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Workshop Modern Numerical Methods in Quantum Mechanics Book of abstracts March 20-22, 2017 2 Contents Winfried Auzinger . . . . . . . . . . . . . . . 3 Weizhu Bao . . . . . . . . . . . . . . . . . . . . . . 3 Sergio Blanes . . . . . . . . . . . . . . . . . . . . 4 Fernando Casas Pérez . . . . . . . . . . . 5 Elena Celledoni . . . . . . . . . . . . . . . . . . 5 Erwin Faou . . . . . . . . . . . . . . . . . . . . . . 5 Ernest Hairer . . . . . . . . . . . . . . . . . . . . 6 Arieh Iserles . . . . . . . . . . . . . . . . . . . . . 6 Othmar Koch . . . . . . . . . . . . . . . . . . . . 6 Karolina Kropielnicka . . . . . . . . . . . 7 Caroline Lasser . . . . . . . . . . . . . . . . . . 7 Christian Lubich (I) . . . . . . . . . . . . . . 7 Christian Lubich (II) . . . . . . . . . . . . . 8 Marcin Napiórkowski . . . . . . . . . . . 8 Hassan Safouhi . . . . . . . . . . . . . . . . . . 9 Pranav Singh . . . . . . . . . . . . . . . . . . . 10 Mechthild Thalhammer . . . . . . . . . 10 Piotr Zgliczyński . . . . . . . . . . . . . . . 10 Recommended restaurants in the surroundings . . . . . . . . . . . . . . 11 Programme of the event . . . . . . . . . . 12 Abstracts Winfried Auzinger (Vienna University of Technology, Austria) Adaptive integrators for Schrödinger-type equations Joint work with: Harald Hofstätter, Othmar Koch, Michael Quell and Mechthild Thalhammer. We give an overview on recent work on a posteriori error estimation for the purpose of time-adaptive integration of evolution equations of Schrödinger type. In particular, splitting techniques are considered. Depending on the problem at hand and the underlying scheme, different techniques can be used. These include optimized embedded pairs of schemes, the use of the adjoint of an (optimized) scheme, and defect-based local error estimators. We concentrate on the construction and efficient implementation of such error estimators and present some examples, including ongoing work on Magnus-type integrators for the case of time-dependent Hamiltonians. Weizhu Bao (National University of Singapore, Singapore) Multiscale methods and analysis for the Dirac equation in the nonrelativistic limit regime Joint work with: Yongyong Cai, Xiaowei Jia, Qinglin Tang and Jia Yin. In this talk, I will review our recent works on numerical methods and analysis for solving the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time and the energy becomes unbounded and indefinite, which bring significant difficulty in analysis and heavy burden in numerical computation. We begin with four frequently used finite difference time domain (FDTD) methods 3 and the time splitting Fourier pseudospectral (TSFP) method and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size and time step as well as the small parameter. Then we consider a numerical method by using spectral method for spatial derivatives combined with an exponential wave integrator (EWI) in the Gautschi-type for temporal derivatives to discretize the Dirac equation. Rigorous error estimates show that the EWI spectral method has much better temporal resolution than the FDTD methods for the Dirac equation in the nonrelativistic limit regime. Based on a multiscale expansion of the solution, we present a multiscale time integrator Fourier pseudospectral (MTI-FP) method for the Dirac equation and establish its error bound which uniformly accurate in term of the small dimensionless parameter. Numerical results demonstrate that our error estimates are sharp and optimal. Finally, these methods and results are then extended to the nonlinear Dirac eqaution in the nonrelativistic limit regime. [1] W. Bao, Y. Cai, X. Jia and Q. Tang, Numerical methods and comparison for the Dirac equation in the nonrelativistic limit regime, J. Sci. Comput., to appear (arXiv: 1504.02881). [2] W. Bao, Y. Cai, X. Jia and Q. Tang, A uniformly accurate multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime, SIAM J. Numer. Anal., 54 (2016), pp. 1785-1812. [3] W. Bao, Y. Cai, X. Jia and J. Yin, Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime, Sci. China Math., 59 (2016), pp. 1461-1494. Sergio Blanes (Polytechnic University of Valencia) Time-average Symplectic propagators for the Schrödinger equation with time-dependent Hamiltonian Joint work with: Fernando Casas and Ander Murua. Several symplectic splitting methods of orders four and six are presented for the stepby-step time numerical integration of the Schrödinger equation when the Hamiltonian is a general explicitly time-dependent real operator. They involve linear combinations of the Hamiltonian evaluated at some intermediate points. We provide the algorithm and the coefficients of the methods, as well as some numerical examples showing their superior performance with respect to other available schemes. It is also shown how the schemes can be adapted when the Hamiltonian is evaluation at the nodes of at any quadrature rule. [1] S. Blanes, F. Casas, and A. Murua, Symplectic time-average propagators for the Schdinger equation with a time-dependent Hamiltonian, J. Chem. Phys. (2017). In press. 