Fast Sparse Spectral Methods for Higher Dimensional PDEs

Transcription

1 Fast Sparse Spectral Methods for Higher Dimensional PDEs Jie Shen Purdue University Collaborators: Li-Lian Wang, Haijun Yu and Alexander Alekseenko Research supported by AFOSR and NSF ICERM workshop, June 3-7, 2013

2 Outlines Approximation by hyperbolic cross Sparse grids for bounded domains Sparse grids for unbounded domains Application to electronic Schrödinger equation Adaptive sparse tensor products with multi-wavelets for the Boltzmann collision operator Concluding remarks

3 Motivations Electronic Schrödinger equation: find the eigenvalues and eigenfunctions of the Hamilton operator H = 1 2 N i i=1 N K i=1 ν=1 Z N ν x i a ν + i=1 N j>i 1 x i x j, where x 1,, x N R 3 are coordinates of N electrons. Boltzmann Equation: f t + p m xf = G(f, f ) where x, p are position and momentum of particle in R 3, m is the particle mass and f is the density distribution function.

4 Fokker-Planck Navier-Stokes equations for FENE dumbbell model: t u + (u x )u + x p = ν Re xu + 1 ν Re De x τ p, x u = 0, t f + x (uf ) + q ( x u qf ) = 1 De q (F(q)f ) + 1 De qf, τ p (x, t) = (F(q) q) f (x, q, t)dq. W x Ω R 3, Ω is the fluid space; q W R 3, W is the configuration space; F(q) is the configuration force.

5 A model problem Consider the model elliptic equation: αu u = f in Ω = ( 1, 1) d ; u Ω = 0. Given an approximation space V n and an interpolation operator I n, the (weighted) spectral-galerkin method is to find u n V n such that ( u n, (v n ω)) = (I n f, v n ) ω, v n V n.

6 A model problem Consider the model elliptic equation: αu u = f in Ω = ( 1, 1) d ; u Ω = 0. Given an approximation space V n and an interpolation operator I n, the (weighted) spectral-galerkin method is to find u n V n such that ( u n, (v n ω)) = (I n f, v n ) ω, v n V n. Error estimate: (u u n ) L 2 ω inf (u v n ) L 2 v n V ω + f I n f L 2 ω. n approximation error interpolation error

7 Usual spectral method: Let V n = P n+1 H 1 0 (Ω) where P n+1 is the space of polynomials of degree n + 1 in EACH variable. N = dim(v n ) = n d ; inf vn V n (u v n ) L 2 ω n 1 s u H s ω N (1 s)/d u H s ω. The # of degree of freedom increases exponentially fast with d the convergence rate deteriorates rapidly as d increases. This is the so called Curse of dimensionality!

8 Hyperbolic cross (Korobov, Babenko 57) The hyperbolic cross is defined as { } d K(n) = k Z d + : max(k i, 1) n. i=1 The cardinality of K(n) is of order n(log n) d 1.

9 Hyperbolic cross (Korobov, Babenko 57) The hyperbolic cross is defined as { } d K(n) = k Z d + : max(k i, 1) n. i=1 The cardinality of K(n) is of order n(log n) d 1. d d Set V n = span{ φ ki (x i ) : max(k i, 1) n}, i=1 i=1 in what functional spaces does V n provide good approximations?

10 Table : Comparison of grids and hyperbolic cross n d=2 d=3 d=4 K(n) K(n) Grids K(n) nlog(n) Grids K(n) nlog(n) 2 Grids K(n)

11 Korobov spaces for periodic functions Given f L 2 p(0, 2π) d and its Fourier series f (x) = ˆf (k)e ik x with ˆf (k) = 1 f (x)e ik x dx. Ω k Z d Ω Let us define r(k) := d j=1 max( k j, 1), k = (k 1,..., k d ), then, the Korobov space of order α R is { } Kp α := f L 2 p(ω) : ˆf (k) 2 r(k) 2α < k Z d { q q d = x q x q d d f L 2 p(ω) : 0 q j α, 1 j d }.

