Fast methods for integral equations

Transcription

1 Fast methods for integral equations Matthieu Aussal Research engineer Centre de Mathématiques Appliquées de l École Polytechnique Route de Saclay Palaiseau CEDEX France Journée de rentrée du CMAP - 03 octobre 2018

2 Introduction Integral representation Integral equations Acceleration Gypsilab Conclusion

3 Main research topics and collaborations Spatial audio (3D audio) : VST plug in for sound engineers (F. Salmon, CNSMDP), Audio guidance for blind people (S. Ferrand, MCV ), Classroom sonorization (Mon cartable connecté). Auralization : Numerical room acoustics for archeology (R. Gueguen), Real-time room acoustic for virtual reality, Augmented reality for museum visit (PSC, MusiX) Finite and boundary element method (FEM, BEM) : Gypsilab : Matlab framework for fast prototyping (F. Alouges) Fast methods for High Performance Computing : DDM, H-Matrix, FFM, ray-tracing, etc. Applications (scattering, 3D audio, eeg, Stokes flows, etc.)

4 Past, present and future collaborations

5 Some selected softwares MyBino (freeware) : Professional VST plug in for binaural rendering. Just4rir (open-source) : Blender plug in for room acoustics by ray-tracing. Gypsilab (open-source part) : Matlab framework for fast FEM-BEM prototyping. Gypsilab (private part) : High-Performance Computing for FEM-BEM problems.

6 Introduction Integral representation Integral equations Acceleration Gypsilab Conclusion

7 A boundary problem for acoustic The full space Ω is separated in two subspaces Ω i and Ω e by a boundary Γ, smooth and oriented along normal n. For u : Ω \ Γ C, we define : u i = u Ωi u e = u Ωe, µ = [u] = u i u e, λ = [ n u] = n u i n u e. Note : Γ is a singularity of Ω Distribution theory!

8 Theorem of integral representation If u satisfies Helmholtz equation and Sommerfeld condition : ( u i + k 2 u i ) = 0 in Ω i, ( u e + k 2 u e ) = 0 in Ω e, lim r ( ru e + iku e ) = 0, r + then u satisfies an integral representation x Ω i Ω e : u(x) = Γ G(x, y) G(x, y)λ(y)dγ y µ(y)dγ y, Γ n y with G(x, y) stand for the oscillatory Green kernel : G(x, y) = eik x y 4π x y. Note : Volumic data are fully determined by surfacic unknowns.

9 Evaluation of boundary integrals Regular integration are done using a quadrature rule (y q, γ q ) 1 q Nq for the boundary Γ : Γ N q G(x, y)λ(y)dγ y γ q G(x, y q )λ φ (y q ). q=1 If necessary, singular integrations are done analytically... and it s hard! (A. Lefebvre-Lepot) Note : Regular integration can be seen as a discrete convolution on a non-uniform grid... meshless technics!

10 Numerical application Integral representation of the elementary solution E = eik x 4π x. Boundary Γ is a unit sphere, meshed with triangles and wave number is fixed at k = 5 : u(x) = Γ G(x, y) ne(y)dγ y Γ Reference E(x), computed G(x,y) n y E(y)dΓ y analytically in full space

11 Introduction Integral representation Integral equations Acceleration Gypsilab Conclusion

12 Calderon projectors Following the theorem of integral representation, if u satisfy : u(x) = Γ G(x, y) G(x, y)λ(y)dγ y µ(y)dγ y, Γ n y then u is solution of Helmholtz equation. Boundary traces are given respectively by interior and exterior Calderon projectors x Γ : ( ) ( ) ( ) +Id ui = 2 D S µ n u +Id i N 2 + D t, λ ( ) ( ) ( ) Id ue = 2 D S µ n u Id e N 2 + D t, λ Sλ(x) = Dµ(x) = Γ Γ G(x, y)λ(y)d y, ny G(x, y)µ(y)d y, D t λ(x) = Nµ(x) = Γ Γ nx G(x, y)λ(y)d y, nx ny G(x, y)µ(y)d y.

13 Integral formulation : Example with Dirichlet scattering 1. Formulate problem considering u e = u tot u inc : ( u e + k 2 u e ) = 0 in Ω e, lim r + r ( ru e + iku e ) = 0, u tot = 0 u e = u inc in Γ. 2. Choose the right projector (according to normal n) : ( ) ( ) ( ) Id ue = 2 D S µ n u Id e N 2 + D t. λ 3. Choose a jump extension : 4. Solve integral equation on Γ : u i = u e µ = 0. Sλ = u e = u inc. 5. Compute the total field on Ω e : u tot = Sλ + u inc.

