A robust computational method for the Schrödinger equation cross sections using an MG-Krylov scheme

Transcription

1 A robust computational method for the Schrödinger equation cross sections using an MG-Krylov scheme International Conference On Preconditioning Techniques For Scientific And Industrial Applications June 2015, Eindhoven, The Netherlands University of Antwerp, Applied Mathematics S. Cools and W. Vanroose Contact:

2 Overview [1] Motivation [2] The Helmholtz and Schrödinger equations [3] The classical far field map calculation [4] The complex contour approach for the far field map [5] Schrödinger cross sections: the MG-CCCS preconditioner [6] Conclusions & discussion

3 Motivation Simulation of quantum mechanical molecular break-up reactions. Electron wave character CERN Large Hadron Collider Practical use: determine material composition and structure.

4 Schrödinger equation Full 3D break-up problem description two particle system (d = 2) Schrödinger equation (6-dimensional) (H E) ψ(r 1, r 2) = ζ(r 1, r 2). with outgoing wave ψ i.f.o. r 1, r 2 R 3 and Hamiltonian H = 1 2m r 1 1 2m r 2 + V 1(r 1) + V 2(r 2) + V 12(r 1, r 2). Right-hand side source term ζ(r 1, r 2 ) models incoming electron (electron scattering), or [Rescigno Baertschy Isaacs McCurdy 1999] dipole operator (photo-ionization). [Vanroose Martin Rescigno McCurdy 2005]

5 Partial wave expansion Spherical coordinate representation: r 1 = (ρ 1, Θ 1 ), r 2 = (ρ 2, Θ 2 ) l 1 l 2 ψ(r 1, r 2) = ψ l1 m 1,l 2 m 2 (ρ 1, ρ 2)Y l1 m 1 (Θ 1)Y l2 m 2 (Θ 2), l 1 =0 l 2 =0 m 2 = l 2 m 1 = l 1 leads to a coupled system for the radial wave components ψ l1 m 1,l 2 m 2 (ρ 1, ρ 2) ψ = ψ l 1 m 1 (ρ1, ρ2),l 2 m 2. Coupled partial wave system 2m 1 l 1,l 2 + V l1 m 1 l 2 m 2 ;l 1 m 1 l 2 m 2 E V l1 m 1 l 2 m 2 ;l 1 m 1 l 2 m 2 V l m 1 1 l 2 m 2 ;l 1 m 1 l 2 m 2 2m 1 l 1,l + V l 2 1 m 1 l 2 m 2 ;l 1 m 1 l 2 m E 2. Note: system decouples for V 1, V 2 and V 12 spherically symmetric..... ψ = ζ.

6 Relation to Helmholtz equation Diagonal blocks assume form of 2D driven Schrödinger equation ( 12 + V (x) E ) u(x) = g(x), x = (x, y) R 2. Helmholtz equation (d-dimensional) (equivalent formulation) ( k 2 (x) ) u(x) = f (x), x R d, with k 2 (x) = 2(E V (x)) and f (x) = 2g(x). Properties. potential V (x) and hence wavenumber k(x) is analytic, experimental observations (cross sections) are far field maps. [McCurdy Baertschy Rescigno 2004]

7 Helmholtz equation Representation of the physics behind a wave scattering at an object χ defined on a compact area O located within a domain Ω R d. Scattered wave solution u sc (x) satisfies inhomogeneous Helmholtz ( k 2 (x) ) u sc (x) = f (x), x Ω R d, with f (x) = k 2 0 χ(x)u in(x). Aim: calculate far field amplitude map F (α) u in O α

8 Far field map Analytic solution on whole R d using Green s function: u(x ) = G(x, x ) k0 2 χ(x) [u Ω }{{} in (x) + u sc (x)] dx, x R d. }{{} Green s function scattered wave [Colton Kress 1998] Calculate u in any point x R d outside the numerical domain Ω, using only the information inside the numerical domain. Computation: Split the far field integral into a sum I 1 + I 2, with I 1 = G(x, x )χ(x)u in (x)dx and I 2 = G(x, x )χ(x)u sc (x)dx Ω Ω }{{}}{{} all factors known explicitly requires u sc(x) for x Ω