12 Setting X n := {f : f (x) = ˆf } r(k) n (k)e ik x. Then, one can prove inf v v n K r v p n r s v K s p. (e.g. Griebel & Hamaekers 2007) n X n

13 Setting X n := {f : f (x) = ˆf } r(k) n (k)e ik x. Then, one can prove inf v v n K r v p n r s v K s p. (e.g. Griebel & Hamaekers 2007) n X n Remarks: The estimate still depends on d since N = O(n(log n) d 1 ). It is also possible to remove the dependence on d by using the optimized (energy based) hyperbolic cross space (Bungartz & Griebel 04): K σ (n) = { k Z d + : d i=1 max(k i, 1) k σ n 1 σ }. Then, for σ (0, 1), Card(K σ (n)) = O(n). In this case, the estimate holds only for σ r s.

14 Setting X n := {f : f (x) = ˆf } r(k) n (k)e ik x. Then, one can prove inf v v n K r v p n r s v K s p. (e.g. Griebel & Hamaekers 2007) n X n Remarks: The estimate still depends on d since N = O(n(log n) d 1 ). It is also possible to remove the dependence on d by using the optimized (energy based) hyperbolic cross space (Bungartz & Griebel 04): K σ (n) = { k Z d + : d i=1 max(k i, 1) k σ n 1 σ }. Then, for σ (0, 1), Card(K σ (n)) = O(n). In this case, the estimate holds only for σ r s. These estimates were extended to the non-periodic case and unbounded domains in S. & Wang (SINUM 10).

15 Korobov type spaces for non-periodic functions Let ω α,β = (1 x) α (1 + x) β and J n (α,β) be the Jacobi weight and Jacobi polynomial with index α, β > 1. We define the multi-dimensional Jacobi weights and Jacobi polynomials by ωᾱ, β( x) = Π d i=1ω α i,β i (x i ), β Φᾱ, ( x) = Π k d i=1j α i,β i k i (x i ). We then define (for any integer t 0) K t ᾱ, β := {u : u 2 K t := ᾱ, β r t r 2 u ωᾱ+ r, β+ r < }.

16 Projection errors in the multi-d case Denote Pᾱ, β n Πᾱ, β n = span{φᾱ, β ( x) : Π k d i=1 max(k i, 1) n}, and define be the orthogonal projector defined by : L 2 ωᾱ, β Pᾱ, β n (u Πᾱ, β n u, v K ) ω ᾱ, β = 0, v n Pᾱ, β n. Theorem. (S. & Wang 10) For u K s ᾱ, β and 0 t < s, we have u β Πᾱ, n u K t n t s u K s, ᾱ, β ᾱ, β where u 2 K s ᾱ, β := r =s r u 2 ωᾱ+ r, β+ r.

17 Projection errors in the multi-d case Denote Pᾱ, β n Πᾱ, β n = span{φᾱ, β ( x) : Π k d i=1 max(k i, 1) n}, and define be the orthogonal projector defined by : L 2 ωᾱ, β Pᾱ, β n (u Πᾱ, β n u, v K ) ω ᾱ, β = 0, v n Pᾱ, β n. Theorem. (S. & Wang 10) For u K s ᾱ, β and 0 t < s, we have where u β Πᾱ, n u K t n t s u K s, ᾱ, β ᾱ, β u 2 K s := ᾱ, β r =s r u 2 ωᾱ+ r, β+ r. Approximation results in usual H 1 -norm and for elliptic equations are also established in S. & Wang 10.

18 Figure : Spectrum of a highly anisotropic exact solution

19 Figure : Convergence history: comparison with the full grid

20 Figure : Spectrum of an isotropic exact solution

21 L2 error of solution e-06 d=2 full d=3 full d=4 full d=5 full d=1 d=2 sparse d=3 sparse d=4 sparse d=5 sparse L2 error 1e-08 1e-10 1e-12 1e-14 1e e+06 # of grid points Figure : Convergence history: comparison with the full grid

22 Sparse grids Sparse grid construction (Smolyak 63): Start from a sequence of (preferably nested) univariate quadrature/interpolation formulas: n i Q i f := f (xj i )ωj i, i = Q i Q i 1 with Q 0 = 0. j=1 Define a multidimensional quadrature rule by: Q (d) l f := ( i1 id )f. d i l

