Optimal Control of the Schrödinger Equation on Many-Core Architectures

Transcription

1 Optimal Control of the Schrödinger Equation on Many-Core Architectures Manfred Liebmann Institute for Mathematics and Scientific Computing University of Graz July 15, 2014

2 Motivation We investigate efficient parallel algorithms for the optimal control problem of the timedepent Schrödinger equation on modern many-core architectures. We consider controls for multi-component Schrödinger systems with external laser fields: { min(1 φ, ψ(t ) 2 ) + γ 1 u 2 + γ 2 u 2 subject to i t ψ = ( 1 2m + V + u(t)b)ψ, ψ(0) = ψ 0 Where φ L 2 (R d, C N ) is the desired target state to be achieved at time T. The external laser field amplitude u : [0, T ] R is the control. The regularizing terms, which penalize overall field energy or overall field fluctuation, are measured in suitable norms and weighted by the regularization parameters γ 1, γ 2 > 0. The time-depent Schrödinger equation describes the effective quantum motion of nuclei with average mass 0 < m 1 and general initial conditions ψ 0 L 2 (R d, C N ). The dipole operator B initiates the transfer between the potential energy surfaces of the matrix-valued potential V : R d C N N. (1) Optimal Control of the Schrödinger Equation on Many-Core Architectures 1

3 Principal Investigators The international Research Training Group (IGDK 1754) Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures is a joint project of Technische Universität München, Universität der Bundeswehr München, Karl-Franzens- Universität Graz, and Technische Universität Graz. P 12 Optimal control for Schrödinger dynamics on multiple potential energy surfaces Graduate Students Felix Henneke David Sattlegger Principal Investigators Gero Friesecke Caroline Lasser Karl Kunisch Optimal Control of the Schrödinger Equation on Many-Core Architectures 2

4 Introduction Numerical integrators for time-depent Hamiltonians in the Schrödinger equation i dψ dt = H(t)ψ, ψ(t 0) = ψ 0 (2) can be constructed as a combination of a generalized Suzuki-Trotter (GSTM) approximation scheme for the exponential operator and the Magnus expansion to capture the timedepence of the Hamiltonian. The general initial value problem ψ = Aψ for a time-indepent linear operator A with initial state ψ(0) = ψ 0 has the solution: ψ(t) = exp(ta)ψ 0 (3) Optimal Control of the Schrödinger Equation on Many-Core Architectures 3

5 Construction Theorem Theorem 1. Let B be a Banach algebra and A B an operator. Let Q 1 (t) B be an analytic operator function and an approximation of first order to the exponential operator exp(ta), exp(ta) Q 1 (t) = o(t) (t 0) (4) and let r > 1 be a fixed integer number. Then an analytic approximation of order m > 1 to the exponential operator is given by Q m (t) = r j=1 Q m 1 (p m,j t). (5) Where the coefficients p m,j for 1 j r are restricted by the two constraints r p m,j = 1, j=1 r p m m,j = 0. (6) j=1 Optimal Control of the Schrödinger Equation on Many-Core Architectures 4

6 Explicit Approximation Scheme We use the construction theorem to build a fractal approximation scheme for the operator function Q 1 (t) = 1 + ta and with r = 2. The coefficients p m,1, p m,2 are determined by the two restrictions The solution of this nonlinear system is given by p m,1 + p m,2 = 1, p m m,1 + p m m,2 = 0. (7) p m,1 = e iπ/m, p m,2 = 1. (8) 1 + eiπ/m This can also be written as p m = i 2 tan(π/2m), p m,1 = p m, p m,2 = p m (9) Optimal Control of the Schrödinger Equation on Many-Core Architectures 5