14 Evaluation of boundary operators Considering a finite elements basis (φ n (y)) 1 n Ndof boundary functions : to represent λ(y) λ φ (y) = N dof n=1 λ n φ n (y) y Γ, Integral operators can be discretized using one of the following methods : Collocation : S ij = G(x i, y)φ j (y)d y, Γ Galerkin : S ij = φ i (x)g(x, y)φ j (y)d y. Γ Note : Using a discrete quadrature (y q, γ q ) 1 q Nq of the boundary Γ, both methods need the computation of a full matrix G xy. Γ

15 Numerical application Integral representation of the solution of an integral equation, satisfying Dirichlet condition. Boundary Γ is a unit sphere, meshed with triangles and wave number is fixe at k = 5 : µ = 0 Sλ = u inc λ = 0 Note : Galerkin method were used with P 1 elements. ( ) Id 2 D µ = u inc

16 Introduction Integral representation Integral equations Acceleration Gypsilab Conclusion

17 Example with a regular kernel All integral formulations need discrete convolutions with Green kernel on non-uniform grids (x i ) i [1,Nx ] and (y j ) j [1,Ny ] in R 3 : N y u i = G(x i, y j )v j. j=1 For example, considering a regular Green kernel : K(x, y) = x y 2 = x 2 2x y + y 2, Convolution by K can be rewritten for all i : N y u i = x i 2 v j 2x i y j v j + y j 2 v j j=1 Separate x and y : Complexity O(N x N y ) O(N y ) + O(Nx)! N y j=1 N y j=1

18 Example with a singular kernel Context : G(x, y) = eik x y 4π x y is locally singular ( x y < ɛ). Idea : Use "smoothing" properties issued from far distances. Method : Divide and conquer with distance criterion. Algorithm(s) : Hierarchical tree and leaves compression. Initialization : Considering a space sampling (x i ) i [1,Nx ] and (y j ) j [1,Ny ] in R 3, a Green kernel matrix is given by : G xy = G(x 1, y 1 )... G(x 1, y Ny ) G(x Nx, y 1 )... G(x Nx, y Ny ) = [G 0 ]

19 Hierarchical tree G xy = [G 0 ] Step 0

20 Hierarchical tree [ ] G1 G G xy = 2 G 3 G 4 Step 1

21 Hierarchical tree G xy = G 11 G 12 G 21 G 22 G 13 G 14 G 21 G 22 G 31 G 32 G 41 G 42 G 33 G 34 G 43 G 44 Step 2 At level n, each n-tuple is unique and define the block hierarchy.

22 Hierarchical tree Step 3 G xy = G 111 G G 221 G 222 G 113 G G 223 G G 331 G G 441 G 442 G 333 G G 444 G 444 Green leaves are far away no recursion.

23 Hierarchical tree Step 4

24 Hierarchical tree Step 5

25 Hierarchical tree Step 6

26 Hierarchical tree Step 7

27 Hierarchical tree and finally...

28 Hierarchical tree The last level is reached! Green cells far interactions Red cells close interactions

29 Leaves compression Idea : Use "smoothing" properties issued from far distances to interpolate interactions data compression! Method : Considering G ij = (G(x i, y j )) i I,j J a block of far interactions, indexed by I [1, N x ] and J [1, N y ] : Weak interpolation : G(x, y) = k I k (x, ξ k )G(ξ k, y) G IJ = A Ik B kj. Strong interpolation : G(x i, y j ) = I k (x, ξ k )G(ξ k, ξ l )I l (ξ l, y) G IJ = A Ik T kl B lj. k,l

30 Final matrix form Full Weak Strong

31 Non exhaustive list of fast solvers Method Class Interpolation Complexity Algebra Full - - N 2 (yes) H 1 -matrix weak SVD/ACA N 3/2 yes H 2 -matrix strong Lagrange N log N yes/no SCSD strong Cardinal sine N 6/5 no FMM/FFM strong Plane wave N log N no SVD/ACA : G = AB t with A and B low-rank matrices, Lagrange : G = A T B t, A and B low-rank and T small, Plane wave : G(x, y) = ik lim 16π 2 L + s S 2 e iks xm 1T M L 1 M 2 (s)e iks M2y ds Cardinal sine : G(x, y) = k + (4π) 2 n=0 α n ρ n S 2 e ikρns x e ikρns y ds

32 Numerical application Context : For all particles (x i ) i [1,N] and (y j ) j [1,N] in R 3 : u(x i ) = N j=1 U = M V. with M M N (R), U and V R N. 1 4π x i y j v(y j), Test case : (x i ) i [1,N] and (y j ) j [1,N] are uniformly distributed on the unit sphere S 2. Computer : 12 core at 2.7 GHz, 256 GO de ram, Matlab R2014a.