4 Fernando Casas Pérez (Jaume I University, Spain) An efficient numerical integrator for the neutrino oscillations problem in matter A special purpose numerical method, based on the Magnus expansion, is proposed for the integration of the linear three neutrino oscillations equations in matter. The computations are speeded up to two orders of magnitude with respect to other general numerical integrators, a fact that might facilitate the massive numerical integration needed in connection with experimental data analyses. Illustrations about the numerical procedure and computer time costs are provided. Elena Celledoni (Norwegian University for Science and Technology, Norway) Shape analysis on Lie groups and homogeneous manifolds Shape analysis is ubiquitous in problems of pattern and object recognition and has developed considerably in the last decade. The use of shapes instead of curves is natural in applications where one wants to compare curves independently of their parametrisation. Shapes are in fact unparametrized curves, evolving on a vector space, on a Lie group or on a manifold. One popular approach to shape analysis is by the use of the Square Root Velocity Transform (SRVT). To manipulate or compare two shapes with values in vector spaces, taking two parametrised curves as their representatives, one transforms the curves via the SRVT, replacing the curves by appropriately scaled tangent vector fields along these curves. The curves are compared computing geodesics in the L2 metric between the transformed curves in place of the original ones. Notably, the scaling can be chosen suitably to yield reparametrization invariance. In this talk we consider the generalisation of the SRVT, from vector spaces and Lie groups to homogeneous manifolds. This approach takes advantage of the Lie group acting transitively on the homogeneous manifold. Examples of problems of computations with shapes and curves on Lie groups and homogeneous manifolds will be presented. Erwin Faou (University of Rennes 1, France) On the long time stability of travelling wave for the discrete nonlinear Schrödinger equation Joint work with: Dario Bambusi, Joackim Bernier, Benoı̂t Grébert and Alberto Maspero. I will discuss the possible existence of travelling wave solutions in discrete nonlinear Schrödinger equations on a grid. I will show the influence of the nonlinearity in this problem and give some partial results for the long time existence and stability. 5 Ernst Hairer (University of Geneva, Switzerland) Long-time behaviour of numerical integrators for charged particle dynamics Joint work with: Christian Lubich The Boris algorithm is the most popular time integrator for charged particle motion in electric and magnetic force fields. It is a symmetric one-step method, and it preserves the phase volume exactly. However, it is not symplectic. Nevertheless, numerical experiments confirm an excellent long-time near energy preservation of the system. In this talk we present various modifications of the Boris algorithm for which the near energy preservation can be proved rigorously. They are based on splitting techniques and on Hamiltonian or Poisson formulations of the equation. Emphasis is put on a new explicit multistep extension of the Boris algorithm. Near energy preservation for the underlying one-step method, and the boundedness of parasitic solution components are shown. A rigorous proof for the excellent near energy preservation of the Boris algorithm is still missing. We thank Martin Gander for drawing our attention to this problem. Arieh Iserles (University of Cambridge, United Kingdom) It takes a wave packet to catch a wave packet We are concerned with spectral methods for signals composed of wave packets, e.g. in quantum mechanics. The traditional approach is to use periodic boundary conditions, in which case standard Fourier methods are more than adequate, except that in long-term integration wave packets might reach the boundary and non-physical behaviour ensues. This motivates us to consider approximations on the entire real line. We consider and analyse four candidates: Hermite polynomials, Hermite functions, stretched Chebyshev expansions and stretched Fourier expansions. And the winner is ... Othmar Koch (University of Vienna, Austria) Adaptive Time-splitting FEM discretization of the Schrödinger-Poisson equation Joint work with Winfried Auzinger, Thomas Kassebacher, Mechthild