7 The complete approximation scheme is given as an operator product Q 2 (t) = Q 1 (p 2 t)q 1 ( p 2 t) (10) Q 3 (t) = Q 1 (p 3 p 2 t)q 1 (p 3 p 2 t)q 1 ( p 3 p 2 t)q 1 ( p 3 p 2 t) (11) Q m (t) = Q 1 (p m p 3 p 2 t)q 1 (p m p 3 p 2 t) Q 1 ( p m p 3 p 2 t) (12) and the complete formula for the approximation scheme reads: Q m (t) = z m,n = 2 m 1 n=1 m k=2 (1 + tz m,n A) (13) e iπb k,n/k 1 + e iπ/k (14) b k,n = (n 1)/2 k 2 mod 2 (15) Optimal Control of the Schrödinger Equation on Many-Core Architectures 6

8 Numerical Solution of the Optimal Control Problem The discretized optimal control problem reads min jh (u h ) = 1 u h U h 2 ψh N(u h ), O h ψn(u h h ) H h + α 2 uh 2 U h, (16) with the solution of the state equation calculated by the GSTM ψ h n(u h ) = n p=1 Q p (u h )ψ h 0, n = 1,..., N. (17) Numerical solution methods for the minimazation Newton Trust Region Barzilai-Borwein Optimal Control of the Schrödinger Equation on Many-Core Architectures 7

9 Data: u h 0 Result: optimal u h u h u h 0; ρ 1; r 1; j h functional(u h ); v h gradient(u h ); while v h TOL do δu h subproblem(u h, v h, r); j h new functional(u h + δu h ); w h hessian(u h, δu h ); ρ (j h j h new)/ δu h, v h 0.5w h U h; if ρ < η then r 0.25r; if ρ 0 then u h u h + δu h ; j h j h new; v h gradient(u h ); if ρ η + then r 2r; Algorithm 1: Newton trust region method Optimal Control of the Schrödinger Equation on Many-Core Architectures 8

10 Data: u h, v h, r Result: δu h p 0; res v h ; d res; while res > TOL 2 do w h hessian(u h, d); s d, w h U h; if s 0 then find τ such that p + τd = r; p p + τd; break; α res /s; if p + αd > r then find τ 0 such that p + τd = r; p p + τd; break; p p + αd; β res αw h 2 / res 2 ; res res αw h ; d res + βd; δu h p Algorithm 2: Newton trust region subproblem (Steihaug CG) Optimal Control of the Schrödinger Equation on Many-Core Architectures 9

11 Functional, Gradient and Hessian Evaluation Data: u h Result: j h ψ h ψ h 0 ; for n = 1,..., N do ψ h Q n (u h ) ψ h ; j h 1 2 ψh, O h ψ h H h + α 2 uh, u h U h Algorithm 3: Functional evaluation Optimal Control of the Schrödinger Equation on Many-Core Architectures 10

12 Data: u h Result: v h ψ h ψ h 0 ; X 0; for n = 1,..., N do ψ h Q n (u h )ψ h ; ϕ h O h ψ h ; for n = N..., 1 do assemble rhs gradient(u h, n, X) solve for Z: MZ = X; v h K k=1 Z kh k + αu h ; Algorithm 4: Gradient evaluation Optimal Control of the Schrödinger Equation on Many-Core Architectures 11

13 Data: u h, n, X Result: X ψ h Q n (u h )ψ h ; χ ( h ) ϕ h ; ( ϕ h Q n (u h ) 0 χ h ) ( ϕ h Q n (u h )(δu h ) Q n (u h ) χ h χ h ϕ h ; δx χ h, ψ h H h; X n X n + δx; X n 1 X n 1 + δx; Algorithm 5: assemble rhs gradient for DTO piecewise linear χ h ) ; Optimal Control of the Schrödinger Equation on Many-Core Architectures 12

14 Data: u h, δu Result: ( ) w h ( ) ψ h ψ h ψ h 0 ; 0 Y 0; for ( n = 1 )..., N do ψ h ( ϕ h ϕ h ψ h ) ( Qn (u h ) 0 Q n(u h )(δu h ) Q n (u h ) ( O h ψ h ) ; ) ( ψ h ψ h O h ψ h for n = N..., 1 do assemble rhs hessian(u h, δu h, n, Y ) solve for Z: MZ = Y ; w h K k=1 Z kh k + αδu h ; Algorithm 6: Hessian evaluation ) ; Optimal Control of the Schrödinger Equation on Many-Core Architectures 13

