We did a first step in solving, so-called, skin problem. We developed an efficient H-matrix preconditioner to solve diffusion problem with jumping coefficients

1. H-matrix based preconditioner for the skin problem
B.N.Khoromskij, A.Litvinenko
bokh, litvinen@mis.mpg.de
Max Planck Institute for Mathematics in the Sciences
Leipzig. 18/08/2006
Abstract
In this paper we propose and analyze the new H-Cholesky based preconditioner for the so-called
skin problem [5]. After a special reordering of indices and omitting the coupling, we obtain a
block diagonal matrix which is very suitable for the hierarchical Cholesky (H-Cholesky) factor-
ization. We perform the H-Cholesky factorization of this matrix and use it as a preconditioner
for the cg method. We will show that the new preconditioner requires less memory and com-
putational time than the standard H-Cholesky preconditioner, which is also very cheap and fast.
Key words: skin problem, H-matrix approximation, hierarchical Cholesky, jumping
coefficients, domain decomposition.
1 Introduction
In the series of papers [7], [9], [10] the authors successfully apply the iteration method (cg,
gmres, bicgstab) with H-matrices based preconditioners to different types of second order
elliptic differential problems. In this paper we continue the research in this direction.
Under some definite conditions H-matrices can be used even as a direct solver. There are
results (see, e.g., [11] and references therein) where authors apply additive Schwarz domain
decomposition preconditioners. It is known that for problems with jumping coefficients
(see (1)) the condition number cond(A) is proportional to h−d
sup
x,y∈Ω
α(x)
α(y)
, where α(x)
denotes the jumping coefficient, d the spatial dimension and h the grid step size. This is
why a good preconditioner W is needed so that cond(W−1
A) ≃ 1.
In this paper we consider a diffusion process (see (1)) through the domain as shown in
Fig. 1 (left). This figure shows cells and the lipid layer between them. In this problem
the Dirichlet boundary condition means the presence of some drugs on the boundary γ
of the skin fragment. The right-hand side presents external forces. The zero Neumann
condition on Γγ shows that there is no penetration through the surface Γγ. Typical for
the skin problem are the high jumping coefficients. The penetration coefficient inside the
cells is very low ∼ 10−5
− 10−3
, but it is large between cells.
The diffusion equation has the form:
div(α(x)∇u) = f x ∈ Ω
u = 0 x ∈ γ
∂u
∂n
= g x ∈ Γ γ
(1)
where Γ = ∂Ω, α(x, y) = ε ≪ 1 in cells and α(x, y) = β = 1 in between. The rest of this
1

2. ε
β
z
y
x
Figure 1: (left) A skin fragment consists of cells and of the lipid layer. The penetration
through the cells goes very slowly and very fast through the lipid layer. (right) The
simplified model of a skin fragment contains 8 cells with the lipid layer between them.
Ω = [−1, 1]3
, α(x, y) = ε inside cells and α(x, y) = β = 1 in the lipid layer.
paper is structured as follows. In Section 2 we describe the discretisation which is done
by FEM. We recall the main idea of the H-matrix technique in Section 3. Section 4 is
devoted to the new preconditioner and estimations of its complexity. Numerical tests and
comparisons of different preconditioners are provided in Section 5. Finally, some remarks
conclude the paper.
2 Discretisation (FEM)
Let us choose the triangulation τh which is compatible with the lipid layer, i.e., τh :=
τ1
h ∪ τ2
h , where τ1
h is a triangulation of the lipid layer and τ2
h a triangulation of cells. Let
bj, j = 1..n, be piecewise linear basis functions and
Vh ⊂ H1
(Ω), Vh := span{b1, ..., bn}. (2)
Then the variational formulation of the initial problem is
find uh ∈ Vn, so that a(uh, v) = c(v) for all v ∈ Vn. (3)
Assuming (2), we obtain the equivalent problem
Au = c, where Aij = a(bj, bi) and ci := c(bi), i, j = 1, .., n. (4)
Here
a(bj, bi) = α(∇bj, ∇bi)dx =
Ω
fbjdx +
Γγ
gbjdΓ =: cj. (5)
2

