An RBF–Galerkin approach to the time-dependent Schrödinger equation

The time-dependent Schrodinger equation (TDSE) models the quantum nature of molecular processes. Numerical simulations based on the TDSE help in understanding and predicting the outcome of chemical reactions. This thesis is dedicated to the derivation and analysis of efficient and reliable simulation tools for the TDSE, with a particular focus on models for the interaction of molecules with time-dependent electromagnetic fields.Various time propagators are compared for this setting and an efficient fourth-order commutator-free Magnus-Lanczos propagator is derived. For the Lanczos method, several communication-reducing variants are studied for an implementation on clusters of multi-core processors. Global error estimation for the Magnus propagator is devised using a posteriori error estimation theory. In doing so, the self-adjointness of the linear Schrodinger equation is exploited to avoid solving an adjoint equation. Efficiency and effectiveness of the estimate are demonstrated for both bounded and unbounded states. The temporal approximation is combined with adaptive spectral elements in space. Lagrange elements based on Gauss-Lobatto nodes are employed to avoid nondiagonal mass matrices and ill-conditioning at high order. A matrix-free implementation for the evaluation of the spectral element operators is presented. The framework uses hybrid parallelism and enables significant computational speed-up as well as the solution of larger problems compared to traditional implementations relying on sparse matrices.As an alternative to grid-based methods, radial basis functions in a Galerkin setting are proposed and analyzed. It is found that considerably higher accuracy can be obtained with the same number of basis functions compared to the Fourier method. Another direction of research presented in this thesis is a new algorithm for quantum optimal control: The field is optimized in the frequency domain where the dimensionality of the optimization problem can drastically be reduced. In this way, it becomes feasible to use a quasi-Newton method to solve the problem.

[1]  Marlis Hochbruck,et al.  Exponential Integrators for Large Systems of Differential Equations , 1998, SIAM J. Sci. Comput..

[2]  Joseph D. Ward,et al.  Scattered-Data Interpolation on Rn: Error Estimates for Radial Basis and Band-Limited Functions , 2004, SIAM J. Math. Anal..

[3]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .

[4]  Willy Dörfler,et al.  A time- and spaceadaptive algorithm for the linear time-dependent Schrödinger equation , 1996 .

[5]  E. Kansa,et al.  Exponential convergence and H‐c multiquadric collocation method for partial differential equations , 2003 .

[6]  Tobin A. Driscoll,et al.  Adaptive residual subsampling methods for radial basis function interpolation and collocation problems , 2007, Comput. Math. Appl..

[7]  Robert Schaback,et al.  An improved subspace selection algorithm for meshless collocation methods , 2009 .

[8]  Lloyd N. Trefethen,et al.  Impossibility of Fast Stable Approximation of Analytic Functions from Equispaced Samples , 2011, SIAM Rev..

[9]  T. Carrington,et al.  An improved neural network method for solving the Schrödinger equation1 , 2009 .

[10]  H. Meyer,et al.  TIME-DEPENDENT ROTATED HARTREE APPROACH , 1987 .

[11]  Christian Rieger,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Sampling Inequalities for Infinitely Smooth Functions, with Applications to Interpolation and Machine Learning Sampling Inequalities for Infinitely Smooth Functions, with Applications to Interpolation and Machine Learning , 2022 .

[12]  Emanuel H. Rubensson,et al.  Bringing about matrix sparsity in linear‐scaling electronic structure calculations , 2011, J. Comput. Chem..

[13]  Spectral differentiation matrices for the numerical solution of Schrödinger's equation , 2006 .

[14]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[15]  T. Martínez,et al.  Ab Initio Multiple Spawning: Photochemistry from First Principles Quantum Molecular Dynamics , 2000 .

[16]  V. Thomée.  Galerkin Finite Element Methods for Parabolic Problems (Springer Series in Computational Mathematics) , 2010 .

[17]  Solving the bound-state Schrodinger equation by reproducing kernel interpolation , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[18]  Holger Wendland,et al.  Meshless Galerkin methods using radial basis functions , 1999, Math. Comput..

[19]  Carsten Franke,et al.  Convergence order estimates of meshless collocation methods using radial basis functions , 1998, Adv. Comput. Math..