9 State-of-the-art Helmholtz solvers. Preconditioned Krylov methods Solve M 1 Au = M 1 f using Krylov methods, with M = ϱk(x) 2 (preconditioner) where A = k(x) 2 and Mu = f easily solvable iteratively. ϱ = 1: original Helmholtz operator [von Helmholtz 19th century] ϱ = 0: Laplacian [Bayliss Goldstein Turkel 1983] ϱ < 0: shifted Laplacian or screened Poisson operator [Laird 2001] ϱ C: complex shifted Laplacian (CSL): ϱ = α + βi [Erlangga Vuik Oosterlee 2004]

10 Far field map Complex contour approach. I{z} For u and χ analytical the far field integral I 2 = G(x, x )χ(x)u sc (x)dx Ω Z 1 Z 2 γ R{z} can be calculated over a complex contour Z = Z 1 + Z 2, rather than over the real domain Ω, i.e. I 2 = G(x, x )χ(z)u sc (z)dz + Z 1 G(x, x )χ(z)u sc (z)dz. Z 2

11 Far field map Complex contour approach. I{z} For u and χ analytical the far field integral I 2 = G(x, x )χ(x)u sc (x)dx Ω Z 1 Z 2 γ R{z} can be calculated over a complex contour Z = Z 1 + Z 2, rather than over the real domain Ω, i.e. I 2 = G(x, x )χ(z)u sc (z)dz + G(x, x )χ(z)u sc (z)dz. Z } 1 Z {{} 2 requires u sc(z) for z Z 1

12 Helmholtz on complex contour Complex shifted Laplacian (CSL) system with shift parameter β R ( (1 + iβ)k 2 (x) ) u(x) = f (x) is efficiently solvable using multigrid. [Erlangga Oosterlee Vuik 2004] FD discretization: ( 1h 2 L (1 + iβ)k2 ) u h = f h, with Laplacian stencil matrix L. Division by (1 + iβ) yields ( ) 1 (1 + iβ)h L 2 k2 u h = f h 1 + iβ, the original Helmholtz system with complex h = 1 + iβ h = ρe iγ h. [Reps Vanroose bin Zubair 2010]

13 Contour approach - 2D validation Object of interest 5-point FD stencil, n x n y = I{z} Z1 γ Z2 R{z} Real domain with ECS χ(x) (θ ECS = 45 ) Complex contour χ(z) (γ = 14.6 )

14 Contour approach - 2D validation Scattered wave solution 5-point FD stencil, n x n y = I{z} Z1 γ Z2 R{z} Real domain with ECS u(x) LU factorization Complex contour u(z) V(1,1) cycles (tol res = 10 6 )

15 Contour approach - 2D validation Far field amplitude map 5-point FD stencil, n x n y = I{z} Z1 γ Z2 R{z} Real domain with ECS F (α) LU factorization Complex contour F (α) V(1,1) cycles (tol res = 10 6 )

16 Contour approach - 3D validation 3D damped Helmholtz solver (γ = 10 ) n x n y n z k 0 = 1/4 k 0 = 1/2 k 0 = 1 k 0 = 2 k 0 = 4 10 (0.79s.) 9 (4.65s.) 9 (44.2s.) 9 (352s.) 9 (2778s.) (0.92s.) 10 (4.96s.) 10 (48.3s.) 10 (390s.) 9 (2797s.) (0.62s.) 13 (6.59s.) 11 (54.6s.) 10 (387s.) 10 (3079s.) (0.28s.) 8 (4.24s.) 13 (63.9s.) 11 (428s.) 10 (3006s.) (0.20s.) 2 (1.35s.) 7 (36.1s.) 13 (503s.) 11 (3306s.) GMRES(3)-smoothed V(1,1) cycles (tol res = 10 6 ) k 0h = 0.625

17 Contour approach - 3D validation 3D damped Helmholtz solver (γ = 10, k 0 = 1) n x n y n z CPU time 0.20 s s s s. 462 s. r 2 3.3e-5 7.9e-5 2.7e-5 1.1e-5 4.6e s. 1.0e-5 GMRES(3)-smoothed FMG(1,1) cycle [Vasseur 2012] Serial implementation, Intel Core i7-2720qm 2.20GHz CPU, 6MB Cache, 8GB RAM.