3. The lipid layer between the cells defines the natural decomposition of Ω. The width of
this layer is proportional to the grid step size h. Note that after the reordering of indices,
we can represent the global stiffness in the following form:
A11 εA12
εA21 εA22
. (6)
Here A11, A22 are the stiffness matrices which correspond to the lipid layer and to the
rest of domain accordingly. A12, A21 are coupling matrices. To simplify the model we will
consider Ω as in Fig. 1 (right).
3 Hierarchical Matrices
The hierarchical matrices (H-matrices) were introduced in 1998 by Hackbusch [2] and
since then, H-matrices have been applied in a wide range of applications. They provide a
format for the data-sparse representation of fully-populated matrices. Suppose there are
two matrices A ∈ Rn×k
and B ∈ Rm×k
, k ≪ min(n, m), so that ABT
= R ∈ Rn×m
. We
say then that R is the rank-k matrix. The main idea of H-matrices is to approximate
certain subblocks of a given matrix by rank-k matrices. The admissible partitioning
indicates which blocks can be approximated by rank-k matrices. The storage requirement
for matrices A and B is k(n + m) instead of n · m for matrix R. One of the biggest
advantages of H-matrices is that the complexity of the H-matrix addition, multiplication
and inversion is not bigger than Ckn logq
n, q = 1, 2 (see [2], [13]). The lack is that the
constant C is large. For example for 3D case it can be bigger than 120.
To build an H-matrix one needs an admissible block partitioning (see Fig. 2). To build
this partitioning one needs an admissibility condition and a block cluster tree. To build
the block cluster tree a cluster tree is necessary. The cluster tree requires grid data. For
more details see [2] or [13].
H-matrixverices
finite elements
cluster tree
block
cluster tree
admissibility
condition
admissible
partitioning
H-Cholesky
factorization
Figure 2: The schema of building an H-matrix and its H-Cholesky factorisation.
Definition 3.1 We define the set of H-matrices with the maximal rank k as follows
H(TI×J , k) := {M ∈ RI×J
| rank(M |t×s) ≤ k for all admissible leaves t × s of TI×J}.
3

4. Algorithm of the H-Cholesky factorization
Our aim is to compute the H-Cholesky factorization of the stiffness matrix which appears
after discretisation of the Laplace operator. Suppose that
A =
A11 A12
A21 A22
=
L11 0
L21 L22
U11 U12
0 U22
then the algorithm is as follows
1. compute L11 and U11 as H-Cholesky decomposition of A11.
2. compute U12 from L11U12 = A12 (use a recursive block forward substitution).
3. compute L21 from L21U11 = A21 (use a recursive block backward substitution).
4. compute L22 and U22 as H-Cholesky decomposition of L22U22 = A22 ⊖ L21 ⊙ U12.
All the steps are executed in the class of H-matrices.
4 New Preconditioner
The H-Cholesky factorization of the stiffness matrix produces H-matrix as shown in Fig.
3 (left). After reordering of the index set I(Ω) and omitting the coupling between cells and
the lipid layer we obtain H-matrix as shown in Fig. 3 (right). As a new preconditioner
we use the H-Cholesky decomposition of
A11 0
0 εA22
. (7)
Remark 4.1 Note that W−1
A := (LLT
)
−1
A = L−T
L−1
A = L−T
AL−1
, i.e., W−1
A is
positive definite and symmetric. Thus, for solving the initial problem (4) we apply the pcg
method with the H-Cholesky preconditioner.
Below we prove that omitting of the coupling for small ε is possible.
Lemma 4.1 For a symmetric and positive definite matrix A =
A11 A12
A21 A22
and any
vector v =
v1
v2
it is hold (A12v1, v2) ≤ A
1/2
11 v1 · A
1/2
22 v2 .
Proof: From Cauchy inequality for any vectors u, v it follows
uT
Av = (u, v) A ≤ u A · v A.
Construct two vectors u = (v1, 0)T
and v = (0, v2)T
, then uT
Av = (A12v2, v1). It means
that
(A12v2, v1) ≤ v1 A · v2 A = A
1/2
11 v1 · A
1/2
22 v2 .
4