23 Figure : Q (2) 5 based on Chebyshev-Gauss-Lobatto quadrature.

24 Sparse grids in frequency space Similarly, we can define sparse grids in frequency space. For example, by splitting the 1-D basis {P 0 (x), P 1 (x), P 2 (x),...} into 1 = {P 0, P 1, P 2 }, 2 = {P 3, P 4 }, 3 = {P 5, P 6, P 7, P 8 }, we can define sparse grids in frequency space by H (d) l := ( i1 id ). d i l

25 Figure : H (2) 5 in frequency space.

26 Optimized hyperbolic cross/sparse grid

27 Hierarchical basis and interpolation on sparse grid The key ingredient for constructing a fast transform between the space grids in physical and frequency spaces is the Hierarchical basis.

28 Hierarchical basis and interpolation on sparse grid The key ingredient for constructing a fast transform between the space grids in physical and frequency spaces is the Hierarchical basis. Given a sparse grid generated by nested 1-D quadratures with the index set I i = {0, 1,, n i 1.} A set of function {φ k } form a hierarchical bases if φ k (x i j ) = 0, j if k n i. Then, we can rearrange {x i j } as {x j} and define an interpolation operator by f (x j ) = k I i b k φ k (x j ), for any j I i, i = 1, 2,... where {b k } do not depend on the level index i.

29 Figure : Left: The 1-d linear hierarchical finite element bases; Right: The 2-d construction of sparse grid (from Bungartz and Griebel 2004)

30 Figure : Hierarchical basis functions with Chebyshev-Gauss-Lobatto quadrature: φ k = T k, for k I 1 ; φ k = T k T 2i k, for k Ĩ i = I i \I i 1, i > 1.

31 Multi-dimensional case: Define the multi-d index set corresponding to the sparse grid I (d) l := (Ĩ i1 Ĩ id ), d i l and the multi-dimensional hierarchical bases by φ k (x) = Π d i=1φ ki (x i ). Then, we can define the interpolation operator U (d) l on the sparse grid by (U (d) l f )(x j ) = b k φ k (x j ), j I (d) l. k I (d) l The transform between the values at the sparse grid and the coefficients of the hierarchical basis can be performed with quasi-optimal computational complexity!

32 Implementation of hyperbolic-cross/sparse-grid solver αu u = f, x Ω = ( 1, 1) d, u Ω = 0. Sparse grid in frequency space: I q d = { (k 1, k 2,..., k d ) : 0 k s < 2 is 1, i s 0, i 1 = q }, { } V q d = φ k, k I q d ; Sparse grid in physical space: J q d = { (k 1, k 2,..., k d ) : 0 k s < 2 is + 1, i s 0, i 1 = q }, X q d = { x j, j J q } d ; U q d the interpolation on X q d. Find u q d V q d such that α(u q d, v) ω ( u q d, v) ω = (U q d f, v) ω, v V q d.

33 Choice of basis functions for V q d Remark: While the hierarchical basis plays a critical role in fast transform, it lacks good orthogonality property.

34 Choice of basis functions for V q d Remark: While the hierarchical basis plays a critical role in fast transform, it lacks good orthogonality property. Chebyshev-Galerkin method: φ k (x) = T k (x) T k+2 (x), ω(x) = (1 x 2 ) 1/2 1-d mass matrix: banded; 1-d stiff matrix: triangular; the multi-d stiffness matrix is not sparse.

35 Choice of basis functions for V q d Remark: While the hierarchical basis plays a critical role in fast transform, it lacks good orthogonality property. Chebyshev-Galerkin method: φ k (x) = T k (x) T k+2 (x), ω(x) = (1 x 2 ) 1/2 1-d mass matrix: banded; 1-d stiff matrix: triangular; the multi-d stiffness matrix is not sparse. Legendre-Galerkin method: φ k (x) = L k (x) L k+2 (x), ω(x) = 1 1-d mass matrix: banded; 1-d stiff matrix: diagonal; Legendre-Gauss type points not nested; lack of fast transform.

