], the authors present the theory of one- and
two-level additive Schwarz preconditioners. To bound the largest eigenvalue of the
preconditioned system they use the algebraic properties of the additive Schwarz pre-
conditioner. However, to bound the smallest eigenvalue, they benefit from the dis-
cretization of the underlying PDE. In the environment of the finite element method,
they construct local matrices corresponding to the integral of the operator in the
overlapping subdomain. For each subdomain, the expanded matrix has the form
Pi˜
AiP>
i=
Ri,0AR>
i,0Ri,0AR>
i,δ
Ri,δAR>
i,0˜
Ai
δ
0
,
18 H. AL DAAS, L. GRIGORI
where ˜
Ai
δcorresponds to the integral of the operator in the overlapping region with
neighbors of the subdomains i. This matrix is SPSD since the global operator is
SPD. Since the integral over the subdomain is always smaller than the integral over
the global domain (positive integrals), the following inequality holds
0u>˜
Aiuu>Au, uRn.
Hence, Lemma 3.3 confirms that the matrix ˜
Aicorresponds to an algebraic local
SPSD splitting of Arelated to the subdomain i. Thus, GenEO is a member of the
class of preconditioners that are based on the algebraic local SPSD splitting of A. We
note that the parameter km, defined in (3.10), with the algebraic local SPSD splitting
of Acorresponding to GenEO can be shown to be equal to the maximum number of
subdomains sharing a DOF.
4.2. Extremum efficient coarse space. In this section we discuss the two
obvious choices to have algebraic local SPSD splitting of A. We show how in practice
these two choices are costly. However, they have two advantages. The first is that
one of these choices gives an answer to the following question that appears in domain
decomposition. How many local vectors must be added to the coarse space in order
to bound the spectral condition number by a number defined a priori? We are able to
answer this question in the case where the additive Schwarz preconditioner is to be
used. We note that the answer is given without any analytic information. Only the
coefficients of the matrix Ahave to be known. The second advantage is that both
choices give an idea of constructing a non-costly algebraic approximation of an ALS.
In the following discussion we disregard the impact of the parameter km. Numer-
ical experiments in section 5 demonstrate that the impact of this parameter can be
negligible. We note that this parameter depends only on the algebraic local SPSD
splitting and it is bounded by N.
Suppose that we have two SPSD splittings of Arelated to a subdomain i,˜
A(1)
i,˜
A(2)
i,
such that:
u>˜
A(1)
iuu>˜
A(2)
iu, uRn.
We want to compare the number of vectors that contribute to the coarse space for
each SPSD splitting. It is clear that a τ-filtering subspace associated to ˜
A(1)
iis a
τ-filtering subspace associated to ˜
A(2)
i. Thus, the following inequality holds,
dim(˜
Z(1)
τ,i )dim(˜
Z(1)
τ,i ),
where ˜
Z(1)
τ,i ,˜
Z(2)
τ,i are the smallest τ-filtering subspaces associated to ˜
A(1)
i,˜
A(2)
i, respec-
tively. Therefore, Lemma 3.3 shows that closer we are to the upper bound in (3.5) less
vectors will contribute to ALS. Moreover, closer we are to the lower bound in (3.5)
more vectors will contribute to ALS. Indeed, the set of algebraic local SPSD splitting
of Arelated to a subdomain iadmits a relation of partial ordering.
M1M2u>M1uu>M2u, u.
This set admits obviously a smallest and a largest element defined by the left and the
right bounds in (3.5), respectively.
Hence, the best ALS corresponds to the following algebraic local SPSD splitting
CELCPCS 19
of A, for i= 1, . . . , N ,
(4.4)
Pi˜
AiP>
i=
Ri,0AR>
i,0Ri,0AR>
i,δ
Ri,δAR>
i,0Ri,δAR>
i,δ Ri,δAR>
i,cRi,c AR>
i,c1Ri,c AR>
i,δ0
.
The dimension of the subspace ˜
Zτ,i associated to ˜
Ai(4.4) is minimal over all possible
algebraic local SPSD splittings of Arelated to the subdomain i. We remarke that this
splitting is not a choice in practice since it includes inverting the matrix Ri,cAR>
i,c
which is of large size (approximately corresponding to N1 subdomains). We will
refer to (4.4) as the upper bound SPSD splitting, the associated coarse space will be
referred to as the upper ALS.
