FastMMLib: a generic Fast Multipole Method library. Eric DARRIGRAND. IRMAR Université de Rennes 1

Transcription

1 FastMMLib: a generic Fast Multipole Method library Eric DARRIGRAND joint work with Yvon LAFRANCHE IRMAR Université de Rennes 1

2 Outline Introduction to FMM Historically SVD principle A first example of use: CFIE+OSRC A second example: Hamiltonian systems FastMMLib Mathematical form of the FMM in FastMMLib Specificities of FastMMLib Structure of the library The user s classes State of the art

3 Introduction to FMM Efficient evaluation of matrix-vector products when the matrix is defined by a kernel or a potential G with a matrix [M] of size N N: e.g. or [M] i j = Γ Γ G(x, y)ϕ j (y)ϕ i (x)dγ(y)dγ(x), [M] i j = G(x i, y j ), with complexity O(N 3/2 ), O(N 4/3 ) or O(N log N).

4 What is FMM? Historically N-body problems Laplace equation via integral formulation G(x, y) = 1/ x y. Multipole expansion / local expansion: using Spherical Harmonics. COIFMAN, GREENGARD, ROKHLIN 1987, Integral equations for acoustic and electromagnetism problems Green kernel G(x, y) = e ik x y /(4π x y ). Gegenbauer addition theorem, Funk-Hecke formula. ROKHLIN et al. (1990), CHEW et al. (1994), DARVE (1997),... Improvements for molecular dynamics Aim: Reduces the cost of the translation operator ELLIOTT et al. (1996), ROKHLIN et al. (1997), SCUSERIA et al. (1998), CHALLACOMBE et al. (1997) Improvements for acoustic and electromagnetism Aim: Deal with unstabilities of the translation operator DARVE and HAVÉ (2004), LIU and CHEW (2010)

5 What is FMM? SVD principle Let A be a matrix of R N N. Let X be a vector of R N. Aim: offer a low-cost matrix-vector product Principle: {σ l } l=1,,l, singular values of A. {V l } l=1,,l, eigenvectors of A A. {U l } l=1,,l, eigenvectors of AA. A = UΣV A = L l=1 σ l U l V l Product AX: with the cost O(NL) AX = L l=1 σ l U l (V l X)

6 A first example of use: CFIE+OSRC+FMM (within the ANR project Microwave) M. DARBAS, E. D., Y. LAFRANCHE. Combining analytic preconditioner and fast multipole method for the 3-D Helmholtz equation. JCP, 236: , Ω + Ω u inc u + =? Γ u + + k 2 u + = 0, in Ω +, n + u + Γ = n + u inc Γ, on Γ, ( u + lim x x + x x iku+ ) = 0, with u inc (x) = e ikθinc x, θ inc the incidence direction, k the wavenumber.

7 CFIE+OSRC+FMM integral formulation Find ϕ = γ + (u + u inc ) H 1/2 (Γ) such that ( ) I 2 K ÑW ϕ = γ + u inc + Ñ + n u inc, on Γ. with K and W, two integral operators, Ñ an approximation of the NtD operator N ex which is derived from On-Surface Radiation Condition methods: Ñ = 1 ( 1 + ) 1/2 Γ ik kε 2, with k ε = k + iε. Essential remarks: Ñ evaluated using a Padé approximation. Ñ involves only differential operators like Γ. spectral properties strongly improved thanks to I/2 K N ex W = I the equation still involves the integral operators K and W.

8 CFIE+OSRC+FMM radar cross section RCS Sphere ; k=10 ; CFIE+OSRC+FMM Mie series n " = 7 n " = Mie series CFIE+OSRC+FMM RCS Unit Sphere ; k=47.4 Normalized RCS n " = 15 Normalized RCS ! !

9 GMRES convergence 200 Unit sphere, n! = Unit sphere, k= GMRES iterations CFIE CFIE+FMM CFIE+OSRC+FMM GMRES iterations CFIE CFIE+FMM CFIE+OSRC+FMM k vs. k (n λ = 10) vs. n λ (k = 10) n!

10 CPU costs k Total CPU time Total CPU time Total CPU time CFIE CFIE + SLFMM CFIE + SLFMM + OSRC min min 47 2 min h 43 min 4 h 33 min 32 min > 15 days 214 h 44 min 6 h 20 min h 48 min Asymptotic behavior of the SLFMM Cost related to OSRC operators << cost related to integral operators

11 A second example: a Regularized FMM for Hamiltonian systems P. CHARTIER, E. D., E. FAOU. A regular fast multipole method for geometric numerical integrations of Hamiltonian systems. BIT, 50(1):23 40, L = 3, N L = 7, N o = 2, R reg = 0.45

12 FMM discontinuities: Far interactions of B T 1 C BS1 y j 2 C B S2 y j1 Far interactions of B T 2 x i1 C BT 1 x i 2 CBT 2 x i1 CBT Close interactions of x i1 B T 1 Close interactions of x i2 B T 2 regularization by considering overlapping FMM boxes

13 Mathematically FastMMLib Let us come back to the matrix-vector products with an interaction matrix [M] of size N N: e.g. or which is dense. [M] i j = Γ Γ G(x, y)ϕ j (y)ϕ i (x)dγ(y)dγ(x), [M] i j = G(x i, y j ),

14 The essential ingredient is the choice of an expansion of the interaction function G: G(x, y) S p=1 c p g (p) x,b S p=1 T (p, p) (p, p) f B, B y, B, where S and S: truncation or discretization parameters; c p depends only on p; f (p, p) named far moment; y, B T (p, p) B, B named translation operator, with B and B containing resp. x and y; g (p) x,b named local moment.

