![4
Example of multiscale problems
(a)macroscopic scale (b)microscopic scale
Different scales in a porous medium.[Bastian 99].
10 s
-6
10 s
-3
10 s
0
10 s
3
10 m
-12
10 m
-9
10 m
-6
10 m
-3
Atom Protein Cell Tissue
molecular events
(ion channel gating)
diffusion cell
signalling
mitosis
Example of time and length scales for modeling tumor growth.[Alarcon,
Byrne, Maini 05]](https://image.slidesharecdn.com/talk082006-170124082121/85/Application-H-matrices-for-solving-PDEs-with-multi-scale-coefficients-jumping-and-strongly-oscillatory-coefficients-4-320.jpg)
![6
Multiscale methods
The equation is :
− (a(x) u) = f in Ω,
u = 0 on ∂Ω.
(1)
Homegenisation [Babuska 75], [Bensoussan, Lions, Papanicolau 78],
[Jikov, Kozlov, Oleinik 94]
Solution is
uε
(x) = u0(x) + εu1(x,
x
ε
) + O(ε2
).
u0 is the solution of the homogenized equation
a∗
u0 = f in Ω, u0 = 0 on ∂Ω, (2)
Resonance effect in MsFEM [T.Hou, X. Wu 97]
u − uh
0,Ω = O(h2
+ ε/h). (3)
Heterogeneous multiscale method [Weinan E, B.Engquist 03]](https://image.slidesharecdn.com/talk082006-170124082121/85/Application-H-matrices-for-solving-PDEs-with-multi-scale-coefficients-jumping-and-strongly-oscillatory-coefficients-6-320.jpg)
![27
(left) Skin problem, (right) model of a cell.
a b
c
Lipid layer
α
β
0 10.25 0.75
0.5
1
4h
[Khoromskij, Wittum 02]](https://image.slidesharecdn.com/talk082006-170124082121/85/Application-H-matrices-for-solving-PDEs-with-multi-scale-coefficients-jumping-and-strongly-oscillatory-coefficients-27-320.jpg)
Application H-matrices for solving PDEs with multi-scale coefficients, jumping and strongly oscillatory coefficients
- 1. 1 Application of H-matrices for solving multiscale problems Litvinenko Alexander, Dissertation work Max-Planck-Institut f¨ur Mathematik in den Naturwissenschaften, Leipzig, 10 August, 2006. www.hlib.org www.mis.mpg.de
- 2. 2 H-matrices Integral Equations, BEM3D Parallel Impl. of H-matrices Helmholz Equation Convection- Diffusion Problems Multigrid+ H-matrices H-Matrix Approximation of sign(A), exp(A), etc Aposteriory Err. Est.+ efficient H-matrix update Lyapunov, Riccati Equations DD methods Schur Complement Methods Hierarchical Domain Decomposition for Multiscale Problems * 3D Skin problem* Multidimensional Problems Fig. 1 – Main directions of applications H-matrices. The sym- bol refers to the projects in which I took part.
- 3. 3 Contents 1. Examples of multiscale problems 2. Multiscale methods 3. HDD method 4. Hierarchical matrices 5. Application of H-matrices to HDD 6. Complexity and storage of HDD 7. Modifications of HDD – Two scales – Truncation of the small scales 8. Numerical results
- 4. 4 Example of multiscale problems (a)macroscopic scale (b)microscopic scale Different scales in a porous medium.[Bastian 99]. 10 s -6 10 s -3 10 s 0 10 s 3 10 m -12 10 m -9 10 m -6 10 m -3 Atom Protein Cell Tissue molecular events (ion channel gating) diffusion cell signalling mitosis Example of time and length scales for modeling tumor growth.[Alarcon, Byrne, Maini 05]
- 5. 5 0,6 0,2 -0,6 0,4 0 x 621 -0,4 -0,2 3 4 50 Fig. 2 – Fine properties of the solution are out of interest.