In the same manner, we can find the worst ALS. The corresponding algebraic local
SPSD splitting of Arelated to the subdomain iis the following
(4.5) Pi˜
AiP>
i=
Ri,0AR>
i,0Ri,0AR>
i,δ
Ri,δAR>
i,0Ri,δAR>
i,0Ri,0AR>
i,01Ri,0AR>
i,δ0
.
On the contrary of the best splitting (4.4), this splitting is not costly. It includes
inverting the matrix Ri,0AR>
i,0which is considered small. However, the dimension
of ˜
Zτ,i associated to ˜
Ai(4.5) is maximal. It is of dimension δiat least. Indeed, a
block-LDLT factorization of Ri˜
AiR>
ishows that its null space is of dimension δi. We
will refer to (4.5) as the lower bound SPSD splitting the associated coarse space will
be referred to as the lower ALS.
Remark 4.7. A convex linear combination of the lower bound and the upper
bound of the SPSD splitting is also an algebraic local SPSD splitting.
α×the upper bound SPSD splitting + (1 α)×the lower bound SPSD splitting
We refer to it as α-convex SPSD splitting, We refer to the corresponding ALS as the
α-convex ALS.
In the following section we propose a strategy to compute an approximation of rea-
sonable ALS that is not costly.
4.3. Approximate ALS. As mentioned in subsection 4.2, the extremum cases
of ALS are not practical choices. We recall that the restriction matrix Ri,c is a
associated to the DOFs outside the overlapping subdomain i. The bottleneck in
computing the upper bound SPSD splitting is the computatation of the term
Ri,δAR>
i,cRi,c AR>
i,c1Ri,c AR>
i,δ
since it induces inverting the matrix Ri,cAR>
i,c. To approximate the last term, we
look for a restriction matrix Ri,˜csuch that
Ri,δAR>
i,cRi,c AR>
i,c1Ri,c AR>
i,δRi,δAR>
i,˜cRi, ˜cAR>
i,˜c1Ri,˜cAR>
i,δ,
Ri,δAR>
i,˜cRi, ˜cAR>
i,˜c1Ri,˜cAR>
i,δis easy to compute.
One choice is to associate Ri,˜cto the DOFs outside the overlapping subdomain i
that have the nearest distance from the boundary of the subdomain ithrough the
20 H. AL DAAS, L. GRIGORI
graph of A. In practice, we fix an integer d1 such that the matrix Ri,˜cAR>
i,˜chas a
dimension dimid×ni. Then we can take a convex linear combination of the lower
bound SPSD splitting and the approximation of the upper bound SPSD splitting. For
instance, the error bound on this approximation is still an open question. Numerical
experiments show that ddoes not need to be large.
5. Numerical experiments. In this section we present numerical experiments
for ALS. We denote ASMALS the two-level additive Schwarz combined with ALS.
If it is not specified, the number of vectors deflated by subdomain is fixed to 15.
We use the preconditioned CG implemented in MATLAB 2017R to compare the
preconditioners. The threshold of convergence is fixed to 106. Our test matrices
arise from the discretization of two types of challenging problems: linear elasticity
and diffusion problems [5,1,15]. Our set of matrices are given in Table 5.1. The
matrices SKY2D and SKY3D arise from the boundary value problem of the diffusion
equation on Ω, the (2-D) unit square and the (3-D) unit cube, respectively:
div(κ(x)u) = fin Ω,
u= 0 on ΓD,
∂u
∂n = 0 on ΓN.
(5.1)
They correspond to skyscraper problems. The domain Ω contains several zones of
high permeability. These zones are separated from each other. The tensor κis given
by the following relation:
κ(x) = 103([10x2] + 1) if [10xi] is odd, i = 1,2,
κ(x) = 1 otherwise.
ΓD= [0,1]×{0,1}in the (2-D) case. ΓD= [0,1]×{0,1}×[0,1] in the (3-D) case. ΓN
is chosen as ΓN=\ΓDand ndenotes the exterior normal vector to the boundary
of Ω. The linear elasticity problem with Dirichlet and Neumann boundary conditions
is defined as follows
div(σ(u)) + f= 0 in Ω,
u= 0 on ΓD,
σ(u)·n= 0 on ΓN,
(5.2)
Ω is a unit cube (3-D). The matrix El3D corresponds to this equation discretized
using a triangular mesh with 65 ×9×9 vertices. ΓDis the Dirichlet boundary, ΓN
is the Neumann boundary, fis a force, uis the unknown displacement field. The
Cauchy stress tensor σ(.) is given by Hooke’s law: it can be expressed in terms of
Young’s modulus Eand Poisson’s ration ν.ndenotes the exterior normal vector to
the boundary of Ω. We consider discontinuous Eand ν: (E1, ν1) = (2 ×1011 ,0.45),
(E2, ν2) = (107,0.25). Data elements of this problem are obtained by the application
FreeFem++ [7]. Table 5.2 presents a comparison between one-level ASM and ASM2
with the upper bound ALS. As it is known, the iteration number of CG preconditioned
by ASM increases by increasing the number of subdomains. However, we remark that
the iteration number of the CG preconditioned by ALS is robust when the number of
subdomain increases.
