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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.
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.
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.
(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
(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 recent years, it has been shown that many modern iterative algorithms (multigrid schemes, multilevel preconditioners, domain decomposition methods etc.) for solving problems resulting from the discretization of PDEs can be interpreted as additive (Jacobi-like) or multiplicative (Gauss-Seidel-like) subspace correction methods. The key to their analysis is the study of certain metric properties of the underlying splitting of the discretization space into a sum of subspaces and the splitting of the variational problem on into auxiliary problems on these subspaces. In this paper, we propose a modification of the abstract convergence theory of the additive and multiplicative Schwarz methods, that makes the relation to traditional iteration methods more explicit. The analysis of the additive and multiplicative Schwarz iterations can be carried out in almost the same spirit as in the traditional block-matrix situation, making convergence proofs of multilevel and domain decomposition methods clearer, or, at least, more classical. In addition, we present a new bound for the convergence rate of the appropriately scaled multiplicative Schwarz method directly in terms of the condition number of the corresponding additive Schwarz operator. These results may be viewed as an appendix to the recent surveys [X], [Ys].
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