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How accurate is molecular dynamics?
Weierstrass Institute for Applied Analysis and Stochastics.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
Dept. of Mathematical Sciences, University of Delaware.
Visa övriga samt affilieringar
2012 (Engelska)Rapport (Övrigt vetenskapligt)
Abstract [en]

Born-Oppenheimer dynamics is shown to provide an accurate approximation of time-independent Schrödinger observables for a molecular system with an electron spectral gap, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on a Hamiltonian system interpretation of the Schrödinger equation and stability of the corresponding hitting time Hamilton-Jacobi equation for non ergodic dynamics, bypasses the usual separation of nuclei and electron wave functions, includes caustic states and gives a different perspective on theBorn-Oppenheimer approximation, Schrödinger Hamiltonian systems and numerical simulation in molecular dynamics modeling at constant energy.

Ort, förlag, år, upplaga, sidor
2012. , s. 47
Serie
TRITA-NA, ISSN 0348-2952 ; 2012:8
Nyckelord [en]
Born-Oppenheimer approximation, WKB expansion, caustics, Fourier integral operators, Schrödinger operators
Nationell ämneskategori
Matematisk analys Atom- och molekylfysik och optik
Identifikatorer
URN: urn:nbn:se:kth:diva-94125OAI: oai:DiVA.org:kth-94125DiVA, id: diva2:525383
Forskningsfinansiär
Swedish e‐Science Research Center
Anmärkning

QC 20120508

Tillgänglig från: 2012-05-08 Skapad: 2012-05-07 Senast uppdaterad: 2016-12-22Bibliografiskt granskad
Ingår i avhandling
1. Complexity and Error Analysis of Numerical Methods for Wireless Channels, SDE, Random Variables and Quantum Mechanics
Öppna denna publikation i ny flik eller fönster >>Complexity and Error Analysis of Numerical Methods for Wireless Channels, SDE, Random Variables and Quantum Mechanics
2012 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

This thesis consists of the four papers which consider different aspects of stochastic process modeling, error analysis, and minimization of computational cost.

     In Paper I, we construct a Multipath Fading Channel (MFC) model for wireless channels with noise introduced through scatterers flipping on and off. By coarse graining the MFC model a Gaussian process channel model is developed. Complexity and accuracy comparisons of the models are conducted.

     In Paper II, we generalize a multilevel Forward Euler Monte Carlo method introduced by Mike Giles for the approximation of expected values depending on solutions of Ito stochastic differential equations. Giles' work proposed and analyzed a Forward Euler Multilevel Monte Carlo (MLMC) method based on realizations on a hierarchy of uniform time discretizations and a coarse graining based control variates idea to reduce the computational cost required by a standard single level Forward Euler Monte Carlo method. This work is an extension of Giles' MLMC method from uniform to adaptive time grids. It has the same improvement in computational cost and is applicable to a larger set of problems.

     In paper III, we consider the problem to estimate the mean of a random variable by a sequential stopping rule Monte Carlo method. The performance of a typical second moment based sequential stopping rule is shown to be unreliable both by numerical examples and by analytical arguments. Based on analysis and approximation of error bounds we construct a higher moment based stopping rule which performs more reliably.

     In paper IV, Born-Oppenheimer dynamics is shown to provide an accurate approximation of time-independent Schrödinger observables for a molecular system with an electron spectral gap, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on a Hamiltonian system interpretation of the Schrödinger equation and stability of the corresponding hitting time Hamilton-Jacobi equation for non ergodic dynamics, bypasses the usual separation of nuclei and electron wave functions, includes caustic states and gives a different perspective on the Born-Oppenheimer approximation, Schrödinger Hamiltonian systems and numerical simulation in molecular dynamics modeling at constant energy.

Ort, förlag, år, upplaga, sidor
Stockholm: KTH Royal Institute of Technology, 2012. s. vii, 65
Serie
Trita-CSC-A, ISSN 1653-5723 ; 2012:06
Nyckelord
Wireless Channels; SDE; Monte Carlo Methods, Molecular Dynamics, Quantum Mechanics
Nationell ämneskategori
Beräkningsmatematik Sannolikhetsteori och statistik
Identifikatorer
urn:nbn:se:kth:diva-94150 (URN)978-91-7501-350-3 (ISBN)
Disputation
2012-05-30, F3, Lindstedtsvägen 26, Kungliga Tekniska högskolan, Stockholm, 10:15 (Engelska)
Opponent
Handledare
Forskningsfinansiär
Swedish e‐Science Research Center
Anmärkning

QC 20120508

Tillgänglig från: 2012-05-08 Skapad: 2012-05-08 Senast uppdaterad: 2013-04-09Bibliografiskt granskad

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Hoel, HåkonKadir, AshrafulSandberg, MattiasSzepessy, Anders
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Numerisk analys, NA (stängd 2012-06-30)
Matematisk analysAtom- och molekylfysik och optik

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