In Table 5.3 we compare three ALS, the upper bound, α1-convex, and α2-convex,
where α1= 0.75 and α2= 0.25. Table 5.3 shows the efficiency of three ALS related
CELCPCS 21
Matrix name Type n NnZ κ
SKY3D Skyscraper 8000 53000 105
SKY2D Skyscraper 10000 49600 106
EL3D Elasticity 15795 510181 3 ×1011
Table 5.1
Matrices used for tests. nis the size of the matrix, N nZ is the number of non-zero elements.
HPD stands for Hermitian Positive Definite. κis the condition number related to the second norm.
Matrix n N nuC nASM
4 23 29
8 25 35
SKY3D 8000 16 25 37
32 22 55
64 24 79
128 24 -
4 18 54
8 19 -
SKY2D 10000 16 20 -
32 22 -
64 26 -
128 31 -
4 38 -
8 43 -
EL3D 15795 16 51 -
32 51 -
64 67 -
128 92 -
Table 5.2
Comparison between ASM2with the upper ALS and one-level additive Schwarz, nis the di-
mension of the problem, Nis the number of subdomains, nuC is the iteration number of CG pre-
conditioned by ASM2, and nAS M is the iteration number of CG preconditioned by one-level ASM .
The sign means that the method did not converge in fewer than 100 iteration.
to different SPSD splittings. The iteration count corresponding to each coarse space
increases slightly by increasing the number of subdomains. The main reason behind
this increasing, is that the predifined parameter τprovides an overestimation of the
upper bound on the spectral condition number; see Table 5.5.
To illustrate the impact of the parameter km, when increasing the number of
subdomains, on bounding the spectral condition number, we do the following. We
choose τas
τ=1
2(˜κ
kc+ 1 2)(2kc+ 1)1,
i.e., we suppose that kmhas no impact on τ. The resulting spectral condition number
will be affected only by the parameter km; see Table 5.5.Table 5.4 and Table 5.5
present results for ALS variants for ˜κ= 100. We perform this test on the elasticity
problem (5.2) where we could also compare against the GenEO coarse space [17,3].
We note that when GenEO is applied on the elasticity problem (5.2), the domain
decomposition performed by freefem++ [7], for all tested values of N, is such that
any DOF belongs to at most two subdomains and hence km(GenEO) = 2. This
22 H. AL DAAS, L. GRIGORI
Matrix n N nuC nα1nα2
4 23 22 22
8 25 25 23
SKY3D 8000 16 25 24 24
32 22 22 22
64 24 23 21
128 24 24 22
4 18 18 17
8 19 19 19
SKY2D 10000 16 20 19 19
32 22 21 18
64 26 24 20
128 31 28 20
4 38 38 38
8 43 43 43
EL3D 15795 16 51 51 51
32 51 51 51
64 67 67 67
128 92 92 92
Table 5.3
Comparison between ALS variants, the upper bound ALS, the α1-convex ALS, and the α2-
convex CosBALSS, nis the dimension of the problem, Nis the number of subdomains, the subscript
uC refers to the upper bound ALS, n.is the iteration number of ASM2,αrefers to the coefficient
in the convex linear combination, α1= 0.75 and α2= 0.25.
NdimuC nuC dimα1nα1dimα2nα2dimGen nGen
4 82 20 92 19 120 18 106 20
8 179 23 209 20 240 20 229 24
16 304 37 394 30 480 28 391 38
32 447 53 583 45 960 36 614 42
64 622 84 769 73 1920 51 850 55
128 969 131 1096 112 3834 77 1326 61
Table 5.4
Matrix El3D, ALS variants and GenEo coarse space with the minimum number of deflated
vectors disregarding the parameter km,Nis the number of subdomains, the subscript uC refers to
the upper bound ALS. dim.is the dimension of ALS, n.is the iteration number of ASM2,αrefers
to the coefficient in the convex ALS, α1= 0.75 and α2= 0.25, the subscript Gen stands for the
GenEO coarse space. See Table 5.5
means that the hyposthesis that kmhas no impact on the selected τis true for
the coarse space GenEO. Nevertheless, this might be false for the other coarse spaces.