36 Choice of basis functions for V q d Remark: While the hierarchical basis plays a critical role in fast transform, it lacks good orthogonality property. Chebyshev-Galerkin method: φ k (x) = T k (x) T k+2 (x), ω(x) = (1 x 2 ) 1/2 1-d mass matrix: banded; 1-d stiff matrix: triangular; the multi-d stiffness matrix is not sparse. Legendre-Galerkin method: φ k (x) = L k (x) L k+2 (x), ω(x) = 1 1-d mass matrix: banded; 1-d stiff matrix: diagonal; Legendre-Gauss type points not nested; lack of fast transform. Chebyshev-Legendre-Galerkin method (Don & Gottlieb 94, S. 96)

37 Linear algebraic system Denote φ k (x) = φ k1 (x 1 ) φ kd (x d ), then V q d = { φ k, k I q d }. The Galerkin method leads to where (αm + S)u = f, M = (m k1,j 1 m kd,j d ) k,j I q, d S = (s k1,j 1 m k2,j 2 m kd,j d ) k,j I q + + (m k1,j 1 m k2,j 2 s kd,j d ) d k,j I q, d m k,j = (φ k, φ j ) ω, s k,j = ( φ k, φ j ) ω f = (f k ) k I q d, f k = (U q d f, φ k).

38 (a) (b) (c) Figure : Sparsity of Legendre-Galerkin method: (a) stiff X 8 2 ; (b) system X 8 2 ; (c) stiff X 9 3 ; (d) system X 9 3. (d)

39 d q n nnz 0 nnz 1 cond 1 pcond Table : nnz 0 : number of non-zeros of stiff matrix; nnz 1 : number of non-zeros of system matrix (α = 1); cond 1 : the condition number of system matrix (α = 1); pcond 1 : the condition number of system matrix (α = 1) with diagonal pre-conditioner.

40 PCG with fast matrix-vector product algorithm For the Chebyshev sparse Galerkin method: The evaluation of f can be performed using the fast transform algorithm. The stiff matrix S is not sparse but the matrix-vector product Mu, Su can be performed with quasi-optimal computational complexity. Solve the linear system by a preconditioned CG type method How to construct an optimal pre-conditioner is still an open problem.

41 PCG with fast matrix-vector product algorithm For the Chebyshev sparse Galerkin method: The evaluation of f can be performed using the fast transform algorithm. The stiff matrix S is not sparse but the matrix-vector product Mu, Su can be performed with quasi-optimal computational complexity. Solve the linear system by a preconditioned CG type method How to construct an optimal pre-conditioner is still an open problem. Not optimal computational complexity, but memory requirement is small since it does not need explicit formation of stiffness/mass matrices.

42 Direct sparse solvers In order to use an efficient sparse solver, we need to form explicitly the system matrix only possible with Chebyshev-Legendre sparse grid method The evaluation of f is done by the fast Chebyshev-Legendre transform (by FMM, Alpert & Rokhlin 91). The sparse mass and stiff matrices can be assembled with a fast algorithm. Solve the linear system by using an efficient sparse solver UMFPack (T. Davis) competitive with PCG in some cases AMG (Y. Notay) competitive with PCG in some cases SuperMF (J. Xia et al.) can lead to quasi-optimal computational complexity.

43 Direct sparse solvers In order to use an efficient sparse solver, we need to form explicitly the system matrix only possible with Chebyshev-Legendre sparse grid method The evaluation of f is done by the fast Chebyshev-Legendre transform (by FMM, Alpert & Rokhlin 91). The sparse mass and stiff matrices can be assembled with a fast algorithm. Solve the linear system by using an efficient sparse solver UMFPack (T. Davis) competitive with PCG in some cases AMG (Y. Notay) competitive with PCG in some cases SuperMF (J. Xia et al.) can lead to quasi-optimal computational complexity. Quasi-optimal computational complexity, but memory requirement is large.