15 Data: u h, δu h, n, Y ( Result: ) Y ( ) ( ψ h Qn (u ψ h h ) 0 ψ h Q n (u h )(δu h ) Q n (u h ) ψ h χ h 2 ϕ h ; χ ( h 3 ) ϕ h ; ( ) ( ϕ h Q χ h n (u h ) 0 ϕ h 2 Q n (u h )(h n ) Q n (u h ) χ h 2 χ h 2 χ h 2 ϕ h ; ϕ h χ h 3; χ h 1 ϕ h ; ϕ h ϕ h χ h 1 χ h 3 Q n (u h ) Q n (u h )(δu h ) Q n (u h ) 0 0 Q n (u h )(h n ) 0 Q n (u h ) 0 Q n (u h )(δu h, h n ) Q n (u h )(h n ) Q n (u h )(δu h ) Q n (u h ) χ h 1 χ h 1 ϕ h ; χ h 3 χ h 3 ϕ h χ h 2; δy χ h 1, ψ h H h + χ h 2, ψ h H h + χ h 3, ψ h H h; Y n Y n + δy ; Y n 1 Y n 1 + δy ; Algorithm 7: assemble rhs hessian for DTO piecewise linear ) ; ) ; Optimal Control of the Schrödinger Equation on Many-Core Architectures 14 ϕ h ϕ h χ h 1 χ h 3 ;

16 Time Optimal Control 1D Model Problem Figure 1: Two potential energy surfaces V 11, V 22, V 12 = V 21 = 0 on the left. Components of the magnetic dipole field B 11 = B 22, B 12 = B 21 on the right. Spatial grid size is 256. Optimal Control of the Schrödinger Equation on Many-Core Architectures 15

17 Newton Trust Region Algorithm Figure 2: Initial control function for the laser field u on the left. Control function calculated by the Newton trust region algorithm after 30 iterations on the right. The functional was reduced from to Temporal grid size is Compute time seconds on Intel Core2Duo 2.8GHz. Optimal Control of the Schrödinger Equation on Many-Core Architectures 16

18 Manfred Liebmann July 15, 2014 Figure 3: Visualization of the complex wavefunctions ψ1, ψ2 and the corresponding probability distribution in a space time diagram. Optimal Control of the Schro dinger Equation on Many-Core Architectures 17

19 Parallelization of the Optimal Control Problem One-dimensional problems are too small for effective parallelization although the compute time is already quite high. Two-dimensional test problem on a spacial grid and 2048 point temporal grid with a similar simulation setup to the 1D problem. Cores GNU g++ Intel icpc compact KMP AFFINITY 4KB page mm malloc More parallel code Table 1: Parallel scaling with OpenMP parallelization on Intel Xeon 2.00GHz for 10 iterations of the Barzilai-Borwein method. Optimal Control of the Schrödinger Equation on Many-Core Architectures 18

20 Many-Core Parallelization for the Optimal Control Problem 2x Intel Xeon E Intel Xeon Phi 5110P Nvidia Tesla K20 16 Cores 60 Cores / 240 Threads 2496 Cuda Cores Table 2: Benchmark for Newton trust region method for 10 iterations on a space-time grid. Timings are given in seconds. Optimal Control of the Schrödinger Equation on Many-Core Architectures 19

21 Conclusions Optimal Control of the Schrödinger Equation Optimal control problems are well suited for many-core architectures Algorithms are essentially bound by the performance of the PDE solver. Flexible C++ framework for different state equations and solution methods Future work: Complete framework for MPI/OpenMP/SSE/AVX/GPU/PHI Optimal Control of the Schrödinger Equation on Many-Core Architectures 20

22 Thank you! Optimal Control of the Schrödinger Equation on Many-Core Architectures 21