Thalhammer We discuss the adaptive numerical solution of the Schrödinger-Poisson equation on a truncated finite domain with an underlying space discretization by conforming piecewise polynomial finite elements, where we truncate to a sufficiently large finite domain and impose homogeneous Dirichlet boundary conditions. The motivation for this approach is the possibility to treat the Poisson equation separately by dedicated solvers for the arising linear equations. The classical convergence orders in both the time and space discretization are established theoretically under natural assumptions on the regularity of the exact solution and illustrated by numerical experiments. Adaptive time-stepping relying on a defect-based error estimator is shown to correctly reflect the solution behaviour. 6 Karolina Kropielnicka (Polish Academy of Sciences, Poland) Commutator-free Magnus based methods In this talk, I will present methods relevant for linear time-dependent Schrödinger equation. We will focus on the case of Schrödinger equation with time dependent potential, where we introduce a commutator free Magnus expansion. A commutator free expansion is possible here due to the deferral of the semidiscretisation to the very last stage of our algorithm, and due to working in an appropriate Lie algebraic setting. This Magnus expansion can be combined with Lanczos iterations or symmetric Zassenhaus decompositions. The resulting methods also happen to be efficient in the semiclassical regime. Numerical simulations will be provided. Caroline Lasser (Technical University Munich, Germany) Computational semiclassics We discuss computational semiclassics for solving high-dimensional quantum dynamics in the high frequency regime. Talk I concentrates on the computation of the wave function by Gaussian and Hagedorn wave packets. Talk II addresses the approximation of the unitary propagator and the computation of expectation values. Christian Lubich (Tübingen University, Germany) The Dirac-Frenkel time-dependent variational principle and its applications This is a review talk on the basic approximation principle of quantum dynamics. Geometric properties and approximation properties will be discussed, and its use in molecular quantum dynamics will be illustrated, leading from the full molecular Schroedinger equation to classical molecular dynamics in a series of model reductions, such as the BornOppenheimer approximation, separation of variables (self-consistent field methods), and Gaussian wavepackets. [1] Ch. Lubich, From quantum to classical molecular dynamics: reduced models and numerical analysis. European Mathematical Society, Zurich, 2008. [2] P. Kramer and M. Saraceno, Geometry of the time-dependent variational principle in quantum mechanics. Lecture Notes in Physics 140, Springer, Berlin, 1981. 7 Christian Lubich (Tübingen University, Germany) Numerical integrators for dynamical low-rank approximation Joint work with: Othmar Koch, Bart Vandereycken, Ivan Oseledets, Emil Kieri and Hanna Walach. This talk is concerned with differential equations on manifolds of matrices or tensors of low rank. They serve to approximate, in a low-rank format, large time-dependent matrices and tensors that are either given explicitly via their increments or are unknown solutions of high-dimensional differential equations, such as multi-particle time-dependent Schrödinger equations. Recently developed numerical time integrators are based on splitting the projector onto the tangent space of the low-rank manifold at the current approximation. In contrast to all standard integrators, these projector-splitting methods are robust with respect to the presence of small singular values in the low-rank approximation. This robustness relies on geometric properties of the low-rank manifolds. Marcin Napiórkowski (University of Warsaw, Poland) Recent advances in the derivation of effective dynamics of many boson systems Joint work with: Phan Thanh Nam. The Gross-Pitaevskii (or nonlinear Schrodinger) equation provides an effective description for systems composed of many bosons. However, using this equation one can approximate the full many-body Schrodinger dynamics only in terms of the reduced densities. In my talk I would like to present recent advances in the derivation of effective equations which allow to approximate the many-body dynamics in norm. 8 Hassan Safouhi (University of Alberta, Canada) Eigenvalues of the Schrödinger Equation with Anharmonic Oscillators Joint work with: Philippe Gaudreau and Richard Slevinsky The one dimensional anharmonic oscillator is of great interest to field theoreticians because it models complicated fields in one-dimensional space-time. The study of quantum anharmonic oscillators as potentials in the Schrödinger