- 6. 6 Multiscale methods The equation is : − (a(x) u) = f in Ω, u = 0 on ∂Ω. (1) Homegenisation [Babuska 75], [Bensoussan, Lions, Papanicolau 78], [Jikov, Kozlov, Oleinik 94] Solution is uε (x) = u0(x) + εu1(x, x ε ) + O(ε2 ). u0 is the solution of the homogenized equation a∗ u0 = f in Ω, u0 = 0 on ∂Ω, (2) Resonance effect in MsFEM [T.Hou, X. Wu 97] u − uh 0,Ω = O(h2 + ε/h). (3) Heterogeneous multiscale method [Weinan E, B.Engquist 03]
- 7. 7 Problem setup The Poisson problem : find u ∈ H1 (Ω) s.t. : 1≤i,j≤2 ∂ ∂xi ai,j(x) ∂ ∂xj u = f in Ω u = g on Γ (4) where ai,j ∈ L∞ (Ω) such A(x) = (ai,j)i,j=1,...,d satisfies 0 < λ ≤ λmin(A(x)) ≤ λmax(A(x)) ≤ λ , ∀x ∈ Ω. ⇒ Oscillatory or jumping coefficients are allowed.
- 8. 8 The idea of HDD Find operators : Bh, Ch s.t. uh = Bhfh + Chgh, (5) where fh is the rhs and gh the Dirichlet-boundary values. Composed matrix (Bh, Ch) is the ’inverse’ of the stiffness matrix Ah. Complete inverse (Bh, Ch) is too much of information. We might be interested only in few functionals of the solution. Example : we want to know uh(fh, gh) only for fh in a smaller space VH ⊂ Vh.
- 9. 9 Domain decomposition tree TTh FE discretisation : triangulation Th, Ω = ∪t∈Th t. 1 2 3 4 5 6 7 9 10 11 12 13 14 15 8 5 6 7 11 12 13 14 15 8 1 2 3 4 5 6 7 9 10 3 4 1 9 10 ...... 5 6 11 12 13 14 15 6 7 11 15 8 ...... 2 6 2 6 • Ω is the root of the tree, • TTh is a binary tree, • if ω ∈ TTh has two sons ω1, ω2 ∈ TTh : ω = ω1 ∪ ω2 and γω = ∂ω1 ∩ ∂ω2, • ω ∈ TTh is a leaf, if and only if ω ∈ Th.
- 10. 10 Notations Let ω ∈ TTh , ω = ω1 ∪ ω2. Γω,1 := ∂ω ∩ ω1, Γω,2 := ∂ω ∩ ω2 and γω := ∂ω1∂ω = ∂ω2∂ω ω 1 ω 2 ωPSfrag replacements ∂ γω Γω,1 Γω,2 Γω I = I(Ω) = set of all vertices of ¯Ω. I(ω) = {i ∈ I ; xi ∈ ω}.
- 11. 11 Discretisation Let ω ∈ TTh . Denote dω := (fi)i∈I(ω) , (gi)i∈I(∂ω) . Define the following discrete problem in the variational form : aω(uh, bj) = (fω, bj)L2(ω) ∀ j ∈ I( ◦ ω), uh(xj) = gj ∀ j ∈ I(∂ω). (6) a(bi, bj) = Ω α(x)( bi, bj)dx, (f, bj) = suppbj fbjdx.
- 12. 12 1. Mapping Ψω Ψω(d) = (Ψω(dω))i∈I(∂ω) with (Ψω(dω))i = aω(uh, bi) − (fω, bi)L2(ω) , Ψωdω = Ψf ωfω + Ψg ωgω. 2. Mapping Φω (Φω(dω))i := uh(xi) , ∀i ∈ I(γω). Hence, Φω(dω) is the trace of uh on γω. Goal of HDD is to build the set of mappings : {Φ0, Φ1, Φ2, ..., Φn} which than produce sequentially the solution on {γω0 , γω1 , γω2 ..., γωn }. ω ω ω 1 2 xj γ ω xj
- 13. 13 Construction of the mappings Ψω and Φω Let ω1 and ω2 be two sons of ω ∈ TTh . Let dω1 and dω2 the data associated to ω1 and ω2 s.t. : • (consistency conditions for the Dirichlet data) g1,i = g2,i , ∀i ∈ I(ω1) ∩ I(ω2), (7) • (consistency conditions for the right-hand side) f1,i = f2,i , ∀i ∈ I(ω1) ∩ I(ω2). (8) Let uω1 and uω2 be the local FE solutions of the problem (6) for the data dω1 , dω2 .