Therefore, the impact of kmwill be remarked only on the ALS coarse spaces. Table 5.4
shows the dimension of ALS for each variant as well as the iteration number for
preconditioned CG to reach the convergence tolerance. On the other hand, Table 5.5
shows an estimation of the spectral condition number of the preconditioned system.
This estimation is performed by computing an approximation of the largest and the
smallest eigenvalues of the preconditioned operator by using the Krylov-Schur method
[19] in MATLAB. The same tolerance τis applied for GenEO. In order to avoid a
large-dimension coarse space, 30 vectors at max are deflated per subdomain.
We note that results in Table 5.4 satisfy the discussion in subsection 4.2. Indeed,
CELCPCS 23
NκuC κα1κα2κGen
4 5 4 4 5
8 8 5 5 7
16 15 10 9 15
32 34 25 15 18
64 100 67 30 31
128 231 178 86 39
Table 5.5
Estimation of the spectral condition number of matrix El3D preconditioned by ASM2with ALS
variants and GenEo coarse space, results correspond to Table 5.4,Nis the number of subdomains,
the subscript uC refers to the upper bound ALS, αrefers to the coefficient in the convex ALS,
α1= 0.75 and α2= 0.25, the subscript Gen stands for the GenEO coarse space.
Fig. 5.1.Histogram of the number of deflated vectors by each subdomain for different ALS,
GenEO; uC, the upper bound ALS; α1-convex ALS, α1= 0.75 ;α2-convex ALS, α2= 0.25
the upper bound ALS has the minimum dimension, 0.75- and 0.25-convex ALS follow
the upper bound ALS respectively.
Table 5.5 demonstrates the impact of kmon the bound of the spectral condition
number. We notice that its effect increases when αis closer to 1 (the larger αis,
the larger kmbecomes). We recall that in the algebraic SPSD splitting kmN.
However, when GenEO is applied to the elasticity problem test case (5.2),kmis inde-
pendant of Nand is equal to 2 as explained previously. The values of the estimated
spectral condition number, especially for small number of subdomains (N= 4), show
how τprovides an overestimation of the theoretical upper bound on the spectral con-
dition number, (estimated(κ)100). For this reason, we consider that this slight
augmentaion of the iteration count does not mean that the method is not robust.
In Figure 5.1 we present a histogram of the number of deflated vectors by each
subdomain. We remark that the number of vectors that each subdomain contributes
24 H. AL DAAS, L. GRIGORI
0 20 40 60 80 100 120
Subdomain number
0
2
4
6
8
10
12
14
16
Number ofdeflated vectors
El3d 128 subdomains
uC
GenEO
Fig. 5.2.Comparaison between the number of deflated vectors per subdomain GenEO coarse
space and the upper bound ALS
to the coarse space is not necessarily equal. In the case of α2-convex ALS, most sub-
domains reach the maximum number of deflated vectors, 30, that we fixed. Moreover,
Figure 5.2 compares the number of deflated vectors in each subdomain for the Ge-
nEO subspace and the upper bound ALS. This figure illustrates the relation of partial
ordering between the SPSD splitting as discussed in subsection 4.2.
In Table 5.6 we show the impact of the approximation strategy that we proposed
in subsection 4.3. The distance parameter related to the approximation, see sub-
section 4.3, is fixed for each matrix. It is obtained by tuning. The convex linear
combination is chosen as α= 0.01. Each subdomain contributes 20 vectors to the
coarse space. We remark that the approximation strategy gives interesting results
with the conviction-diffusion problem matrices SKY2D and SKY3D. With a small
factor of the local dimension d= 2 and d= 3, respectively, the approximate ALS is
able to perform relatively as efficient as the upper bound ALS. For the elasticity prob-
lem with a larger factor d= 5, the approximate ALS reduces the iteration number,
however, we remark that the latter increases by increasing the number of subdomains.