44 Sparse grid by Hermite Gauss quadrature The natural choice : Construct Sparse grids from the Hermite-Gauss points and Laguerre-Gauss points. The problem: The Hermite-Gauss and Laguerre-Gauss points are not nested leads to too many points in the sparse grid so it is not computationally effective Figure : 5 Left: Hermite sparse grid X2 5 ; Right: the corresponding index

45 Mapped Chebyshev method Given a mapping x = x(ξ) : ( 1, 1) R and its inverse g ξ = ξ(x) : R ( 1, 1). The mapped Chebyshev functions ˆT k (x) = T k (ξ(x))µ(ξ(x)), µ(ξ) = ω(ξ)/x (ξ) with ω(ξ) = 1/ 1 ξ 2 which satisfies ( ˆT k, ˆT j ) = R ˆT k (x) ˆT j (x)dx = 1 1 T k (ξ)t j (ξ)ω(ξ)dξ = δ kj. The convergence rate will depend on the rate of decay at infinity A class of rational mapping: ( 1, 1) R: x (ξ) = L (1 ξ 2, r 0, ) 1+r/2

46 Two most useful cases: r = 0, 1: { L 1+ξ 2 log 1 ξ x(ξ) =, r = 0 { tanh( 1 Lξ, r = 1 ; ξ(x) = L x) r = 0 x, r = 1. 1 ξ 2 x 2 +L Figure : Mapped Chebyshev sparse grids. Left: X2 5 with r = 1. X 5 2 with r = 0; Right:

47 Let us define our hyperbolic approximation space by with X d N := span{ Tk : k Υ H N}, Υ H N = {k N d 0 : 1 k mix := d j=1 } max{1, k j } N. We define the orthogonal projection π d N : L2 (R d ) X d N, and the mapped derivative D x u := a(x) dû dx dx u(x) with a(x) =, û(x) = dξ µ(ξ(x)), d (1 ξj 2 ) (1+r)/2+k j. D k x u = D k 1 x 1 D k d x d u, ϖ (1+r)/2+k = j=1

48 Approximation results Theorem. For r 0 and any u K m (R d ), we have and π d N u u L 2 (R d ) CN m u K m (R d ), m 0, (π d N u u) L 2 (R d ) CN 1 m u K m (R d ), m 1. where K m (R d ) is the Korobov-type space K m (R d ) := { u : Dx k u L 2 (R d ), 0 k ϖ (1+r)/2+k m }, m N 0, and ( u K m (R d ) = k =m D k x u L 2 ϖ (1+r)/2+k(Rd ) ) 1 2.

49 Mass matrix M = I ; Stiff matrix S: banded leads to system matrix with much less non-zero entries. S = S M M + M S M M + M M S Fast transform available on mapped Chebyshev sparse grids. Figure : Sparsity of the mapped Chebyshev method: Left: d = 2, q = 8;

50 The linear system for Poisson type equations can be solved efficiently by using the sparse fast transform and PCG or a sparse solver such as CHOLMOD L 2 error d=1 d=2 full d=3 full 10 5 d=4 full d=2 sparse d=3 sparse d=4 sparse number of grid points Figure : Comparison between the MCFG method and MCSG method (with r = 1) for solving Poisson-type equation with an anisotropic solution with algebraic delay.

51 L 2 error d=1 d=2 full d=3 full d=2 sparse d=3 sparse number of grid points Figure : Left: the optimal frequency index set for tensor product of 1-D function with geometric convergence; Right: Comparison between the MCFG method and MCSG method (with r = 1).

52 L 2 error d=1 d=2 full d=3 full d=4 full d=2 sparse d=3 sparse d=4 sparse number of grid points Figure : Left: the optimal frequency index set for tensor product of 1-D function with subgeometric convergence; Right: Comparison between the MCFG method and MCSG method (with r = 1).

53 Problems with variable coefficients Use a constant coefficient problem as preconditioner While the system matrix is full, but the matrix-vector product can be performed in quasi-optimal complexity Ex. 1: r = 0 Ex. 1: r = 1 Ex. 2: r = 0 d q iter# CPU iter# CPU iter# CPU (-3) (-3) (-3) (-3) (-3) (-3) ) (-2) (-2) (-3) (-2) (-2) (-2) (-1) (-1) (-2) (-1) (-1) (-2) (-1) (-1) (-1) (-0) (-0) (-1) (-0) (-0) (-0) Table : The numbers of iterations and CPU time of using BICGSTAB solving equations with non-constant coefficients.