- 14. 14 ω ω ω 1 2 xj γ ω xj If uω1 , uω2 satisfy to the Neu- mann condition γ Ψω1 (dω1 ) + γ Ψω2 (dω2 ) = 0, Then, uω defined by uω(xi) := uω1 (xi) for i ∈ I(ω1) uω2 (xi) for i ∈ I(ω2) (9) is solution of (6) for the data dω := (fω, gω) given by fω := f1,i for i ∈ I(ω1) f2,i for i ∈ I(ω2) gω := g1,i for i ∈ I(∂ω1) g2,i for i ∈ I(∂ω2)
- 15. 15 γ Ψγ ω1 + γ Ψγ ω2 gγ = −Ψf ω1 f1 − ΨΓ ω1 g1,Γ − Ψf ω2 f2 − ΨΓ ω2 g2,Γ. We set M := −( γ Ψγ ω1 + γ Ψγ ω2 ), compute M−1 and solve for gγ : gγ = M−1 (Ψf ω1 f1 + ΨΓ ω1 g1,Γ + Ψf ω2 f2 + ΨΓ ω2 g2,Γ). For given mappings Ψω1 , Ψω2 , defined on the sons ω1, ω2, we can compute Φω and Ψω for the father ω. This recursion process ends as soon as ω = Ω.
- 16. 16 Hierarchical Process 1. Leaves to Root 1. Compute Ψω on all leaves (3 × 3 matrices). 2. Recursion from the leaves to the root : (a) Compute and store Φω and Ψω from Ψω1 , Ψω2 . (b) Delete Ψω1 , Ψω2 . 2. Root to Leaves 1. Given dω = (fω, gω), compute the solution uh on the interior boundary γω by Φω (dω). 2. Build the data dω1 = (fω1 , gω1 ), dω2 = (fω2 , gω2 ) from dω = (fω, gω) and gγ := Φω (dω).
- 17. 17 Rank-k matrices 1. R ∈ Rn×m , R = ABT , where A ∈ Rn×k , B ∈ Rm×k , k min(n, m). The storage A and B is k(n + m) instead of n · m. = A B T * R k k n m n m H-matrices (Hackbusch ’98) 2. Grid → cluster tree (TI) → blockclus- ter tree (TI×J ) + admissibility condition → admissible partitioning → H-matrix → H-matrix arithmetics . 4 2 2 2 3 3 3 4 2 2 2 4 2 2 2 4
- 18. 18 3. Let I := I(Ω), t, s ∈ TI, (t × s) ∈ TI×I. Admissibility : max{diam(t), diam(s)} ≤ η · dist(t, s). if(adm=true) then M|t×s is a rank-k matrix block if(adm=false) then divide M|t×s further or define as a dense matrix block. Q Qt S dist H= t s ... I I I I I I I I I I I 1 1 2 2 11 12 21 22 I11 I12 I21 I22
- 19. 19 Definition 0.1 H(TI×J , k) := {M ∈ RI×J | rank(M |t×s) ≤ k for all admissible leaves t × s of TI×J }. n := max(|I|, |J|, |K|). Operation Sequential Compl. Parallel Complexity (R.Kriemann 2005) building(M) N = O(n log n) N p + O(|V (T)L(T)|) storage(M) N = O(kn log n) N Mx N = O(kn log n) N p αM ⊕ βM N = O(k2 n log n) N p αM M ⊕ βM N = O(k2 n log2 n) N p + O(Csp(T)|V (T)|) M−1 N = O(k2 n log2 n) N p + O(nn2 min) LU N = O(k2 n log2 n) N H-LU N = O(k2 n log2 n) N p + O(k2 n log2 n n1/d )
- 20. 20 Application of H-matrices to HDD Let ω = ω1 ∪ ω2, γω = ∂ω1∂ω. Suppose Ψg ω1 , Ψg ω2 → Ψg ω =: A and Ψf ω1 , Ψf ω2 → Ψf ω =: F. A11 A12 A21 A22 x1 x2 = F1 F2 b. Eliminate internal nodal points : A11 − A12A−1 22 A21 0 A21 A22 x1 x2 = F1 − A12A−1 22 F2 F2 b. Ψg ωx1 := (A11 − A12A−1 22 A21)x1 = (F1 − A12A−1 22 F2)b = Ψf ωb x2 = A−1 22 F2b − A−1 22 A21x1 =: Φf ωb + Φg ωx1,
