Kernel Independent FMM

Transcription

1 Kernel Independent FMM FMM Issues FMM requires analytical work to generate S expansions, R expansions, S S (M2M) translations S R (M2L) translations R R (L2L) translations Such analytical work leads to barriers in creating FMM for new problems Previous lecture one approach Analytically connect polyharmonic and Stokes flow kernels to Laplace FMM Analytically connect Maxwell s kernel to Helmholtz FMM 1

2 Kernel Independent Approach Lexing Ying, George Biros, Denis Zorin. A kernelindependent adaptive fast multipole algorithm in two and three dimensions Journal of Computational Physics 196 (2004) Walter Dehnen A Hierarchical O(N) Force Calculation Algorithm Journal of Computational Physics 179, (2002) Cristopher Cecka, Pierre-David L etourneau, and Eric Darve. Fast Multipole Method using the Cauchy Integral Formula, BIRS 2010 P. G. Martinsson and V. Rokhlin ``An accelerated kernelindependent fast multipole method in one dimension, SIAM J. Sci. Comput. Vol. 29, no. 3, pp , 2007 Goal of KIFMM Given a known kernel function G(x,y) Assume that this function is FMM able However, you don t know expansions or translations Only that such expansions (multipole, local) exist That translations exist Can write a routine to evaluate this function for given pair of arguments. Build an FMM-like algorithm for this function 2

3 Check Surfaces and Equivalent Densities - Multipole Set of sources y i in a box. We wish to compute (1) Let us compute this on a set of points (blue open circles) placed on a check surface enclosing the box Indicated with dotted lines Points are outside sphere enclosing the box Denoted x B,u Choose a set of different source points on an equivalent sphere (orange) These sources are fictitious but chosen to achieve the same potential on the check surface points for the box Check Surfaces and Equivalent Densities - Local Set of sources y i outside box. We wish to compute (1) Let us compute this on a set of points (blue open circles) placed on a check surface enclosing the box Indicated with dotted lines Points are outside sphere enclosing the box Denoted x B,u Choose a set of different source points on an equivalent sphere (orange), outside this surface These sources are fictitious but chosen to achieve the same potential on the check surface points for the box 3

4 M2M (S S) translation. Potential on parent box check surface for parent equiv. density should match that from child equiv. density M2L (S R) translation. Potential on check surface of evaluation box, from source-box density and evaluation box densities should match R R translation. Potential on check surface of parent for equivalent densities of parent and child must match Solving the Equation We are solving integral equations, where the surface is the enclosing circle/sphere or box To make solution stable, solve Keeping equivalent density surface and check surface separate avoids any singular integrals These inverses can be precomputed 4

5 ;u the for 12/2/2011 Algorithm ASSUME N is the total number of points s is the maximum number of points allowed in leaf box STEP 1 TREE CONSTRUCTION for each box B in preorder traversal of the tree do subdivide B if B has more than s points in it for each box B in preorder traversal of the tree do construct L B U, L B V, L B W and L B X for B STEP 2 UPWARDS PASS for each leaf box B in postorder traversal of the tree do evaluate q B,u at x B,u using {φ i, i in I BS } solve for φ B,u at y B,u that matches q B,u at x B,u for each non-leaf box B in postorder traversal of the tree do add to q B,u at x B,u contribution from φ C,u C;u each child C of B solve for φ B,u at y B,u that matches q B,u at x B,u STEP 3 DOWNWARDS PASS for each non-root box B in preorder traversal of the tree do add to q B,d at x B,d the contribution from φ V,u for each box V in L B V add to q B,d at x B,d the contribution from {φ i, in I X s} for each box X in L B X add to q B,d at x B,d the contribution from φ P,d, where P is the parent of B solve for φ B,d at y B,d that matches q B,d at x B,d 5

6 Complexity Ignore solution costs, since grids and kernels are fixed. Just count rhs evaluation costs 6

7 Error bounds Error introduced by the solution procedure Analysis of the integral equation technique and regularization presented in the paper Not as strong as the FMM Error 7

8 Timing 8