uu.seUppsala University Publications

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Efficient and Reliable Simulation of Quantum Molecular DynamicsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2012 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Acta Universitatis Upsaliensis, 2012. , p. 52
##### Series

Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 970
##### Keywords [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, id: diva2:549981
##### Public defence

2012-10-19, Room 2446, Polacksbacken, Lägerhyddsvägen 2D, Uppsala, 10:00 (English)
##### Opponent

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##### Supervisors

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#####

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##### Projects

eSSENCE
##### Funder

eSSENCE - An eScience CollaborationAvailable from: 2012-09-27 Created: 2012-09-01 Last updated: 2013-01-23Bibliographically approved
##### List of papers

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.

1. Accurate time propagation for the Schrödinger equation with an explicitly time-dependent Hamiltonian$(function(){PrimeFaces.cw("OverlayPanel","overlay43951",{id:"formSmash:j_idt925:0:j_idt935",widgetVar:"overlay43951",target:"formSmash:j_idt925:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Global error control of the time-propagation for the Schrödinger equation with a time-dependent Hamiltonian$(function(){PrimeFaces.cw("OverlayPanel","overlay432117",{id:"formSmash:j_idt925:1:j_idt935",widgetVar:"overlay432117",target:"formSmash:j_idt925:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. A time–space adaptive method for the Schrödinger equation$(function(){PrimeFaces.cw("OverlayPanel","overlay548653",{id:"formSmash:j_idt925:2:j_idt935",widgetVar:"overlay548653",target:"formSmash:j_idt925:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Error control for simulations of a dissociative quantum system$(function(){PrimeFaces.cw("OverlayPanel","overlay359737",{id:"formSmash:j_idt925:3:j_idt935",widgetVar:"overlay359737",target:"formSmash:j_idt925:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Communication-efficient algorithms for numerical quantum dynamics$(function(){PrimeFaces.cw("OverlayPanel","overlay375913",{id:"formSmash:j_idt925:4:j_idt935",widgetVar:"overlay375913",target:"formSmash:j_idt925:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

6. Parallel finite element operator application: Graph partitioning and coloring$(function(){PrimeFaces.cw("OverlayPanel","overlay475824",{id:"formSmash:j_idt925:5:j_idt935",widgetVar:"overlay475824",target:"formSmash:j_idt925:5:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

7. An RBF–Galerkin approach to the time-dependent Schrödinger equation$(function(){PrimeFaces.cw("OverlayPanel","overlay549957",{id:"formSmash:j_idt925:6:j_idt935",widgetVar:"overlay549957",target:"formSmash:j_idt925:6:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

8. A Fourier-coefficient based solution of an optimal control problem in quantum chemistry$(function(){PrimeFaces.cw("OverlayPanel","overlay331737",{id:"formSmash:j_idt925:7:j_idt935",widgetVar:"overlay331737",target:"formSmash:j_idt925:7:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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