Fast Methods for Integral Equations. Jacob White. Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy

Transcription

1 Introduction to Simulation - Lecture 23 Fast Methods for Integral Equations Jacob White Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy

2 Outline Solving Discretized Integral Equations Using Krylov Subspace Methods Fast Matrix-Vector Products Multipole Algorithms Multipole Representation. Basic Hierarchy Algorithmic Improvements Local Expansions Adaptive Algorithms Computational Results

3 Exterior Problem in Electrostatics potential + v 2 Ψ=0 Outside - Ψ is given on Surface Dirichelet Problem First Kind Integral Equation For Charge: 1 Ψ ( x) = σ ( x ) ds x x { surface Ch arg e potential Green' s Density Function SMA-HPC 2003 MIT

4 Drag Force in a Microresonator Courtesy of Werner Hemmert, Ph.D. Used with permission. Resonator Discretized Structure Computed Forces Bottom View Computed Forces Top View SMA-HPC 2003 MIT

5 3-D Laplace s Equation Integral Equation: Ψ x = Basis Function Approach Piecewise Constant Basis ( ) ( ) surface 1 x x σ x ds Discretize Surface into Panels Represent n σ ( x) α ( ) i ϕi x { i= 1 Basis Functions Panel j ϕ j x = x ϕ = j ( ) 1 if is on panel j ( x) 0 otherwise SMA-HPC 2003 MIT

6 3-D Laplace s Equation Basis Function Approach Centroid Collocation Put collocation points at panel centroids ( ) (, ) c α j c Ψ = i c i x Collocation point n x G x x ds i j= 1 panel j A i, j ( x ) c A1,1 L L A1, n α1 Ψ 1 M O M M M = M O M M M A ( ) n,1 L L An, n α n Ψ x cn SMA-HPC 2003 MIT

7 3-D Laplace s Equation Basis Function Approach Calculating Matrix Elements z y x x Collocation c i point Panel j A i, j 1 = panel j x x c i ds One point quadrature Approximation A i, j Panel Area x c i x centroid j Four point quadrature Approximation SMA-HPC 2003 MIT A i, j * Area x j= 1 c i x point j

8 3-D Laplace s Equation Basis Function Approach Calculating Self-Term z y x x Collocation c i point Panel i A ii, 1 = panel i x x c i ds One point quadrature Approximation SMA-HPC 2003 MIT A ii, Panel Area x c i i Aii, = ds is an integrable singularity x x panel i c i 0 x c

9 3-D Laplace s Equation z y Integrate in two pieces x Panel i Basis Function Approach x Collocation c i point Disk of radius R surrounding collocation point Calculating Self-Term Tricks of the trade A ii, 1 = panel i x x c i ds 1 1 Aii, = ds + ds x x x x disk c i rest of panel c i Disk Integral has singularity but has analytic formula disk x c i 1 x R 2π 1 ds = rdrdθ = 2π R r 0 0 SMA-HPC 2003 MIT

10 3-D Laplace s Equation z y x Panel i Basis Function Approach x Collocation c i point Calculating Self-Term Other Tricks of the trade A ii, = panel i x c 1 x i Integrand is singular ds 1) If panel is a flat polygon, analytical formulas exist 2) Curve panels can be handled with projection SMA-HPC 2003 MIT

11 3-D Laplace s Equation Basis Function Approach Galerkin (test=basis) n ( ) ( ) ( ) (, ) ( ) ϕi x Ψ x ds = α j ϕi x G x x ϕ j x dsds j= b A i For piecewise constant Basis n 1 Ψ ( x) ds = α j dsds panel i panel j j= 1 x x b i A i, j i, j SMA-HPC 2003 MIT A1,1 L L A1, n α1 b1 M O M M M = M O M M M An,1 L L An, n α n b n

12 3-D Laplace s Equation Basis Function Approach Problem with dense matrix Integral Equation Method Generate Huge Dense Matrices ( x ) c A1,1 L L A1, n α1 Ψ 1 M O M M M = M O M M M A ( ) n,1 L L An, n α n Ψ x cn Gaussian Elimination Much Too Slow! SMA-HPC 2003 MIT

13 Solving Discretized Integral Equations The Generalized Conjugate Residual Algorithm The kth step of GCR compute α = k ( k ) T r ( Apk ) T ( Ap ) ( Ap ) k Apk k For discretized Integral equations, A is dense Determine optimal stepsize in kth search direction x = x + α p r r Ap k 1 k k k k + 1 k = αk k ( k 1) T ( ) k + Ar Ap k + 1 j pk + 1 = r p T j j= 0 SMA-HPC 2003 MIT Update the solution and the residual ( Ap ) ( ) j Ap j Compute the new orthogonalized search direction

14 Solving Discretized Integral Equations The Generalized Conjugate Residual Algorithm Complexity of GCR compute α = k+ 1 k T ( r ) ( Apk ) k T ( Apk) ( Apk) k 1 k x + = x + αkpk k 1 k r + = r α Ap ( 1) T k k + Ar 1 ( Ap k + j ) = T j= 0 ( Ap j) ( Ap j) p r p SMA-HPC 2003 MIT Apk k k Dense Matrix-vector product costs O(n 2 ) Vector inner products, O(n) Vector Adds, O(n) j O(k) inner products, total cost O(nk) Algorithm is O(n 2 ) for Integral Equations even though # iters (k) is small!

15 Solving Discretized Integral Equations The Generalized Conjugate Residual Algorithm Fast Matrix Vector Products exactly compute Apk Dense Matrix-vector product costs O(n 2 ) approximately compute Apk Reduces Matrix-vector product costs to O(n) or O(nlogn) SMA-HPC 2003 MIT

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42 Summary Solving Discretized Integral Equations GCR plus Fast Matrix-Vector Products Multipole Algorithms Multipole Representation. Basic Hierarchy Algorithmic Improvements Local Expansions Adaptive Algorithms Computational Results Precorrected-FFT Algorithms