15 y 21y22 C 2 y 21y22 B T 1 y 23 B T 1 y 23 x 11 x 12 x interactions B S2 B S3 y 31 y 32 y 33 x 11 x 12 x 13 C 1 11 interactions B S2 B S3 y 31 C 3 y 32 y 33

16 Case of Helmholtz Green kernel in the framework of integral operators: e.g. [M] i j = Γ Γ G(x, y)ϕ j (y)ϕ i (x)dγ(y)dγ(x), i, j where G is the Helmholtz fundamental solution, G(x, y) = eik x y 4π x y, k the wavenumber, (ϕ i ) i a set of finite-element basis functions.

17 An efficient calculation of the matrix-vector product [M]X given thanks to such an expansion for i far from j : [M] i j S p=1 c p g (p) i,b T (p) (p) f B, B j, B, B/B suppϕ i B/ B suppϕ j with g (p) i,b = f (p) j, B = c p = B suppϕ i B suppϕ j ik (4π) 2 w p, e ik<s p,x C B > ϕ i (x)dγ(x), e ik<s p,y C B> ϕ j (y)dγ(y), and T (p) B, B the translation operator from the FMM box B to the FMM box B...

18 ... the translation operator is given by the expression and T (p) L B, B = ( i) l (2l + 1)h (1) l (k C B C B )P l (cos(s p, C B C B)), l=1 w p, s p are the quadrature rule on the unit sphere due to Funk-Hecke formula, S p=1 comes from the discretization of the Funk-Hecke formula, L l=1 is a truncation of the Gegenbauer addition theorem, C B is the center of the FMM box B, h (1) l is the spherical Hankel function of the first kind of degree l, P l is the Legendre polynomial of degree l, L and S are estimated thanks to the empirical formulae L = kd + C(kd) 1/3 S = (L + 1)(2L + 1) with d = diam. of the boxes.

19 Case of Coulomb potential in the framework of particles: [M] i j = G(x i, y j ) with G(x, y) = 1 x y. For i far from j : [M] i j where g (l,k) i,b L l=0 l k= l g (l,k) i,b L n n=0 m= n = ρl i Y k l (α i, β i ) and f (n,m) j, B T ((l,k),(n,m)) B, B f (n,m) j, B = ρ n j Y m n (α j, β j ),, with (ρ i, α i, β i ) (x i C B ) and (ρ j, α j, β j ) (y j C B), and with A m n = T ((l,k),(n,m)) B, B = i k m k m A m n A k l ( 1) n A m k l+n ( 1) n (n m)!(n+m)! and (ρ st, α st, β st ) (C B C B). Y m k l+n (α st, β st ) ρ l+n+1 st

20 p=1 Specificities of FastMMLib Aim: a generic FMM library written in C++ Initial idea: the library should work with any expansion of the form S S G(i, j) = c p g (p) i,b T (p, p) (p, p) f B, B j, B, for all (i, j) such that i far from j; B/i B B/j B... should deal with particles or finite-elements d.o.f.;... should integrate the regularized form (RFMM); p=1 P. CHARTIER, E. D., E. FAOU. A regular fast multipole method for geometric numerical integrations of Hamiltonian systems. BIT, 50(1):23 40, should enable a particle or a d.o.f. to belong to several boxes. A FMM box may be identified by several centroids. L. YING, G. BIROS, D. ZORIN. A kernel-independent adaptive fast multipole algorithm in two and three dimensions. J. Comput. Phys., 196: , 2004.

21 Crucial choices The library deals with the translations, the geometrical structure, the interpolations,... the organization of the fast calculation. FastMMLib creates a geometrical object and provides usual Green kernels; The user can specify his own kernel (and then the expansion); FastMMLib provides a generic particle structure; The user should specify his own particles. To take in consideration the user s context (particles, FE-d.o.f.). The far moments and local moments are calculated using user s functions: or g (l,k) i,b = ρl i Y k l (α i, β i ) with (x i C B ) (ρ i, α i, β i ) g (p) i,b = the same for the close interactions. B suppϕ i e ik<s p,x C B > ϕ i (x)dγ(x),

22 Structure of FastMMLib Classes related to the geometry: particles, FMM boxes, levels of the oc-tree Classes related to the physics: Green kernels Classes defining the FMM object: far/local moments, far/local fields, translation operators, FMM product

23 User-connected classes in FastMMLib Geometry: NVParticle and IEMParticle which derive from Particle, SetOfParticles, ListOfLevelsParameters, Levels. Physics: Helm3DKernelGeg which derives from GreenKernel. FMM object: FMM precalculation: call of MomentContribution, TransferOperator, CloseInteraction, matrix-vector product: call of FieldContribution.