54 Application to electronic Schrödinger equation Much effort have been devoted to this problem using density function theory, but not much work on solving the equation directly; It has been shown (Yserentant 04) that its solution lives in the Korobov space; The problem is high-dimensional and set in unbounded domains; previous work (Gribel & Hamaekers 07) was based on domain truncation and Fourier method the effect of domain truncation is not easy to quantify; Our fast sparse spectral method in unbounded domain is well suited for this problem.

55 Consider the 1-D electronic Schrödinger equation: with Hψ = λψ H = T + V := 1 2 N i + ( N N N Z x i x i x j ). i=1 i=1 i=1 j>i We look for the ground state energy which is the smallest eigenvalue of H.

56 Consider the 1-D electronic Schrödinger equation: with Hψ = λψ H = T + V := 1 2 N i + ( N N N Z x i x i x j ). i=1 i=1 i=1 j>i We look for the ground state energy which is the smallest eigenvalue of H. Sparse spectral-galerkin approximation: Find ψ V q N, λ R such that (T ψ, φ)+ < V ψ, φ > disc = λ(ψ, φ), φ V q N, where V q N consists of mapped Chebyshev polynomials.

57 Using the mapped Chebyshev polynomials, the above reduces to: where (S + V)E = λme the mass matrix M is diagonal, the stiffness matrix S is sparse, the matrix V is full, but the matrix-vector product V x can be performed in quasi-optimal complexity thanks to the fast Chebyshev transform.

58 How to find the smallest eigenvalue? The direct application of Arnoldi method for computing the smallest eigenvalue converges very slowly; To accelerate the Arnoldi method, we employed a shifted-inverse technique which requires solving (S + V δi ) x = f ; We used S δi to precondition the above system. The above approach appears to be very efficient and robust. For example, six-dimensional problem with q = 12 can be solved on a laptop using single core within 3 hours.

59 q dof L = 0.5 L = 0.6 L = 0.75 L = Table : Two electrons: the smallest eigenvalue should lie in [ , ] with a reletive error of

60 q dof Memory L = 0.5 L = 0.6 L = 0.75 L = M M M M M M M Table : Six electrons: the smallest eigenvalue should lie in [ , ] with a reletive error of

61 q dof memory L = 0.5 L = 0.6 L = 0.75 L = M M M M M Table : Eight electrons: the smallest eigenvalue should lie in [11.339, ] with a relative error of

62 N q DoF δ n 1 n 2 N q DoF δ n 1 n Table : Numbers of iterations used in ARPack and BICGSTAB.

63 Adaptive sparse tensor products using multi-wavelets The one domain approach is not suitable for problems with local features, such as those in many challenging applications. For this type of problems, an efficient adaptive procedure, e.g., wavelets, is needed.

64 Adaptive sparse tensor products using multi-wavelets The one domain approach is not suitable for problems with local features, such as those in many challenging applications. For this type of problems, an efficient adaptive procedure, e.g., wavelets, is needed. Standard wavelets are generated by a single function: the order of accuracy is low and fixed.

65 Adaptive sparse tensor products using multi-wavelets The one domain approach is not suitable for problems with local features, such as those in many challenging applications. For this type of problems, an efficient adaptive procedure, e.g., wavelets, is needed. Standard wavelets are generated by a single function: the order of accuracy is low and fixed. We propose to use the Legendre multi-wavelets (Alpert 93), which allow spectral accuracy, and we propose to use a sparse tensor product with these multi-wavelets so that it is feasible for higher-dimensional problems.

66 Legendre multi-wavelets in L 2 ( 1, 1) Let φ j (x) be the Legendre polynomial of degree j and V0 k = span{φ j(x)} : 0 j k 1}. For n 1, we define Vn k as the space of piece-wise polynomials of degree less than k on a regular mesh with h = 2 1 n. Then, a set of basis function for Vn k is given by φ n jl (x) = 2 n 2 φj (2 n x l), j = 0,..., k 1, l = 0,..., 2 n 1.

67 Legendre multi-wavelets in L 2 ( 1, 1) Let φ j (x) be the Legendre polynomial of degree j and V0 k = span{φ j(x)} : 0 j k 1}. For n 1, we define Vn k as the space of piece-wise polynomials of degree less than k on a regular mesh with h = 2 1 n. Then, a set of basis function for Vn k is given by φ n jl (x) = 2 n 2 φj (2 n x l), j = 0,..., k 1, l = 0,..., 2 n 1. Let W k 0 = V k 0 and define the multi-wavelet subspace W k n, n = 1, 2,... as the L 2 -orthogonal compliment of V k n 1 in V k n, i.e., Then, we have V k n 1 W k n = V n k. V k n = W k 0 W k 1... W k n.