[20]  T. Driscoll,et al.  Observations on the behavior of radial basis function approximations near boundaries , 2002 .

[21]  M. Feit,et al.  Solution of the Schrödinger equation by a spectral method , 1982 .

[22]  Erwan Faou,et al.  Gauss–Hermite wave packet dynamics: convergence of the spectral and pseudo-spectral approximation , 2009 .

[23]  Holger Wendland,et al.  Scattered Data Approximation: Conditionally positive definite functions , 2004 .

[24]  Askar,et al.  Solution of the quantum fluid dynamical equations with radial basis function interpolation , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[25]  Victor S. Batista,et al.  Matching-pursuit for simulations of quantum processes , 2003 .

[26]  Holger Wendland,et al.  Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree , 1995, Adv. Comput. Math..

[27]  Robert Schaback,et al.  Error estimates and condition numbers for radial basis function interpolation , 1995, Adv. Comput. Math..

[28]  E. Kansa,et al.  Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary , 2002 .

[29]  R. Kosloff,et al.  A fourier method solution for the time dependent Schrödinger equation as a tool in molecular dynamics , 1983 .

[30]  Jie Shen,et al.  Spectral and Pseudospectral Approximations Using Hermite Functions: Application to the Dirac Equation , 2003, Adv. Comput. Math..

[31]  Bengt Fornberg,et al.  Locality properties of radial basis function expansion coefficients for equispaced interpolation , 2007 .

[32]  Bengt Fornberg,et al.  The Runge phenomenon and spatially variable shape parameters in RBF interpolation , 2007, Comput. Math. Appl..

[33]  Erik Lehto,et al.  Rotational transport on a sphere: Local node refinement with radial basis functions , 2010, J. Comput. Phys..

[34]  Eric J. Heller,et al.  Frozen Gaussians: A very simple semiclassical approximation , 1981 .

[35]  Katharina Kormann,et al.  Stable difference methods for block-structured adaptive grids , 2011 .

[36]  Margot Gerritsen,et al.  Stability at Nonconforming Grid Interfaces for a High Order Discretization of the Schrödinger Equation , 2012, J. Sci. Comput..

[37]  Tobin A. Driscoll,et al.  Eigenvalue stability of radial basis function discretizations for time-dependent problems , 2006, Comput. Math. Appl..

[38]  Christian Ochsenfeld,et al.  Locality and Sparsity of Ab Initio One-Particle Density Matrices and Localized Orbitals , 1998 .

[39]  Gregory E. Fasshauer,et al.  Meshfree Approximation Methods with Matlab , 2007, Interdisciplinary Mathematical Sciences.

[40]  D. Shalashilin,et al.  Multidimensional quantum propagation with the help of coupled coherent states , 2001 .

[41]  I. Degani.  Observations on Gaussian bases for Schrodinger's equation , 2007, 0707.4587.

[42]  E. Villaseñor.  Introduction to Quantum Mechanics , 2008, Nature.

[43]  R. Platte.  How fast do radial basis function interpolants of analytic functions converge , 2011 .

[44]  Martin D. Buhmann,et al.  Radial Basis Functions: Theory and Implementations: Preface , 2003 .

[45]  Elisabeth Larsson,et al.  Stable Computations with Gaussian Radial Basis Functions , 2011, SIAM J. Sci. Comput..

[46]  Christophe Besse,et al.  A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations , 2008 .

[47]  S. Blanes,et al.  The Magnus expansion and some of its applications , 2008, 0810.5488.

[48]  Erwan Faou,et al.  Computing Semiclassical Quantum Dynamics with Hagedorn Wavepackets , 2009, SIAM J. Sci. Comput..

[49]  Shmuel Rippa,et al.  An algorithm for selecting a good value for the parameter c in radial basis function interpolation , 1999, Adv. Comput. Math..

[50]  B. Fornberg,et al.  Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions , 2003 .

[51]  S. Sarra,et al.  A linear system‐free Gaussian RBF method for the Gross‐Pitaevskii equation on unbounded domains , 2012 .

[52]  Mehdi Dehghan,et al.  A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions , 2007, Comput. Math. Appl..