Application of Hierarchical Matrices For Solving Multiscale

Problems

Thesis Summary

In this dissertation we consider an elliptic boundary value problem

Lu = f in Ω,u = g on ∂Ω,

(1)

whose coefficients may contain a non-smooth parameter, e.g.,

L = −2

∑

i,j=1

∂

∂jαij

∂

∂i(2)

with αij = αji(x) ∈ L∞(Ω) such that the matrix function A(x) = (αij)i,j=1,2 satisfies 0 < λ ≤λmin(A(x)) ≤ λmax(A(x)) ≤ λ for all x ∈ Ω ⊂ R

2. This setting allows oscillatory coefficients aswell as jumping ones. Such problem has relevance to several important problems such as flowin porous media, mechanical properties of composite materials etc.The motivation and goals

Suppose the solution on the boundary ∂Ω (Fig. (a)) is given. One is interested in the fast

Ω ΩΩ

a) b) c)

u|∂Ωγ

ω

u|∂ω

u|γ = Φ(u|∂Ω, f)

numerical approach which produces the solution u|γ on the interface γ. The solution u|γ dependson the right-hand side and u|∂Ω, i.e. u|γ = Φ(u|∂Ω, f). In another problem setup (Fig. (b)) thesolution in a small subdomain ω ⊂ Ω is of interest. To solve the problem in ω the boundaryvalues on ∂ω are required. How to produce them efficiently from the global boundary data∂Ω and the given right-hand side? The third possible problem setup is as follows. For solvingthe initial problem in parallel the solution on the interface (see Fig. (b)) is required. How tocompute this solution effectivelly?

Numerical methods for solving the discrete counterpart of problem (1) are iterative methods(multigrid, PCG, GMRES), direct methods (Gauss elimination, method of H-matrices, directdomain decomposition) and a combination of the previous methods (H-LU factorisation aspreconditioning for CG). Iterative methods produce approximations un converging to the exactsolution u, but do not compute the matrix A−1. Direct methods (Gauss, etc) compute A−1, buthave large complexity O(n3).The method of H-matrices was developed by Hackbusch and others in 1999. A hierarchicalmatrix M ∈ R

n×m (notation H-matrix) is a matrix which consists mostly of low-rank subblockswith maximal rank k, where k ≪ minn,m. Such matrices require only O(kn log n) (w.l.o.g.

1

n ≥ m) units of memory. The complexity of all arithmetic operations with H-matrices isO(kβn logβ n), where β = 1, 2, 3. The accuracy of the approximation depends on the rankk. Recently Bebendorf and Hackbusch proved the existence of H-matrix approximants to theinverse FE-matrix of elliptic operators with L∞ coefficients in (2).

Domain decomposition methods have other advantages. A large domain can be decom-posed into subdomains and a direct method is applied simultaneously to each subdomain (Hack-busch/Khoromskij/Kriemann).

In this work we present a new method (HDD method), which combines the advantages ofthe H-matrix technique and the advantages of the hierarchical domain decomposition. The ideaof this method belongs to Hackbusch (Multiscale Workshop, MPI, Leipzig 2002).For numerical tests we apply the HDD method to multiscale problems. In order to betterunderstand the HDD method, we now list its properties.

1. The complexities of the one-scale version and two-scale version of HDD are O(k2nh log3 nh)and O(k2√nhnH log3 √nhnH) respectively.

2. The storage requirements of the one-scale version and two-scale version of HDD areO(knh log2 nh) and O(k

√nhnH log2 √nhnH).

3. HDD allows one to compute two discrete solution operators Fh and Gh such that:

uh := Fhfh + Ghgh, (3)

where uh(fh, gh) ∈ Vh is the FE solution of (1), fh the FE right-hand side, and gh the FEDirichlet boundary data.

4. The fast H-matrix technique is applied for the approximation Fh and Gh.

5. HDD allows one to compute functionals of the solution. The examples are:

(a) Neumann data ∂uh

∂nat the boundary,

(b) mean values∫

ωuhdx, ω ⊂ Ω, the solution at a point, the solution in a small subdomain

ω,

(c) flux∫

C∇u−→n dx, where C is a curve in Ω,

6. HDD also allows one to compute uh(fh, gh) for fh in a smaller space VH ⊂ Vh.

7. It provides the possibility of finding uh restricted to a coarser grid with reduced compu-tational resources.

8. HDD shows big advantages in complexity for problems with multiple right-hand sides andmultiple Dirichlet data.

9. HDD is easy parallelizable.

10. If a problem contains repeated patterns then the computational resources can be reduceddrastically.

2

..... .....

..... .....

Root to

Leaves

u=g, on Γ

..... .....

..... .....