equation has been on the edge of thrilling and exciting research during the past three decades [1–3]. Numerous approaches which have been proposed to solve this problem and while several of these methods yield excellent results for specific cases, it would be favorable to have one general method that could handle efficiently and accurately any anharmonic potential. In this talk, we present a method based on the double exponential Sinc collocation method (DESCM) for numerically solving the Schrödinger equation with anharmonic oscillator. The Sinc collocation methods (SCM) have been used extensively during the last three decades to solve many problems in numerical analysis [4]. Their applications include numerical integration, linear and non-linear ordinary differential equations. The double exponential transformation yields optimal accuracy for a given number of function evaluations when using the trapezoidal rule in numerical integration. Recently, combination of the SCM with the double exponential (DE) transformation has sparked great interest [5]. Using DESCM, the eigenvalues are computed to unprecedented accuracy and efficiency [6–8]. DESCM starts by approximating the wave function as a series of weighted Sinc functions in the eigenvalue problem and evaluating the expression at several collocation points spaced by a given mesh size h, we obtain a generalized eigensystem which can be transformed into a regular eigenvalue problem. The proposed method is successfully applied to Coulombic anharmonic oscillator potentials that describe the interaction between charged particles and consistently arises in physical applications. These applications include interactions in atomic, molecular and particle physics, and between nuclei in plasma. We will also show how the DESCM can be applied to harmonic oscillators perturbed by a rational function DESCM leading to an unprecedented accuracy in computing the energy eigenvalues. [1] J. Zamastil, J. Cı́zek, and L. Skála. Renormalized perturbation theory for quartic anharmonic oscillator. Ann. Phys. (NY), 276:39–63, 1999. [2] P. Gaudreau, R.M. Slevinsky, and H. Safouhi. An asymptotic expansion for energy eigenvalues for anharmonic oscillators. Ann. Phys., 337:261–277, 2013. [3] A.V. Turbiner. Double well potential: Perturbation theory, tunneling, WKB (beyond instantons). Int. J. Mod. Phys. A, 25:647–658, 2010. [4] F. Stenger. Summary of Sinc numerical methods. J. Comput. Appl. Math., 121:379–420, 2000. [5] K. Tanaka, M. Sugihara, and K. Murota. Function classes for successful DE-Sinc approximations. [6] P. Gaudreau and H. Safouhi. Centrosymmetric matrices in the Sinc collocation method for Sturm-Liouville problems. European Physical Journal, 108, 01004, 2016. [7] P. Gaudreau, R. Slevinsky and H. Safouhi. The Double Exponential Sinc Collocation Method for Singular Sturm- Liouville Problems. Journal of Mathematical Physics, In press, 2016. [8] P. Gaudreau, R. Slevinsky and H. Safouhi. Computing Energy Eigenvalues of Anharmonic Oscillators using the Double Exponential Sinc collocation Method. Annals of Physics., 360, 520–538, 2015. 9 Pranav Singh (University of Oxford, United Kingdom) An algebraic theory for higher-order methods in computational quantum mechanics We have recently devised a series of high-order methods for solving the time-dependent Schrödinger equation which are particularly effective in the semiclassical regime. These are the nested-commutator-free symmetric Zassenhaus, Magnus–Lanczos and Magnus– Zassenhaus methods. In this talk, we will see how the effectiveness of these methods arises from the structural properties of the Lie algebra of certain symmetrised differential operators. The study of this algebra provides a theoretical underpinning for the development and analysis of our methods as well as paving a way for their application to other equations of quantum mechanics. Mechthild Thalhammer (University of Innsbruck, Austria) Commutator-free quasi-Magnus exponential integrators combined with operator splitting methods and their areas of application In this talk, I shall introduce the class of commutator-free quasi-Magnus exponential integrators for non-autonomous linear evolution equations and identify different areas of application. Commutator-free quasi-Magnus exponential integrators are (formally) given by a composition of several exponentials that comprise certain linear combinations of the values of the defining operator at specified nodes. Avoiding the evaluation of commutators, they provide a favourable alternative to standard Magnus integrators. Non-autonomous linear evolution equations also arise as a part of more complex problems, for instance in connection with nonlinear evolution equations of the form u0 (t) = A(t)u(t)+B(u(t)). A natural