- 21. 21 13 4 4 4 5 5 8 5 5 8 2 2 8 5 5 16 5 5 8 5 5 8 5 5 16 5 5 8 1 1 8 5 5 8 5 5 15 5 5 16 5 5 15 Matrix Ψg with the weak admissibility condition 9 3 8 3 3 3 8 3 8 3 3 3 8 3 8 3 3 3 9 3 8 3 3 3 8 3 8 3 3 3 8 3 8 3 3 3 3 8 3 3 3 3 3 3 3 3 8 3 3 8 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 8 3 3 3 3 3 3 3 8 3 8 3 3 3 8 3 8 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 8 3 12 8 3 3 12 8 4 4 3 8 3 3 3 3 3 3 4 4 3 8 3 3 3 3 3 3 3 3 8 3 3 8 3 3 3 8 3 8 8 3 3 3 3 3 8 3 3 3 3 8 3 8 3 3 3 8 3 3 3 3 3 3 3 3 3 3 8 3 3 3 3 3 3 8 3 8 3 3 3 8 3 3 3 3 3 3 3 8 3 3 8 8 3 3 3 3 3 8 4 4 3 8 8 8 3 3 8 8 4 4 8 8 8 3 8 3 8 3 8 3 8 3 8 3 3 3 8 3 8 3 3 8 3 7 3 3 3 9 8 9 8 8 8 1 8 8 8 8 8 3 8 8 8 3 8 8 8 8 3 8 8 8 3 8 3 3 3 3 8 3 3 3 8 3 3 8 3 3 3 8 3 3 3 3 8 3 8 8 8 3 8 8 8 8 3 7 8 8 3 7 3 8 3 3 8 3 3 3 3 3 Matrix Ψf with the standard admissibility condition
- 22. 22 Building (Ψg ω)H ∈ R512×512 from (Ψg ω1 )H and (Ψg ω2 )H ∈ R384×384 . 25 5 5 8 6 6 16 6 6 16 6 6 32 7 7 32 1 1 32 6 6 16 6 6 32 6 6 16 6 6 32 11 11 32 6 6 16 6 6 32 5 5 16 6 6 32 1 1 32 6 6 32 5 5 16 6 6 16 5 5 31 25 5 8 6 16 6 16 6 32 7 32 32 16 32 16 32 6 32 6 16 6 32 5 16 6 32 32 32 16 16 31 19 32 5 32 6 32 5 31 25 8 16 16 32 32 1 32 6 16 6 32 6 16 6 32 5 32 16 32 16 32 1 32 6 32 5 16 6 16 5 31 20 32 5 32 6 32 5 31 25 7 7 8 9 9 16 10 10 16 11 11 32 18 18 32 15 15 32 17 17 16 10 10 32 8 8 16 11 11 32 19 19 32 11 11 32 14 14 32 12 12 31 17 8 8 16 11 11 16 8 8 32 10 10 16 17 17 32 14 14 32 16 16 17 6 6 16 9 9 16 10 10 16 8 8 31 20 20 32 12 12 32 13 13 32 11 11 31 25 5 5 8 6 6 16 6 6 16 6 6 32 7 7 32 1 1 32 6 6 16 6 6 32 6 6 16 6 6 32 11 11 32 6 6 16 6 6 32 5 5 16 6 6 32 1 1 32 6 6 32 5 5 16 6 6 16 5 5 31 32 32 32 10 10 32 12 12 32 10 10 31 PSfrag replacements (Ψg ω1 )H (Ψg ω2 )H (Ψg ω1 )H |I×I (Ψg ω2 )H |I×I (Ψg ω)H ∈ H(TI×I, k) ˜H
- 23. 23 Complexity and storage storage complexity Ψg - O(k3√ nhnH log √ nhnH) Ψf - O(k3√ nhnH log2 √ nhnH) Φg O(k √ nhnH) O(k2√ nhnH) Φf O(k2√ nhnH log2 √ nhnH) O(k3√ nhnH log √ nhnH)
- 24. 24 Prolongation of the right-hand side on the fine grid h H, fH ∈ VH ⊂ Vh is given ⇒ to build fh. Mappings Ψf , Φf can be compressed. H h . = g replacements Φfh ωΦfH ω Ph←H ω
- 25. 25 Truncation of the small scales : S(Φω) = S(Φg ω) + S(Φf ω) = O(k2√ nhnH log √ nhnH). Ω h HPSfrag replacements T≥H Th TTh T<H Th Fig. 3 – Domain decomposition tree TTh and its parts.