24 The classes NVParticle and IEMParticle : Data members int tag ; // numbering in the corresponding initial geometry. const fmm real t* centroid ; // centroid of the geometrical particle. fmm real t diam ; // charateristic length of the particle. public Particle Member functions NVParticle(fmm real t* centroid, fmm real t diam); // constructor; const fmm real t* centroid(); fmm real t diam(); bool contained(const fmm real t* box centroid, fmm real t box diam );.../...

25 void MomentComputation( void f(const fmm real t* y, fmm complex t* res), or fmm complex t* res ); // define the operator: f B suppϕ i f f(this->centroid()) f(x)ϕ i (x)dγ(x) = K B suppϕ i K f(x)ϕ i (x)dγ(x) with f the function associated with the definition of the moments of the kernel expansion, e.g. f(x) = ρ l x Y k l (α x, β x ) with (x C B ) (ρ x, α x, β x ) or f(x) = e ik<s p,x C B > B

26 The class ListOfLevelsParameters: Data members fmm real t magnitude exponent ; fmm number t min level, max level ; fmm real t initial dilatation ; fmm number t neighbor order ; fmm real t rfmm dilatation ; Member functions A unique constructor with default values set...(), get...(); // set and access functions

27 The class Helm3DKernelGeg : public GreenKernel Data members from GreenKernel const Levels& geom levels ; fmm number t space dim, far dim, local dim, close dim ; KernelFMMFormat kernel format ; vector<fmm number t> S, St ; Specific data members fmm real t wave number ; fmm number t truncation parameter ; // coefficient to define the truncation parameter of the Gegenbauer theorem. fmm real t* vx y ; // local working variable. std::vector<std::vector<fmm real t> > sphere directions ; // for the Funk-Hecke formula.

28 The class Helm3DKernelGeg : public GreenKernel Member functions Helm3DKernelGeg(const Levels& userlevels, const fmm real t wavenumber, // a constructor. const fmm real t truncationcoefflog=0, const fmm real t truncationcoeff1 3=2.6);.../...

29 void KernelDefFunc(const fmm real t* vx, const fmm real t* vy, fmm complex t* res); // defines the kernel function. void FarMomentDefFunc(const fmm number t& nsp, const fmm real t* rx, fmm complex t* res); // defines the far-moment function. void LocalMomentDefFunc(const fmm number t& nsp, const fmm real t* rx, fmm complex t* res); // defines the local-moment function. void F2LtransOpDefFunc(const fmm number t& nsp, const fmm number t& ndir, fmm complex t* res); // defines the operator involved in the F2L transfers.

30 An example of use: #include "fmm config.h"... int main() {... vector<nvparticle*> MyParticles; //... define them here... SetOfParticles MySetsofParticles(...,MyParticles); ListOfLevelsParameters MyListOfParams(...); Levels MyLevels(MySetsofParticles, MyListOfParams); Helm3DKernelGeg MyKernel(MyLevels); FMM MyFMM(MyKernel);... V = MyFMM.MatVec(U);... }

31 State of the art Geometry: validated Physics: Helm3DKernelGeg validated SLFMM object: single level FMM validated MLFMM object: some field constructors still have to be implemented A first public version expected this year ;-) XLiFE++ use: to be plugged into XLiFE++ nextly

32 SLFMM convergence validation for Helmholtz kernel expansion based on Gegenbauer addition theorem Interaction between two particles on a same surface (1/2) FMM level 2 2 FMM relative error, case k=3 2 FMM relative error; case k= Log10(err) -4-6 Log10(err) Truncation parameter L Truncation parameter L

33 Interaction between two particles on a same surface (2/2) FMM level 2 on left, FMM level 3 on right 2 FMM relative error; case k=30-2 FMM relative error; case k= Log10(err) -4-6 Log10(err) Truncation parameter L Truncation parameter

34 Interaction between two particles on two surfaces FMM level 2 on left, FMM level 3 on right 0 FMM relative error, case k=3-2 FMM relative error; case k= Log10(err) -6-8 Log10(err) Truncation parameter L Truncation parameter

35 What else? the transient problems: the FMM expansion derived in A. A. ERGIN, B. SHANKER, E. MICHIELSSEN. Fast Evaluation of Three-Dimensional Transient Wave Fields Using Diagonal Translation Operators. J. Comput. Phys., 146: , other fast methods integration of High Order method and H-matrices O. P. BRUNO, L. A. KUNYANSKY. A Fast High Order Algorithm for the Solution of Surface Scattering Problems: Basic Implementation, tests, and Applications. J. Comput. Phys., 169(1):80 110, May W. HACKBUSCH. A Sparse Matrix Arithmetic Based on H-Matrices. Part I: Introduction to H-Matrices. Computing, 62:89 108, 1999.

36 Thank you for your attention and your participation to the school