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Efficient and Reliable Simulation of Quantum Molecular Dynamics
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Scientific Computing. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Computational Science.
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The time-dependent Schrödinger 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 Schrödinger 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.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2012. , 52 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 970
Keyword [en]
time-dependent Schrödinger equation, quantum optimal control, exponential integrators, spectral elements, radial basis functions, global error control and adaptivity, high-performance computing implementation
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-180251ISBN: 978-91-554-8466-8 (print)OAI: oai:DiVA.org:uu-180251DiVA: diva2:549981
Public defence
2012-10-19, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 10:00 (English)
Opponent
Supervisors
Projects
eSSENCE
Funder
eSSENCE - An eScience Collaboration
Available from: 2012-09-27 Created: 2012-09-01 Last updated: 2013-01-23Bibliographically approved
List of papers
1. Accurate time propagation for the Schrödinger equation with an explicitly time-dependent Hamiltonian
Open this publication in new window or tab >>Accurate time propagation for the Schrödinger equation with an explicitly time-dependent Hamiltonian
2008 (English)In: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 128, 184101:1-11 p.Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-16180 (URN)10.1063/1.2916581 (DOI)000255983500008 ()
Available from: 2008-05-10 Created: 2008-05-10 Last updated: 2017-12-08Bibliographically approved
2. Global error control of the time-propagation for the Schrödinger equation with a time-dependent Hamiltonian
Open this publication in new window or tab >>Global error control of the time-propagation for the Schrödinger equation with a time-dependent Hamiltonian
2011 (English)In: Journal of Computational Science, ISSN 1877-7503, E-ISSN 1877-7511, Vol. 2, 178-187 p.Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-156538 (URN)10.1016/j.jocs.2011.02.003 (DOI)000208808200011 ()
Projects
eSSENCE
Available from: 2011-02-18 Created: 2011-07-31 Last updated: 2017-12-08Bibliographically approved
3. A time–space adaptive method for the Schrödinger equation
Open this publication in new window or tab >>A time–space adaptive method for the Schrödinger equation
2012 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-023
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-180182 (URN)
Projects
eSSENCE
Available from: 2012-08-31 Created: 2012-08-31 Last updated: 2012-09-27Bibliographically approved
4. Error control for simulations of a dissociative quantum system
Open this publication in new window or tab >>Error control for simulations of a dissociative quantum system
2010 (English)In: Numerical Mathematics and Advanced Applications: 2009, Berlin: Springer-Verlag , 2010, 523-531 p.Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
Berlin: Springer-Verlag, 2010
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-132929 (URN)10.1007/978-3-642-11795-4_56 (DOI)978-3-642-11794-7 (ISBN)
Projects
eSSENCE
Available from: 2010-10-29 Created: 2010-10-29 Last updated: 2012-09-27Bibliographically approved
5. Communication-efficient algorithms for numerical quantum dynamics
Open this publication in new window or tab >>Communication-efficient algorithms for numerical quantum dynamics
2012 (English)In: Applied Parallel and Scientific Computing: Part II, Berlin: Springer-Verlag , 2012, 368-378 p.Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
Berlin: Springer-Verlag, 2012
Series
Lecture Notes in Computer Science, 7134
National Category
Computer Science Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-135980 (URN)10.1007/978-3-642-28145-7_36 (DOI)000309716000036 ()978-3-642-28144-0 (ISBN)
Conference
PARA 2010: State of the Art in Scientific and Parallel Computing
Projects
eSSENCEUPMARC
Available from: 2012-02-16 Created: 2010-12-09 Last updated: 2014-11-21Bibliographically approved
6. Parallel finite element operator application: Graph partitioning and coloring
Open this publication in new window or tab >>Parallel finite element operator application: Graph partitioning and coloring
2011 (English)In: Proc. 7th International Conference on e-Science, Los Alamitos, CA: IEEE Computer Society, 2011, 332-339 p.Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
Los Alamitos, CA: IEEE Computer Society, 2011
National Category
Computer Science Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-166267 (URN)10.1109/eScience.2011.53 (DOI)978-1-4577-2163-2 (ISBN)
Projects
eSSENCE
Available from: 2012-01-09 Created: 2012-01-11 Last updated: 2014-03-14Bibliographically approved
7. An RBF–Galerkin approach to the time-dependent Schrödinger equation
Open this publication in new window or tab >>An RBF–Galerkin approach to the time-dependent Schrödinger equation
2012 (English)Report (Other academic)
Series
Technical report / Department of Information Technology, Uppsala University, ISSN 1404-3203 ; 2012-024
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-180388 (URN)
Projects
eSSENCE
Available from: 2012-09-05 Created: 2012-09-05 Last updated: 2012-09-27Bibliographically approved
8. A Fourier-coefficient based solution of an optimal control problem in quantum chemistry
Open this publication in new window or tab >>A Fourier-coefficient based solution of an optimal control problem in quantum chemistry
2010 (English)In: Journal of Optimization Theory and Applications, ISSN 0022-3239, E-ISSN 1573-2878, Vol. 147, 491-506 p.Article in journal (Refereed) Published
National Category
Computational Mathematics Computer Science
Identifiers
urn:nbn:se:uu:diva-128817 (URN)10.1007/s10957-010-9735-9 (DOI)000283509600006 ()
Projects
eSSENCE
Available from: 2010-07-21 Created: 2010-07-25 Last updated: 2017-12-12Bibliographically approved

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Citation style
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Output format
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