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Complexity and Error Analysis of Numerical Methods for Wireless Channels, SDE, Random Variables and Quantum Mechanics
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
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 [en]
Wireless Channels; SDE; Monte Carlo Methods, Molecular Dynamics, Quantum Mechanics
Nationell ämneskategori
Beräkningsmatematik Sannolikhetsteori och statistik
Identifikatorer
URN: urn:nbn:se:kth:diva-94150ISBN: 978-91-7501-350-3 (tryckt)OAI: oai:DiVA.org:kth-94150DiVA, id: diva2:525531
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: 2022-06-24Bibliografiskt granskad
Delarbeten
1. Gaussian Coarse Graining of a Master Equation Extension of Clarke's Model
Öppna denna publikation i ny flik eller fönster >>Gaussian Coarse Graining of a Master Equation Extension of Clarke's Model
2012 (Engelska)Rapport (Övrigt vetenskapligt) [Forskning på konstnärlig grund]
Abstract [en]

We study the error and computational cost of generating outputsignal realizations for the channel model of a moving receiver in a scatteringenvironment, as in Clarke’s model, with the extension that scatterers randomlyflip on and off. At micro scale, the channel is modeled by a Multipath FadingChannel (MFC) model, and by coarse graining the micro scale model we derivea macro scale Gaussian process model. Four algorithms are presented for gen-erating stochastic signal realizations, one for the MFC model and three for theGaussian process model. A computational cost comparison of the presentedalgorithms indicates that Gaussian process algorithms generate signal realiza-tions more efficiently than the MFC algorithm does. Numerical examples ofgenerating signal realizations in time independent and time dependent scatter-ing environments are given, and the problem of estimating model parametersfrom real life signal measurements is also studied.

Ort, förlag, år, upplaga, sidor
Stockholm: KTH Royal Institute of Technology, 2012. s. 30
Serie
Trita-NA, ISSN 0348-2952 ; 2012:5
Nyckelord
Wireless channel modeling; signal theory; master equations; Gaussian processes
Nationell ämneskategori
Sannolikhetsteori och statistik
Identifikatorer
urn:nbn:se:kth:diva-94105 (URN)
Anmärkning
Funded by Centre for Industrial and Applied Mathematics (CIAM). QC 20120508Tillgänglig från: 2012-05-08 Skapad: 2012-05-07 Senast uppdaterad: 2022-06-24Bibliografiskt granskad
2. Implementation and Analysis of an Adaptive Multilevel Monte Carlo Algorithm
Öppna denna publikation i ny flik eller fönster >>Implementation and Analysis of an Adaptive Multilevel Monte Carlo Algorithm
2012 (Engelska)Rapport (Övrigt vetenskapligt)
Abstract [en]

This work generalizes a multilevel Monte Carlo (MLMC) method in-troduced in [7] for the approximation of expected values of functions depending on the solution to an Ito stochastic differential equation. The work [7] proposed and analyzed a forward Euler MLMC method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a standard, single level, forward Euler Monte Carlo method from O( TOL^(−3) ) to O( TOL^(−2) log( TOL^(−1))^2 ) for a meansquare error of size 2 . This work uses instead a hierarchy of adaptivelyrefined, non uniform, time discretizations, generated by an adaptive algo-rithm introduced in [20, 19, 5]. Given a prescribed accuracy TOL in theweak error, this adaptive algorithm generates time discretizations basedon a posteriori expansions of the weak error first developed in [24]. Atheoretical analysis gives results on the stopping, the accuracy, and thecomplexity of the resulting adaptive MLMC algorithm. In particular, it isshown that: the adaptive refinements stop after a finite number of steps;the probability of the error being smaller than TOL is under certain as-sumptions controlled by a given confidence parameter, asymptotically asTOL → 0; the complexity is essentially the expected for MLMC methods,but with better control of the constant factors. We also show that themultilevel estimator is asymptotically normal using the Lindeberg-FellerCentral Limit Theorem. These theoretical results are based on previouslydeveloped single level estimates, and results on Monte Carlo stoppingfrom [3]. Our numerical tests include cases, one with singular drift andone with stopped diffusion, where the complexity of uniform single levelmethod is O TOL−4 . In both these cases the results confirm the theoryby exhibiting savings in the computational cost to achieve an accuracy of O(TOL), from O( TOL^(−3) )for the adaptive single level algorithm toessentially O( TOL^(−2) log(TOL−1)^2 ) for the adaptive MLMC.

Förlag
s. 57
Serie
TRITA-NA, ISSN 0348-2952 ; 2012:6
Nyckelord
computational finance; Monte Carlo; multi-level; adaptivity; weak approximation
Nationell ämneskategori
Sannolikhetsteori och statistik Datorteknik
Identifikatorer
urn:nbn:se:kth:diva-94108 (URN)
Anmärkning

QC 20120508

Tillgänglig från: 2012-05-08 Skapad: 2012-05-07 Senast uppdaterad: 2024-03-18Bibliografiskt granskad
3. On non-asymptotic optimal stopping criteria in Monte Carlo simulations
Öppna denna publikation i ny flik eller fönster >>On non-asymptotic optimal stopping criteria in Monte Carlo simulations
2012 (Engelska)Rapport (Övrigt vetenskapligt)
Abstract [en]

We consider the setting of estimating the mean of a random variable by a sequential stopping rule Monte Carlo (MC) method. The performance of a typical second moment based sequential stopping rule MC method is shown to be unreliable in such settings both by numerical examples and through analysis. By analysis and approximations, we construct a higher moment based stopping rule which is shown in numerical examples to perform more reliably and only slightly less efficiently than the second moment based stopping rule.

Förlag
s. 16
Serie
Trita-NA, ISSN 0348-2952 ; 2012:7
Nyckelord
Monte Carlo methods; optimal stopping; sequential stopping rules; non-asymptotic
Nationell ämneskategori
Sannolikhetsteori och statistik Datavetenskap (datalogi)
Identifikatorer
urn:nbn:se:kth:diva-94109 (URN)
Anmärkning
QC 20120508Tillgänglig från: 2012-05-08 Skapad: 2012-05-07 Senast uppdaterad: 2022-06-24Bibliografiskt granskad
4. How accurate is molecular dynamics?
Öppna denna publikation i ny flik eller fönster >>How accurate is molecular dynamics?
Visa övriga...
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.

Förlag
s. 47
Serie
TRITA-NA, ISSN 0348-2952 ; 2012:8
Nyckelord
Born-Oppenheimer approximation, WKB expansion, caustics, Fourier integral operators, Schrödinger operators
Nationell ämneskategori
Matematisk analys Atom- och molekylfysik och optik
Identifikatorer
urn:nbn:se:kth:diva-94125 (URN)
Forskningsfinansiär
Swedish e‐Science Research Center
Anmärkning

QC 20120508

Tillgänglig från: 2012-05-08 Skapad: 2012-05-07 Senast uppdaterad: 2024-03-15Bibliografiskt granskad

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