33 Numerical Application N Time (s) Memory peak Mo Mo Full product (single thread) N Build time (s) MV time (s) LU time (s) Mem Mo Go Go H 1 -matrix product and algebra with ɛ = 10 3 (single-thread)

34 Numerical Application N Time 1 core (s) Time 12 cores (s) Error L2 Mem Mo Mo Mo Go Go Go Fast and Furious product with ɛ = 10 3 (multi-thread) Note : Assembly, full matrix would weigh 8 exa-octets! (double precision : 8N Bytes)

35 Introduction Integral representation Integral equations Acceleration Gypsilab Conclusion

36

37 But what is Gypsilab? Gypsilab is an open-source framework for Matlab, freely available under GPL 3.0, composed of object-oriented libraries : openmsh : Multi-dimensional mesh management. opendom : Numerical integration and variational calculation, joint work with François Alouges. openfem : Finite Element method for 2D and 3D shapes, joint work with François Alouges. openhmx : H-Matrix compression and algebra. openray : Accelerated ray-tracing for HF problems, joint work with Robin Gueguen. Designed to combine industrial requirements with fast prototyping.

38 Licensed add-on for Gypsilab Open source part of Gypsilab is accompanied by licensed add-on, also composed of object-oriented libraries : libhbm : Block matrix computation and algebra. Domain Decomposition Method, H-Matrix Parallelization, multi-physic coupling, out-of core computation, etc. libffm : Fast and Furious Method. High Performance Computing for very large BEM problems.

39 Application 1 : Head-Related Tranfert Function (3D audio) Helmholtz equation Neumann boundary condition P 1 finite element 10 3 compression accuracy dof at 8 khz H-Matrix exact solver Figure Acoustic scatering by human head, mesh from SYMARE database.

40 Application 2 : Radar detection (ISAE 2016) Maxwell equation PEC boundary condition, CFIE RWG finite element 10 3 compression accuracy dof at 2.5 GHz H-Matrix exact solver Figure Electromagnetic scattering by an UAV, workshop ISAE, mesh from Onera.

41 Application 3 : Sonar detection (BeTSSI) Helmholtz equation Neumann boundary condition P 1 finite element 10 3 compression accuracy dof at 1600 Hz Prec. FFM, DDM (32 cores)

42 Application 3 : Sonar detection (BeTSSI) Helmholtz equation Neumann boundary condition P 1 finite element 10 3 compression accuracy dof at 1600 Hz Prec. FFM, DDM (32 cores)

43 Application 4 : Space launcher scattering Maxwell equation PEC boundary condition, CFIE RWG finite element 10 3 compression accuracy dof at 2 GHz Prec. FFM, DDM (32 cores) Note : Without compression, full operator would weigh 57 Po! (complex double precision : 16N Bytes)

44 Application 4 : Space launcher scattering Maxwell equation PEC boundary condition, CFIE RWG finite element 10 3 compression accuracy dof at 2 GHz Prec. FFM, DDM (32 cores) Note : Without compression, full operator would weigh 57 Po! (complex double precision : 16N Bytes)

45 Application 4 : Space launcher scattering Maxwell equation PEC boundary condition, CFIE RWG finite element 10 3 compression accuracy dof at 2 GHz Prec. FFM, DDM (32 cores) Note : Without compression, full operator would weigh 57 Po! (complex double precision : 16N Bytes)

46 Introduction Integral representation Integral equations Acceleration Gypsilab Conclusion

47 Conclusion Communications Kalman filterring with H-Matrix (P. Moireau) Electro-encephalogram with H-Matrix (A. Gramfort et al) Vibro-acoustic with FFM (Naval group) Add numericals methods (Finite volume, finite difference, etc). Mesh generation (2D, 3D, coupling) Reach world s records ( Maxwell, Stokes) Download gypsilab : Il faut se débarrasser des casse-têtes. On ne vit qu une fois. Charles Aznavour,