6. Conclusion. In this paper we reviewed generalities of one- and two-level
additive Schwarz preconditioner. We introduced the algebraic local SPSD splitting of
an SPD matrix A. We characterized all possible algebraic local SPSD splitting. To
study the minimality of the dimension of the coarse space, we introduced the τ-filtering
subspaces. Based on the algebraic local SPSD splitting and inspired by the GenEO
method [17,3], we introduced a class of algebraic coarse spaces that are constructed
locally, ALS. The characterization of algebraic local SPSD splitting of Aand the
associated τ-filtering subspaces makes an algebraic framework for studying the coarse
spaces related to the additive Schwarz method. We proved that the coarse space of
CELCPCS 25
Matrix n N nuC d nap
4 22 22
8 23 23
SKY3D 8000 16 24 2 22
32 22 22
64 24 22
128 22 44
4 17 17
8 18 18
SKY2D 10000 16 20 3 19
32 22 22
64 26 59
128 31 90
4 27 54
8 36 56
EL3D 15795 16 37 5 77
32 43 136
64 61 -
128 83 -
Table 5.6
Comparison between the upper bound ALS and the approximation strategy presented in subsec-
tion 4.3,nis the dimension of the problem, Nis the number of subdomains, nuC is the iteration
number of CG preconditioned by ASM2with the upper bound ALS, dstands for the factor of local
dimension to approximate the upper bound SPSD splitting, as explained in subsection 4.3, and nap is
the iteration number of CG preconditioned by ASM2with approximation of ALS, the convex linear
combination is chosen as (0.01 ×approximation of the upper bound + 0.99 ×lower bound). The sign
means that the method did not converge in fewer than 150 iteration.
GenEO corresponds to a special case of the SPSD splitting. We discussed different
types of ALS and suggested a simple method to approximate a valuable coarse space.
For matrices issued from the conviction-diffusion problem, the simple method that we
proposed gave very interesting results. The algebraic formulation presented in this
paper is particularly important when the theory of GenEO cannot be applied. We
also note that in our ongoing work, we develop a theoretical and practical framework
that will give rise to a three-level additive Schwarz preconditioner combining GenEO
and ALS.
7. Acknowledgement. The authors would like to thank the editor and the
anonymous referees for their useful remarks that helped us improve the clarity of the
paper.
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In this paper we introduce LORASC, a robust algebraic preconditioner for solving sparse linear systems of equations involving symmetric and positive definite matrices. The graph of the input matrix is partitioned by using k-way partitioning with vertex separators into N disjoint domains and a separator formed by the vertices connecting the N domains. The obtained permuted matrix has a block arrow structure. The preconditioner relies on the Cholesky factorization of the first N diagonal blocks and on approximating the Schur complement corresponding to the separator block. The approximation of the Schur complement involves the factorization of the last diagonal block and a low rank correction obtained by solving a generalized eigenvalue problem or a randomized algorithm. The preconditioner can be build and applied in parallel. Numerical results on a set of matrices arising from the discretization by the finite element method of linear elasticity models illustrate the robusteness and the efficiency of our preconditioner.
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We introduce spectral coarse spaces for the balanced domain decomposition and the finite element tearing and interconnecting methods. These coarse spaces are specifically designed for the two‐level methods to be scalable and robust with respect to the coefficients in the equation and the choice of the decomposition. We achieve this by solving generalized eigenvalue problems on the interfaces between subdomains to identify the modes that slow down convergence. Theoretical bounds for the condition numbers of the preconditioned operators, which depend only on a chosen threshold, and the maximal number of neighbors of a subdomain are presented and proved. For the finite element tearing and interconnecting method, there are two versions of the two‐level method: one based on the full Dirichlet preconditioner and the other on the, cheaper, lumped preconditioner. Some numerical tests confirm these results. Copyright © 2013 John Wiley & Sons, Ltd.
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This paper presents an overview of parallel algorithms and their implementations for solving large sparse linear systems which arise in scientific and engineering applications. Preconditioners constitute the most important ingredient in solving such systems. As will be seen, the most common preconditioners used for sparse linear systems adapt domain decomposition concepts to the more general framework of “distributed sparse linear systems”. Variants of Schwarz procedures and Schur complement techniques are discussed. We also report on our own experience in the parallel implementation of a fairly complex simulation of solid-liquid flows.
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The paper studies traces of piecewise linear prolongations of mesh functions given on triangulations of two- and three-dimensional piecewise smooth domains. In particular, it considers condensing triangulations. The way of condensing triangulations may be arbitrary but the widely used assumptions on the triangles of triangulations (the so-called regular triangulations) must be satisfied. Equivalent normalizations of the space of traces of mesh functions and problems of inverting these normalizations are also analysed.