H 2 -matrices with adaptive bases

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1 1 H 2 -matrices with adaptive bases Steffen Börm MPI für Mathematik in den Naturwissenschaften Inselstraße 22 26, Leipzig

2 Problem 2 Goal: Treat certain large dense matrices in linear complexity. Model problem: Discretization of integral operators of the form K[u](x) = κ(x, y)u(y) dy. Ω Discretization: Galerkin method with FE basis (ϕ i ) n i=1 leads to a matrix K R n n with K ij = ϕ i, K[ϕ j ] L 2 = ϕ i (x)κ(x, y)ϕ j (y) dy dx. (1 i, j n) Ω Ω Problem: K is dense. Idea: Use properties of the kernel function to approximate K. Examples: Wavelets, multipole expansion, interpolation.

3 Interpolation 3 Idea: We consider a subdomain τ σ Ω Ω and replace the kernel function κ by an interpolant κ τ,σ (x, y) := k k κ(x τ ν, x σ µ)l τ ν(x)l σ µ(y). (k n) ν=1 µ=1 Result: Local matrix in factorized form K τ,σ ij := ϕ i (x)κ(x, y)ϕ j (y) dy dx = τ k σ k ν=1 µ=1 κ(x τ ν, x σ µ) } {{ } =:Sνµ τ,σ = (V τ S τ,σ (V σ ) ) ij τ ϕ i (x)l τ ν(x) dx τ } {{ } =:Viν τ σ ϕ i (x) κ τ,σ (x, y)ϕ j (y) dy dx ϕ j (y)l σ µ(y) dy σ } {{ } =:Vjµ σ n k n k

4 Local interpolation 4 Problem: Typical kernel functions are not globally smooth. Idea: They are locally smooth outside of the diagonal x = y. Examples: 1/ x y, log x y. Error bound: For m-th order tensor interpolation on axis-parallel boxes B τ τ and B σ σ, we get ( κ τ,σ κ L (B τ B σ ) C apx (m) dist(bτ, B σ ) m ) diam(b τ B σ. ) Rank: m interpolation points per coordinate k = m d. Admissibility: Uniform exponential convergence requires diam(b τ B σ ) 2η dist(b τ, B σ ).

5 Block structure 5 Approach: Split Ω Ω into admissible subdomains and a small remainder. Cluster tree: Hierarchy of subdomains of Ω Nearfield: Small blocks, stored in standard format. Farfield: Admissible blocks, stored in V τ S τ,σ (V σ ) format. Partition: Collection of farand nearfield blocks. Construction of cluster tree and block partition can be accomplished by general algorithms in O(n) operations.

6 Nested bases 6 Goal: Treat the cluster basis (V τ ) τ T efficiently. Idea: For a cluster τ and τ sons(τ) we have L τ ν = k ν =1 L τ ν(x τ ν )Lτ. (1 ν k) ν Nested bases: Consider i τ τ. Setting T τ ν ν := Lτ ν(x τ ν ), we find k V τ iν = τ ϕ i (x)l τ ν(x) dx = ν =1 T τ ν ν τ ϕ i (x)l τ ν (x) dx = (V τ T τ ) iν. Consequence: Store V τ only for leaves and use T τ for all other clusters. Complexity: Storage O(nk), matrixvector O(nk).

7 Orthogonalization 7 Problem: Efficiency depends on the rank k. Can we reduce this rank by eliminating redundant expansion functions? Idea: Orthogonalize columns of V τ. Gram-Schmidt: Find Ṽ τ = V τ Z τ such that (Ṽ τ ) Ṽ τ = I holds. Nested structure: Using P τ with V τ = Ṽ τ P τ, we find V τ = V τ 1 T τ 1 = Ṽ τ 1 P τ 1 T τ 1 = Ṽ τ 1 V τ 2 T τ 2 Ṽ τ 2 P τ 2 T τ 2 Ṽ τ 2 = Ṽ τ V τ Z τ P τ 1 T τ 1, P τ 2 T τ 2 so it is sufficient to orthogonalize a (2k) k matrix. The resulting cluster basis is nested due to Ṽ τ = V τ Z τ = Ṽ τ 1 (P τ 1 T τ 1 Z τ ) = Ṽ τ 1 T τ 1. Ṽ τ 2 (P τ 2 T τ 2 Z τ ) Ṽ τ 2 τ T 2 Result: Complexity O(nk 2 ).