5. Lemma 4.2 For a symmetric and positive definite matrix A =
A11 A12
A21 A22
and any
vector v =
v1
v2
it is hold
2(A12u2, u1) ≤ (A11u1, u1) + (A22u2, u2),
(A12v1, v2) ≤
1
2
A
1/2
11 v1 + A
1/2
22 v2 .
Proof: Let u1 := v1 and u2 = −v2 then u =
u1
−u2
. From the positive definiteness
of A it follows
0 ≤ (Au, u) = (A11u1, u1) − (A12u2, u1) − (A21u1, u2) + (A22u2, u2).
Move negative terms to the left, obtain
(A12u2, u1) + (A21u1, u2) ≤ (A11u1, u1) + (A22u2, u2).
Recall that A is symmetric, obtain 2(A12u2, u1) ≤ (A11u1, u1) + (A22u2, u2) and
2(A12u2, u1) ≤ (A
1/2
11 u1, A
1/2
11 u1) + (A
1/2
22 u2, A
1/2
22 u2),
(A12u2, u1) ≤
1
2
A
1/2
11 u1 + A
1/2
22 u2 .
Lemma 4.3 Let u be a vector and W =
A11 0
0 A22
be a preconditioner, then
(Au, u) ≤ 2(Wu, u). (8)
Proof: Compute both scalar products
(W2u, u) =
A11 0
0 εA22
u1
u2
,
u1
u2
= (A11u1, u1) + ε(A22u2, u2).
(Au, u) =
A11 εA12
εA21 εA22
u1
u2
,
u1
u2
= (A11u1, u1) + 2ε(A12u2, u1) + ε(A22u2, u2) = (Wu, u) + 2ε(A12u2, u1),
From the previous Lemma it follows that (Au, u) ≤ (Wu, u) + (Wu, u).
Remark 4.2 Recall that A and W are spectral equivalent if c1 · I ≤ W−1
A ≤ c2 · I,
∀u ∈ Rn
.
Lemma 4.4 Matrices A and W are spectral equivalent with I ≤ W−1
A ≤ 2cdotI.
Proof: We will write A ≥ B if A − B is semi-positive definite. From Lemma 4.3 follows
(Au, u) ≤ 2(Wu, u), u ∈ Rn
. Move everything in the left part, obtain ((A−2W)u, u) ≤ 0.
Since the last holds for ∀u than A − 2W ≤ 0 or W−1
A ≤ 2.
From the construction of W it is clear that A − W ≥ 0, i.e. W−1
A ≥ I.
Thus, I ≤ W−1
A ≤ 2 · I.
5

