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

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1 1 Fast evaluation of boundary integral operators arising from an eddy current problem Steffen Börm MPI MIS Leipzig Jörg Ostrowski ETH Zürich

2 Induction heating 2 Induction heating: Time harmonic eddy current problem div E = 0, in Ω + curl 1 µ curl E = iω(σe + j 0) in R 3, [n E] = [n 1 curl E] = 0 in Ω µ

3 Variational Formulation 3 R. Hiptmair Symmetric coupling f. eddy current problems SIAM Representation formula of Stratton-Chu kind x Ω + : curl E(y) n y n y γ D E(y) E(x) = ds(y) + curl ds(y) 4π x y 4π x y Γ (Remark: E n = 0) Apply traces γ D E := n (E n) γ N E := curl E n = grad φ n = curl Γ φ from outside (Ω + ) on representation formula and test equations by using impedance boundary conditions γ D E = (1 i) 1 2σµω curl Γ φ Γ

4 Variational Formulation 4 Goal: Find E V and φ W solving a(v, E) b(v, φ) =f(v) for all v V b(e, ψ) q(ψ, φ) =ζ(ψ) for all ψ W with the bilinear forms a(v, E) = curl Γ v(x), curl Γ E(y) Φ(x, y) dy dx + sparse, Γ Γ q(ψ, φ) = curl Γ ψ(x), curl Γ φ(y) Φ(x, y) dy dx, Γ Γ b(v, φ) = curl Γ φ(y), v(x) grad x Φ(x, y), n(x) dy dx Γ Γ curl Γ φ(y), n(x) grad x Φ(x, y), v(x) dy dx Γ Γ + sparse, where Φ(x, y) = 1/(4π x y ) is the Laplace singularity function.

5 Discretization 5 Surface edge elements: Let V h = span{b i first order surface edge elements. : i E} be the set of Surface nodal elements: Let W h = span{ψ i of surface nodal elements. : i N } be the set Galerkin discretization: Leads to A B = rhs B Q with A ij = a(b i, b j ), Q ij = q(ψ i, ψ j ) and B ij = b(b i, ψ j ). Problem: All matrices dense Triangles 4 Gbyte

6 Simplification 6 Split bilinear forms into their components M ij := χ i (x)φ(x, y)λ j (y) dy dx Γ Γ G ij := χ i (x) grad i Φ(x, y)λ j (y) dy dx Γ Γ Problem: The singularity function Φ is not local, so M, G are not sparse. High complexity. Still expensive matrix-vector multiplication y = Mx or y = Gx.

7 Tschebyscheff Interpolation 7 Idea: Replace Φ and grad Φ on a sub-domain τ σ Γ Γ by Φ τ,σ (x, y) := m ν=1 µ=1 grad Φ τ,σ (x, y) := grad Φ τ,σ. m Φ(x τ ν, x σ µ)l τ ν(x)l σ µ(y) and = ONLY POINTWISE EVALUATION OF KERNEL Result: Local matrices M and G, for example M τ,σ ij := χ i (x)φ(x, y)λ j (y) dy dx χ i (x) Φ τ,σ (x, y)λ j (y) dy dx = τ m σ m ν=1 µ=1 Φ(x τ ν, x σ µ) } {{ } =:S τ,σ νµ = (V τ S τ,σ (W σ ) ) ij χ i (x)l τ ν(x) dx τ } {{ } =:Viν τ τ σ λ j (y)l σ µ(y) dy σ } {{ } =:Wjµ σ

8 Local interpolation 8 Problem: The function Φ is not globally smooth = m too large. Idea: It is asymptotically smooth, i.e., locally smooth far from the diagonal x = y = m N. Error estimate: For the tensor-product interpolation of order m on axis-parallel boxes B τ τ and B σ σ we have Φ τ,σ Φ L (B τ B σ ) C dist(b τ, B σ ) 3 m ( dist(b τ, B σ ) (m+1) ) diam(b τ B σ. ) Admissibility condition: Convergence for η ]0, [ if max{diam(b τ ), diam(b σ )} 2η dist(b τ, B σ )

9 Cluster tree and block partition 9 Cluster tree: Split Γ into a hierarchy of subdomains. Block partition: Partition P of Γ Γ consisting of admissible subdomains in the farfield P far and non-admissible subdomains in nearfield P near. P far := {τ σ P : τ σ is admissible}, P near := P \ P far. Matrix: M is approximated by M := V τ S τ,σ W σ + τ σ P far τ σ P near M τ,σ. Error decreases exponentially in m, complexity O(Nm 3 log N).

10 H 2 -Matrix Approximation Preparation: Build cluster tree + partition, calculate V, S, W 2. Multiplications Cluster tree by bisection:

11 H 2 -Matrix Approximation 11 Block partition by recursive algorithm. Calculation of V and W for each cluster τ Viν τ := χ i (x)l τ ν(x) dx, Wiν τ := τ and S for admissible blocks τ σ τ λ i (x)l τ ν(x) dx S τ,σ νµ := Φ(x τ ν, x σ µ). Idea of nested basis: For cluster τ with sons τ holds L τ ν = k ν =1 L τ ν(x τ ν ) }{{} =:T τ,τ ν ν L τ ν. Store V τ, W τ, only for leaf clusters. Others: V τ = τ sons(τ) Vτ T τ,τ

12 Fast matrix-vector multiplication 12 y Mx := τ σ P far V τ S τ,σ W σ x + τ σ P near M τ,σ x 1. Forward transform: x σ := W σ x Use W σ in leaves and T σ,σ in remaining clusters. 2. Multiply farfield: y τ := σ,(τ,σ) P far S τ,σ x σ 3. Backward transform: y := τ Vτ y τ Use V τ in leaves and T τ,τ in remaining clusters. 4. Add Nearfield

13 Numerical experiments: Variations in N 13 m = 2, η = 1, Rel. Error := Γ j H 2(x) j St. (x) j St. (x) ds x Standard H 2 N Mem[MB] Time[min] Mem Time Rel. Error

14 Numerical experiments: Variations in m and η 14 N=8836, Operator Err.= K K 2 K 2 η m Build[min] Mem[MB] MVM[s] Op. Err Standard

H 2 -matrices with adaptive bases

1 H 2 -matrices with adaptive bases Steffen Börm MPI für Mathematik in den Naturwissenschaften Inselstraße 22 26, 04103 Leipzig http://www.mis.mpg.de/ Problem 2 Goal: Treat certain large dense matrices