- 27. 27 (left) Skin problem, (right) model of a cell. a b c Lipid layer α β 0 10.25 0.75 0.5 1 4h [Khoromskij, Wittum 02]
- 28. 28 α ucg−˜u 2 ucg 2 ucg − ˜u ∞ ucg − ˜u A 1.0 6.6 ∗ 10−9 7.1 ∗ 10−10 2.3 ∗ 10−7 10−1 2.0 ∗ 10−8 1.4 ∗ 10−8 2.0 ∗ 10−6 10−2 6.6 ∗ 10−8 2.6 ∗ 10−7 1.7 ∗ 10−5 10−3 7.4 ∗ 10−7 1.8 ∗ 10−5 4.2 ∗ 10−4 10−4 4.2 ∗ 10−6 1.8 ∗ 10−3 1.4 ∗ 10−2 10−5 7.0 ∗ 10−5 2.3 ∗ 10−1 9.0 ∗ 10−1 Tab. 1 – Dependence of absolute and relative errors on α. 1292 dofs, εk = 10−8 , β = 1.0, residium Au − f = 10−10 , A = 1.22 ∗ 105 .
- 29. 29 ε ucg−˜u 2 ucg 2 ucg − ˜u ∞ ucg − ˜u A 10−6 4.4 ∗ 10−1 6.67 ∗ 102 1.1 ∗ 103 10−8 7.27 ∗ 10−5 2.3 ∗ 10−1 9.0 ∗ 10−1 10−10 5.1 ∗ 10−7 1.0 ∗ 10−3 3.0 ∗ 10−3 10−12 3.9 ∗ 10−9 1.2 ∗ 10−5 2.9 ∗ 10−5 10−14 1.2 ∗ 10−11 6.6 ∗ 10−7 1.2 ∗ 10−7 10−16 1.6 ∗ 10−12 1.1 ∗ 10−8 1.7 ∗ 10−8 Tab. 2 – Dependence of absolute and relative errors on εk. 1292 dofs, α = 10−5 , β = 1.0, residium Au − f = 10−10 , A 2 = 1.22 ∗ 105 . ε is responsible for the H-matrix approximation accuracy. σk ≤ εσ1.
- 30. 30 dofs Φg,Φf ,h,Kb Φg,Φf ,H=0.5,Kb Φg,Φf ,H=0.125,Kb 332 2.45 ∗ 102, 4 ∗ 102 9.1 ∗ 10, 1.7 ∗ 102 2 ∗ 102, 2.8 ∗ 102 652 1.1 ∗ 103, 2.4 ∗ 103 2.9 ∗ 102, 1.2 ∗ 103 7.9 ∗ 102, 1.8 ∗ 103 1292 5 ∗ 103, 1.4 ∗ 104 6.8 ∗ 102, 8 ∗ 103 2.6 ∗ 103, 1.2 ∗ 104 2562 2.1 ∗ 104, 7.86 ∗ 104 1.4 ∗ 103, 4.1 ∗ 104 7.4 ∗ 103, 6.9 ∗ 104 Tab. 3 – Dependence of memory requirements for Φg and Φf on numbers of dofs and size of the interface, f = 4, nmin = 12, u = x2 + y2 and k = 7.
