Download Workshop Modern Numerical Methods in Quantum Mechanics

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
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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.
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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.
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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.
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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.
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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.
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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.
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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
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Recommended restaurants in the surroundings
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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)
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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