approach is thus to apply commutator-free quasi-Magnus exponential integrators combined with operator splitting methods. Relevant applications include Schrödinger equations with space-time-dependent potential describing Bose-Einstein condensation or diffusion-reaction systems modelling pattern formation. Piotr Zgliczyński (Jagiellonian University, Poland) Stabilizing effect of large average initial velocity in forced dissipative PDEs invariant with respect to Galilean transformations We describe a topological method to study the dynamics of dissipative PDEs on a torus with rapidly oscillating forcing terms. We show that a dissipative PDE, which is invariant with respect to the Galilean transformations, with a large average initial velocity can be reduced to a problem with rapidly oscillating forcing terms. We apply the technique to the viscous Burgers’ equation, and the incompressible 2D Navier-Stokes equations with a time-dependent forcing. We prove that for a large initial average speed the equation admits a bounded eternal solution, which attracts all other solutions forward in time. For the incompressible 3D Navier-Stokes equations we establish the existence of a locally attracting eternal solution. This talk is based on paper J. Cyranka and P. Zgliczyński, Stabilizing effect of large average initial velocity in forced dissipative PDEs invariant with respect to Galilean transformations, J. Diff. Eq., 261(2016) 4648– 4708 10 Recommended restaurants in the surroundings • • • • • • • • • A - Aı̈oli (international quisine) D - Sexy Duck (Italian quisine) F - Sofra (Middle Eastern quisine) G - Szkola Gastronomyczna (Polish quisine, practise restaurant of a gastronomy school) K - Hala Koszyki (house of restaurants of many kinds - including Mexican, German, sushi...) L - Lanse (Wiesz, co zjesz (Polish quisine) M - Manekin (pancake place, non-alcoholic beverages) S - Secado (international quisine, mixed drinks) W - U Szwejka (Czech-like quisine, beer) 11 Elena Celledoni (Norwegian University for Science and Technology, Trondheim) Hamiltonian 19:00 16:15 – 17:00 Stabilizing effect of large average initial velocity in 15:45 – 16:15 Workshop Dinner Ernst Hairer (University of Geneva) Arieh Iserles (University of Cambridge) charged particle dynamics Marcin Napiórkowski (University of Warsaw) dynamics of many boson systems Recent advances in the derivation of effective It takes a wave packet to catch a wave packet 30 min 45 min Coffee break Long-time behaviour of numerical integrators for Piotr Zgliczyński (Jagiellonian University, Kraków) Galilean transformations forced dissipative PDEs invariant with respect to Coffee break Hassan Safouhi (University of Alberta) Pranav Singh (University of Oxford) 15:30 – 15:45 Anharmonic Oscillators computational quantum mechanics 2 x 45 min Eigenvalues of the Schrödinger Equation with An algebraic theory for higher-order methods in Winfried Auzinger (Vienna University of Technology) Karolina Kropielnicka (Polish Academy of Sciences) 15:00 – 15:30 Adaptive integrators for Schrödinger-type equations Commutator-free Magnus based methods Lunch Othmar Koch (University of Vienna) Schroedinger-Poisson equation 14:30 – 15:00 Erwan Faou (University of Rennes 1) Fernando Casas (Jaume I University, Castellón) 12:45 Lunch wave for the discrete nonlinear Schrödinger equation oscillations problem in matter Adaptive time-splitting FEM discretization of the Lunch 12:00 – 12:45 On the long time stability of travelling An efficient numerical integrator for the neutrino Mechthild Thalhammer (University of Innsbruck) methods and their areas of application integrators combined with operator splitting Commutator-free quasi-Magnus exponential Coffee break Weizhu Bao (National University of Singapore) [2/2] Break 12:30 – 14:30 12:00 – 12:30 manifolds Schrödinger equation with time-dependent Sergio Blanes (Polytechnic University of Valencia) Shape analysis on Lie groups and Homogeneous Time-average symplectic propagators for the 11:15 – 12:00 Coffee break Coffee break Caroline Lasser (Technical University of Münich) Christian Lubich (Tübingen University) 10:45 – 11:15 [2/2] Dynamical low-rank approximation 10:00 – 10:45 Break Break 9:45 – 10:00 Weizhu Bao (National University of Singapore) equation in the nonrelativistic limit regime [1/2] principle and its applications Christian Lubich (Tübingen University) Multiscale methods and analysis for the Dirac Caroline Lasser (Technical University of Münich) [IM PAN room 321] Computational semiclassics [1/2] Coffee and light refreshments Wednesday, 22nd March The Dirac-Frenkel time-dependent variational [IM PAN room 321] 9:00 – 9:45 Coffee and light refreshments Tuesday, 21st March Coffee and light refreshments [IM PAN room 403] 8:30 – 9:00 Monday, 20th March