- 31. 31 Storage ε LLT (Mb) HDD(Mb) (A−1)H(Mb) 10−3 13.3 19.7 51.0 10−4 14.7 20.1 64.0 10−5 16.0 20.4 75.2 10−6 17.2 20.6 87.4 Tab. 4 – Dependence of memory requirements on ε, 1292 dofs.
- 32. 32 dofs HDD pre,LLT ,cg Inv(A) pre,LLT 332 0.19 0.1=0.03+0.02+0.04 0.24 0.11=0.03+0.08 652 0.96 0.6=0.2+0.1+0.26 3.54 0.5=0.2+0.3 1292 5.6 5=2.6+0.6+1.8 65.8 4.7=2.7+2.0 2572 36.1 53=38.0+3.4+11.4 - 50=38.2+11.7 5122 218 - - - Tab. 5 – Comparison of times for the skin problem with α(x, y) = 10−5 , ε = εcg = 10−8 .
- 33. 33 Oscillatory coefficients global k ˜u40 − ˜uk 2 / ˜u40 2 ˜u40 − ˜uk ∞ 2 7 7 ∗ 10−2 4 2 ∗ 10−2 1.8 ∗ 10−3 6 5.4 ∗ 10−4 4.5 ∗ 10−5 8 6.6 ∗ 10−5 6.3 ∗ 10−6 10 7.6 ∗ 10−6 9 ∗ 10−7 Tab. 6 – α(x, y) = 1 + 0.5sin(50x)sin(50y) ω u40 − ˜uk 2 / u40 2 u40 − ˜uk ∞ 10 1.65 ∗ 10−4 1.76 ∗ 10−5 50 1.8 ∗ 10−4 1.9 ∗ 10−5 450 7.7 ∗ 10−4 10−4 Tab. 7 – 2572 Dofs, f = 1, α(x, y) = 1 + 0.5sin(ωx)sin(ωy).
- 34. 34 Ω α β 0.1 0.2 0.8 0.9 0.1 0.2 0.8 0.9 Fig. 4 – Domain Ω = (0, 1)2 with jumping coefficients α and β.
- 35. 35 ε Au − f 2 cg ; HDD(sec) ucg−˜u 2 ˜u 2 ucg − ˜u ∞ 10−4 2 ∗ 10−4 5.3 ; 8.9 6.7 ∗ 10−1 1.4 10−6 4.8 ∗ 10−7 5.0 ; 10.1 1.8 ∗ 10−4 9.5 ∗ 10−4 10−8 1.4 ∗ 10−8 5.7 ; 11.5 1.1 ∗ 10−6 1.48 ∗ 10−5 10−10 1.45 ∗ 10−8 6.7 ; 12.3 5.3 ∗ 10−7 10−5 10−12 1.2 ∗ 10−8 7.4 ; 13.5 5.2 ∗ 10−7 10−5 Tab. 8 – Dependence of the relative and absolute errors on ε, ˜u is HDD solution from ε, α = 10, β = 0.01, 1292 dofs.
- 36. 36 Properties of HDD : 1. HDD computes uh := Bhfh + Chgh or uh := BHfH + Chgh. 2. Bh, BH and Ch have H-matrix format. 3. The complexities are O(k2 nh log3 nh) and O(k2√ nhnH log3 √ nhnH). 4. The storages are O(knh log2 nh) and O(k √ nhnH log2 √ nhnH). 5. HDD computes functionals of the solution : (a) Neumann data ∂uh ∂n at the boundary, (b) mean values ω uhdx, 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.
- 37. 37 6. HDD for multiple right-hand sides and multiple Dirichlet data. 7. HDD can easily be parallelized. 8. Problems with repeated patterns.
- 38. 38 Thanks for your attention !