Leaves to R

oot

Ψ11 Ψ12

Ψ1Ψ2

Φ11Φ12

Φ1 u|γ

u|γ1u|γ2

Figure 1: HDD contains two main algorithms “Leaves to Root” which computes auxiliary map-pings and “Root to Leaves” which computes the solution.

The L∞ coefficients in (2) allow multiscale coefficients a(x), a(x

ε) etc. We start the dissertation

with the description of the multiscale problems. In Chapter 2, we recall classical theories (vari-ational formulation, discretisation, FEM, error estimations etc). Chapter 3 contains the mainresult of this work - the hierarchical domain decomposition method (HDD) and three modifi-cations. We build the hierarchical domain decomposition tree (see Fig. 1), give the definitionsof the boundary-to-boundary (Ψg), domain-to-boundary (Ψf ), boundary-to-interface (Φg) anddomain-to-interface (Φf ) mappings which are essential for the definition of the HDD method.We then present two main algorithms “Leaves to Root” and “Root to Leaves”. We introducethe H-matrix technique in Chapter 4. We explain in particular how to convert one H-matrix toanother H-matrix with a different block structure. The HDD method from Chapter 3 requiredexact matrix computations. In Chapter 5 we replace the exact matrix arithmetic by the approx-imate H-matrix arithmetic. There is an explanation of how to build H-matrix approximationsof Ψg

ω and Ψfω for the domain ω from H-matrix approximations of Ψg

ωiand Ψf

ωi, where ωi ⊂ ω.

In Chapter 6 we prove that the complexities of the algorithms ”Leaves to Root” and ”Root toLeaves” for the one-scale version are

O(k2nh log3 nh) and O(knh log2 nh)

respectively and the storage requirements are O(knh log2 nh) (see Table 1). We show that thecomplexities of the same algorithms for the two-scale version are

O(k2√nhnH log3 √nhnH) and O(k√

nhnH log2 √nhnH),

where nh, nH are the respective numbers of degrees of freedom of the fine grid and the coarsegrid. For problems with the homogeneous right-hand side the complexity (and the storage) isO(k2√nhnH).For a machine with q := 2r processors and a global memory which can be accessed simultaneouslyby all processors the parallel complexity of the algorithm “Leaves to Root” is estimated by

N(q) ≤ 3r

2r· C ′k2√nh log2 √nh + Ck2nh

q+ C ′′(1 − 3r

4r)√

nhn2min

3

+C1k

2nh log2 nh log q

q+ C2k

2 nh

qlog3 nh

q,

where C,C ′, C ′′, C1, C2, nmin ∈ R+. The parallel complexity of the algorithm “Root to Leaves”on a machine with q processors is estimated by

Ck2nh log nh log q

q+

C ′k2nh log2 nh

q, C,C ′ ∈ R+.

In Chapter 8 we show the implementation of the triangulation, the grid hierarchy and the HDDmethod. We describe the connection between the data structures ’vertex’, ’triangle’, ’edge’,’subdomain’, ’boundary’, ’hierarchical decomposition tree’.We test HDD on a model problem with strong oscillatory and jumping coefficients. The HDDdid not show dependence of the approximation error on the frequency (see Table 2). We applyHDD to the so-called skin problem which has highly jumping coefficients. Three modificationsof the HDD were implemented. HDD1 works with a right-hand side from VH ⊂ Vh, HDD2

computes the solution on the coarse scale with a step size H and HDD3 is a combination ofHDD1 and HDD2. There are examples of computing mean values in different subdomains ofΩ. All numerical results confirm the theoretical estimates of the discretisation error and theH-matrix approximation error, as obtained in Chapters 2 and 6.

dofs Storage(Φg), Kb Storage(Φf ), Kb ‖u − uk‖2/ ‖u‖

2‖u − uk‖∞

332 2.45 ∗ 102 4 ∗ 102 3.3 ∗ 10−5 8.47 ∗ 10−5

652 1.1 ∗ 103 2.4 ∗ 103 5.75 ∗ 10−5 1.0 ∗ 10−4

1292 5 ∗ 103 1.4 ∗ 104 7.4 ∗ 10−5 1.1 ∗ 10−4

2572 2.1 ∗ 104 7.86 ∗ 104 8.3 ∗ 10−5 1.3 ∗ 10−4

Table 1: An almost linear dependence of the memory requirements for Φg and Φf on the numbersof degrees of freedom, rank k = 7. u is the ’exact’ solution, uk is the approximate solution.

ν ‖u− uk‖2/ ‖u‖

2‖u− uk‖∞

10 1.65 ∗ 10−4 1.76 ∗ 10−5

50 1.8 ∗ 10−4 1.9 ∗ 10−5

Table 2: Dependence of the relative and the absolute errors on the frequency ν, 2572 dofs, f = 1,α(x, y) = 1 + 1

2sin(νx)sin(νy).

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