8 Orthogonalization: Simple example 8 Example: Unit sphere, piecewise constant trial and test functions. Interpolation Orthogonalized n Build MVM Mem/n Expl Impl MVM Mem/n H 2 -matrix constructed by cubic interpolation and subsequent orthogonalization, relative error Hardware: SunFire 6800 with 900 MHz UltraSparc IIIcu processors Software: HLib, see

9 Recompression algorithm: Problem 9 Problem: Interpolation does not take into account that a subset of polynomials may be sufficient to approximate the kernel function. Idea: Use local singular value decompositions to find optimal cluster bases Take kernel and geometry into account. Important: Since the cluster bases are nested, the father clusters depend on the son clusters. Consequence: If τ σ is a farfield block and τ sons(τ), we have K τ σ = (V τ S τ,σ (V σ ) ) τ σ = V τ T τ S τ,σ (V σ ), i.e., the choice of V τ its other ancestors. influences all farfield blocks connected to τ or

10 Recompression algorithm: Local problems and recursion 10 Block row: R τ := {σ : τ + τ : τ + σ is admissible}. Goal: Find an orthogonal matrix V τ R τ kτ that maximizes σ R τ (V τ ) K τ σ 2 F. Optimal solution: Use eigenvectors of corresponding Gram matrix. Nested structure: Ensured by bottom-up recursion and projections. Result: Algorithm with complexity O(nk 2 ), truncation error can be controlled directly.

11 Recompression: Simple example 11 Example: Unit sphere, piecewise constant trial and test functions. Interpolation Recompressed n Build MVM Mem/n Build MVM Mem/n H 2 -matrix constructed by cubic interpolation. Recompression with relative error bound 10 3, maximal rank 16. Hardware: SunFire 6800 with 900 MHz UltraSparc IIIcu processors. Software: HLib, see

12 Recompression: Netgen example 12 Example: Grid Crank shaft of the Netgen package. n Build Mem/n Near/n MVM Rel. error 0.05% 0.07% 0.08% DMem/n

Recompression: Netgen geometries and eddy current model 13 Z Z b(v, φ) = Γ Z φ(y), v(x) x Φ(x, y), n(x) dy dx Z φ(y),n(x) x Φ(x, y), v(x) dy dx Γ Γ Γ n 42432

13 Recompression: Netgen geometries and eddy current model 13 Z Z b(v, φ) = Γ Z φ(y), v(x) x Φ(x, y), n(x) dy dx Z φ(y),n(x) x Φ(x, y), v(x) dy dx Γ Γ Γ n Direct MVM Mem/DoF Recompressed MVM Mem/DoF Rel. error

14 Summary 14 H 2 -matrices: Nested structure leads to complexity O(nk) for storage requirements and matrix-vector multiplication. Black box: The approximation is constructed by using only kernel evaluations κ(x τ ν, x σ µ) and works for all asymptotically smooth kernel functions. Recompression: Use O(nk 2 ) algorithm to reduce the rank k, find a quasi-optimal cluster basis adapted to geometry, discretization and kernel function. Work in progress: H 2 -matrix arithmetics, HCA.

15 H 2 -matrix arithmetics 15 Goal: Perform matrix addition and multiplication efficiently. Addition and scaling: For an admissible block τ σ, we have (A + λb) τ σ = V τ (S τ,σ A + λsτ,σ B )(V σ ), so H 2 -matrices form a vector space. Addition and scaling can be performed in O(nk) operations. ( + λ ) Multiplication: Defined recursively. A 11 A 12 B 11 B 12 = A 11B 11 + A 12 B 21 A 11 B 12 + A 12 B 22 A 21 A 22 B 21 B 22 A 21 B 11 + A 22 B 21 A 21 B 12 + A 22 B 22 Use truncation if product does not match prescribed format.