18 Schrödinger cross sections Single ionization amplitude = single ionization probability s n(e) = φ kn (x)φ n(y) [g(x, y) V 12(x, y)u(x, y) ] dx dy. Ω }{{}}{{} Green s function scattered wave Double ionization cross section = double ionization probability d k1,k 2 (E) = φ k1 (x)φ k2 (y) [g(x, y) V 12(x, y)u(x, y) ] dx dy. Ω }{{}}{{} Green s function scattered wave Single ionization Double ionization

19 Schrödinger on complex contour 2D driven Schrödinger equation ( 12 + V1(x) + V2(y) + V12(x, y) E ) u(x, y) = f (x, y), with x, y 0, V i potentials, E R energy of the system. Multigrid convergence factor damped Schrödinger on full complex grid with γ 8.5 GMRES(3)-smoothed multigrid V(1,1) cycles Observation: Poor convergence for 1<E <0.

20 Schrödinger spectral analysis Single ionization Double ionization E < 0 E = 0 E > imag 4 imag 4 imag real real real bound states with near-zero e.v. s destroy multigrid convergence

21 Coupled channel approximation Bound states are characterized by eigenstates of 1D Hamiltonians H 1φ n(x) = λ nφ n(x), H 2ϕ n(y) = µ nϕ n(y), with λ n < 0 and µ n < 0, hence for M n x, L n y approximate M L u(x, y) A m(y)φ m(x) + B l (x)ϕ l (y). m=1 l=1 [Heller Reinhardt 1973] [McCarthy Stelbovics 1983] Coupled channel correction step (CCCS). M L u (k+1) (x, y) = u (k) (x, y) + em(y)φ A m(x) + el B (x)ϕ l (y). m=1 l=1 Determine coefficients as solution of M+L 1D Schrödinger systems.

22 Numerical results: convergence Convergence rate of MG-CCCS as stand-alone solver/preconditioner Exponential model problem Temkin-Poet model problem V1 (x) = 4.5 exp( x 2 ) V1 (x) = 1/x V12 (x, y ) = 2 exp( 0.1(x + y )2 ) V12 (x, y ) = 1/ max(x, y ) Ω = [0, 20]2, nx ny = Ω = [0, 100]2, nx ny =

23 Numerical results: cross sections 2D Temkin Poet model problem Potentials: V1 (x) = 1/x, V2 (y ) = 1/y and V12 (x, y ) = 1/ max(x, y ) Discretization: Ω = [0, 108]2 with nx ny = spectral element grid Solver: LU with θecs = 30 (real) vs. MG-CCCS with γ = 9 (complex) Single differential cross section Total cross section E =1 E [0, 2]

24 In this work we presented... Conclusions Proof-of-concept for complex contour approach proposed in [Cools Reps Vanroose 2013] and application to impact scattering. Coupled Channel Correction Step (CCCS) after each MG V-cycle accounts for presence of localized bound states. Fast and robust method for the computation of the ionization cross sections for electron-impact models (for any energy E). MG-CCCS validated on 2D Temkin-Poet model problem, convergence as solver/preconditioner shows O(N) scalability. Outlook Generalization to 3D Schrödinger partial wave systems. Analysis of bound states and influence of complex rotation γ for general discretizations.

25 References Y.A. Erlangga, C.W. Oosterlee, and C. Vuik, A novel multigrid based preconditioner for heterogeneous Helmholtz problems, SIAM Journal on Scientific Computing 27(4), pp , W. Vanroose, D.A. Horner, F. Martin, T.N. Rescigno and C.W. McCurdy. Double photoionization of aligned molecular hydrogen. Physical Review A, 74(5), pp , O.G. Ernst and M.J. Gander, Why it is difficult to solve Helmholtz problems with classical iterative methods, Numerical Analysis of Multiscale Problems, pp , S. Cools, B. Reps and W. Vanroose, An efficient multigrid method calculation of the far field map for Helmholtz problems, SIAM Journal on Scientific Computing 36(3), pp. B367-B395, S. Cools and W. Vanroose, A fast and robust computational method for the ionization cross sections of the driven Schrödinger equation using an O(N) multigrid-based scheme, under review, arxiv: , 2015.