6. 32
15 32
8 8
8
24
15 24
8 8
8 15
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15 36
8
8 15
12 15
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24
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15 15
9 15
36
15 27
8 8
8
15
12
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12
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9
24
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8 8
8
24
15 24
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36
15 36
15 15
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9
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9
36
15 36
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9 15
27
15 27
8 8
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12
15
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9 1512 12
12
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24
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12
36
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5 5
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4 4
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20
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15 4
12 10
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10 12
7 14
32
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15
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12
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8
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12 18
15 30
27
9 18
15 30
15 15
10
30
15 20
Figure 3: H-Cholesky factorizations of the standard stiffness matrix (left) and the stiffness
matrix without coupling between the lipid layer and cells (right). The dark blocks ∈
R36×36
are dense matrices and the light blocks are low-rank matrices. The steps in the
grey blocks show the decay of the singular values in the logarithmic scale.
5 Numerical tests
Table 5 gives the theoretical estimations of the sequential and parallel complexities of the
H-Cholesky factorization of W1 and W2.
Preconditioner Comp. Complexity Parallel Complexity
W1 := H-Cholesky decomp. of
A11 A12
A21 A22
O(kn log2
n) O(kn log2
n)
W2 := H-Cholesky decomp. of
A11 0
0 A22
O(knI log2
nI) max{O(knI log2
nI),
+O(k(n − nI) log2
(n − nI)) O(kn0 log2
n0)}
Table 1: Complexities of the preconditioners W1 and W2. p is the number of processors,
nI is the number of degrees of freedom in the lipid layer, n0 := n−nI
p−1
.
Remark 5.1 The sparsity constant Csp is an important H-matrix feature and is present
in all H-matrix complexity estimates. This constant depends on the size of the H-matrix.
Since the new preconditioner is simplier the sparsity constant is also smaller. In the frame
of the numerical experiments for Table 5 Csp(W1) = 64 and Csp(W1) = 26. For the model
6

7. 0.01
0.1
1
10
100
1000
10000
0 50 100 150 200 250
alpha=1
alpha=1e-2
alpha=1e-4
"alpha_1"
"alpha_1e-2"
"alpha_1e-4"
Figure 4: Decay of singular values of A for ε = 1, ε = 10−2
and ε = 10−4
.
domain with larger number of cells the difference between sparsity constants will be more
significant.
Table 5 shows the resources requirements for the preconditioners W1 and W2. We see
that W2 requires less resources than W1. It requires less memory (S(W1) > S(W2)) and
time (t(W1) > t(W2)) for the building. Columns 2 and 5 contain the times for computing
the Cholesky factorisations and cg iterations. In Table 5 we compare the solutions ˜u and
k t(W1),sec S(W1),MB iter(W1) t(W2),sec S(W2),MB iter(W2)
1 24 + 10.6 2 ∗ 102
69 8.7+10 102
99
2 70 + 11.3 3.8 ∗ 102
46 21.6+13.3 1.8 ∗ 102
91
4 208 + 12.5 7.5 ∗ 102
17 68+13.5 3.5 ∗ 102
60
6 483.7 + 82 1.1 ∗ 103
11 123+26 5.1 ∗ 102
74
Table 2: Comparison of the preconditioners W1 and W2. 403
dofs, Ax − b = 10−8
,
α = 10−5
.
ucg, obtained with the preconditioners W1 and W2. The solution ucg, obtained with the
preconditioner W1 is considered as ’exact’.
6 Conclusion
The matrix W2 can be successfully used as a preconditioner. The simple structure of
W2 is the reason why it is good parallelisable. The parallel computational complexity is
7

8. k |ucg−˜u|
|˜u|
|ucg − ˜u|∞
1 5.3 ∗ 10−10
4.5 ∗ 10−6
2 5.1 ∗ 10−9
3.5 ∗ 10−8
4 5.8 ∗ 10−10
4.6 ∗ 10−6
6 7.2 ∗ 10−10
2.5 ∗ 10−5
Table 3: Comparison of the solutions ucg and ˜u. 403
dofs, Ax − b = 10−8
, α = 10−5
.
max{O(nI log2
nI), O(nD log2
nD)}, nD := n−nI
p−1
, nI number of degrees of freedom in the
lipid layer. The sequential version of the preconditioner W2 requires less memory. Note
that the more cells domain Ω contains, the bigger the advantages in storage and compu-
tational resources will be (see Table 5). The disadvantage is the relative large number of
pcg iterations, but these iterations require less resources than the standard H-Cholesky
preconditioner W1. In frames of HLIB (see [1]) it is quite easy to implement the offered
preconditioner.
Acknowledgment: The authors wish to thank Prof. Dr. Hackbusch for his correc-
tions as well as Dr. B¨orm and Dr. Grasedyck for HLIB.
8

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9