16 H 2 -matrix multiplication I 16 := Situation: Let A = V τ S τ,ϱ A (V ϱ ), B not admissible. Compute best approximation of AB in the form C = V τ S τ,σ C (V σ ). Orthogonality: If V τ and V σ are orthogonal: S τ,σ C = (V τ ) ABV σ = (V τ ) V τ S τ,ϱ A (V ϱ ) BV σ = S τ,ϱ A (V ϱ ) BV σ. }{{} =: S ϱ,σ B Problem: Computing (V ϱ ) BV σ directly is too expensive. Idea: Prepare S ϱ,σ B := (V ϱ ) BV σ in advance. Matrix forward transformation: Since the cluster bases are nested, all matrices can be prepared in O(nk3 ) operations. S ϱ,σ B

17 H 2 -matrix multiplication II 17 := Situation: Let A = V τ S τ,ϱ A (V ϱ ) and B = V ϱ S ϱ,σ B (V σ ). Advantage: The product AB already has the factorized form AB = V τ S τ,ϱ A (V ϱ ) V ϱ S ϱ,σ B (V σ ) = V τ (S τ,ϱ A Sϱ,σ B )(V σ ). Problem: Converting AB directly to the format of C will lead to inacceptable complexity. S τ,σ C Idea: Store result in multiplication is complete. := S τ,ϱ A Sϱ,σ B and distribute after the Matrix backward transformation: Since the cluster bases are nested, all matrices can be distributed in O(nk3 ) operations. S τ,σ C

18 H 2 -matrix multiplication III 18 := Situation: Let B = V ϱ S ϱ,σ B (V σ ). Approach: Split B and proceed by recursion: B ϱ σ = V ϱ T ϱ S ϱ,σ B (T σ ) (V σ ). }{{} =: bs ϱ,σ B Complexity: Only O(Cspn) 2 such splitting operations are required Complexity O(Cspnk 2 3 ). Result: The best approximation C of AB in a prescribed H 2 -matrix format can be computed in O(C 2 spnk 3 ) operations.

19 H 2 -matrix multiplication: Fixed structure 19 Example: Discretized double layer potential on the unit sphere K, approximate KK. n Build C sp H 2 -MVM H 2 -MMM H-MMM Error < Important: Here, the result matrix is projected to the same cluster bases and the same block structure as K. Question: Can we reduce the error by using adapted cluster bases?

20 H 2 -matrix multiplication: Adapted structure 20 Modification: Choose cluster bases for result matrix adaptively to improve the precision. n C sp Old MMM Error New basis New MMM Error Advantage: Precision significantly improved, computation time only slightly increased. Current work: Improve the efficiency of the construction of adapted cluster bases.

21 Hybrid Cross Approximation 21 Problem: For each admissble block V τ S τ,σ (V σ ) of the H 2 -matrix, the kernel function κ has to be evaluated in m 2d points. Very time-consuming if high precision is required. Approach: Apply adaptive cross approximation to the coefficient matrices S τ,σ to find rank k representation S τ,σ X τ,σ (Y τ,σ ). Result: Only m d k m 2d kernel evaluations required to approximate coefficient matrix S τ,σ. Important: ACA is only applied to coefficient matrices. Construction still linear in n, no problems with quadrature. m Std 24.1s 64.7s 160.8s 399.2s HCA 22.3s 49.2s 90.0s 222.7s

22 Conclusion 22 H 2 -matrices: Data-sparse approximation of certain dense matrices with linear complexity in the number of degrees of freedom. Adaptive cluster bases: Find quasi-optimal expansion systems for arbitrary matrices. In the case of integral operators, point evaluations of the kernel function are sufficient to construct efficient and reliable H 2 -matrix approximations. H 2 -matrix arithmetics: Approximate A + λb and AB in O(nk 2 ) and O(nk 3 ) operations. Next goal: Approximate A 1 and LU. HCA: Reduce the number of kernel evaluations required to construct the H 2 -matrix. Software: HLib package for hierarchical matrices and H 2 -matrices, including applications to elliptic PDEs and integral equations.

23 Interpolation vs. Multipole 23 Construction: Pointwise evaluation and integrals of polynomials. Tν t ν = L t ν(x t ν ), S t,s νµ = κ(x t ν, x s µ), Viν t = ϕ i (x)l t ν(x) dx. Interpolation Multipole - Rank k = p d + Rank k = p d 1 - Low frequencies only + Low and high frequencies + General kernels - Special kernels only + Anisotropic clusters - Spherical clusters only Γ Goal: Reduce rank for interpolation without sacrificing precision.

Fast evaluation of boundary integral operators arising from an eddy current problem. MPI MIS Leipzig

1 Fast evaluation of boundary integral operators arising from an eddy current problem Steffen Börm MPI MIS Leipzig Jörg Ostrowski ETH Zürich Induction heating 2 Induction heating: Time harmonic eddy current