68 let ψ 0,..., ψ k 1 be the orthogonal basis for W k 1. Then, W k n (n 1) is spanned by ψ n jl (x) = 2 n 2 ψj (2 n x l), j = 0,..., k 1, l = 0,..., 2 n 1.

69 let ψ 0,..., ψ k 1 be the orthogonal basis for W k 1. Then, W k n (n 1) is spanned by ψ n jl (x) = 2 n 2 ψj (2 n x l), j = 0,..., k 1, l = 0,..., 2 n 1. We have 1 ψil n (x)ψn i l (x) dx = δ ii δ ll δ nn, 1 and they form a complete orthonormal basis in L 2 ( 1, 1).

70 Sparse tensor products in ( 1, 1) d Let Ω = ( 1, 1) d. The regular tensor product approximation space which can be written as VL k = V k L = V k L V k L, 0 l i L W k l 1 W k l 2 W k l d. However, dim(v k L ) = O(2Ld ) which is not feasible for d large.

71 Sparse tensor products in ( 1, 1) d Let Ω = ( 1, 1) d. The regular tensor product approximation space which can be written as VL k = V k L = V k L V k L, 0 l i L W k l 1 W k l 2 W k l d. However, dim(v k L ) = O(2Ld ) which is not feasible for d large. The sparse tensor product approximation space ˆV L k = Wl k 1 Wl k 2 Wl k d, 0 l 1 + +l d L with dim( ˆV k L ) = O(Ld 1 2 L ).

72 Galerkin approximation of the collision operator with sparse multi-wavelets Let L = ( 1,, l d ) with 0 l l d L. Then φ k L ˆV L k, and the inner product of the collision operator with ˆV L k is I (φ k L R ) = φ k L (v)dv [f (x, v )f (x, w ) f (x, v)f (x, w)] v w dsd 3 R 3 S 2 = f (x, v)f (x, w)dvdw R 3 R 3 v w [φ k L 2 (v ) + φ k L (w ) φ k L (v) φk L (w)]ds S 2 = f (x, v)f (x, w)a(v, w; φ k L )dvdw, R 3 where R 3 A(v, w; φ k v w L ) = 2 S 2 [φ k L (v ) + φ k L (w ) φ k L (v) φk L (w)]ds.

73 Proposed algorithm In practice, we can replace the sparse Legendre multi-wavelets by corresponding Lagrangian basis functions on a sparse grid. Given a sparse grid for v and w, we pre-compute A(v, w; φ k L ) for all φ k L ˆV k L. The total # of entries in this trilinear tensor is of order (L d 1 2 L ) 3. However, due to the local feature of φ k L, most of A(v, w; φ k L ) are zeros. We are in the process of working out the detail, and hopefully this will be a feasible alternative approach for the collision operator.

74 Concluding remarks Fast sparse spectral algorithms for elliptic equations in high-dimensional bounded and unbounded domains fast transforms between mapped Chebyshev sparse grids and corresponding hyperbolic cross space; fast solvers for the resulting linear system.

75 Concluding remarks Fast sparse spectral algorithms for elliptic equations in high-dimensional bounded and unbounded domains fast transforms between mapped Chebyshev sparse grids and corresponding hyperbolic cross space; fast solvers for the resulting linear system. Preliminary results to electronic Schrödinger equation indicate that this approach is promising for solving moderately high-dimensional PDEs.

76 Concluding remarks Fast sparse spectral algorithms for elliptic equations in high-dimensional bounded and unbounded domains fast transforms between mapped Chebyshev sparse grids and corresponding hyperbolic cross space; fast solvers for the resulting linear system. Preliminary results to electronic Schrödinger equation indicate that this approach is promising for solving moderately high-dimensional PDEs. A Galerkin sparse tensor product approach with multi-wavelets is proposed for dealing with the collision operator. Thank You!