Infinite dimensional stochastic inverse problems
| dc.contributor.author | Yang, Lei, author | |
| dc.contributor.author | Estep, Donald, advisor | |
| dc.contributor.author | Breidt, F. Jay, committee member | |
| dc.contributor.author | Tavener, Simon, committee member | |
| dc.contributor.author | Zhou, Wen, committee member | |
| dc.date.accessioned | 2018-09-10T20:05:53Z | |
| dc.date.available | 2018-09-10T20:05:53Z | |
| dc.date.issued | 2018 | |
| dc.description.abstract | In many disciplines, mathematical models such as differential equations, are used to characterize physical systems. The model induces a complex nonlinear measurable map from the domain of physical parameters to the range of observable Quantities of Interest (QoI) computed by applying a set of functionals to the solution of the model. Often the parameters can not be directly measured, and people are confronted with the task of inferring information about values of the parameters given the measured or imposed information about the values of the QoI. In such applications, there is generally significant uncertainty in the measured values of the QoI. Uncertainty is often modeled using probability distributions. For example, a probability structure imposed on the domain of the parameters induces a corresponding probability structure on the range of the QoI. This is the well known Stochastic Forward Problem that is typically solved using a variation of the Monte Carlo method. This dissertation is concerned with the Stochastic Inverse Problems (SIP) where the probability distributions are imposed on the range of the QoI, and problem is to compute the induced distributions on the domain of the parameters. In our formulation of the SIP and its generalization for the case where the physical parameters are functions, main topics including the existence, continuity and numerical approximations of the solutions are investigated. Chapter 1 introduces the background and previous research on the SIP. It also gives useful theorems, results and notation used later. Chapter 2 begins by establishing a relationship between Lebesgue measures on the domain and the range, and then studies the form of solution of the SIP and its continuity properties. Chapter 3 proposes an algorithm for computing the solution of the SIP, and discusses the convergence of the algorithm to the true solution. Chapter 4 exploits the fact that a function can be represented by its coefficients with respect to a basis, and extends the SIP framework to allow for cases where the domain representing the basis coefficients is a countable cube with decaying edges, referred to as the infinite dimensional SIP. We then discusses how its solution can be approximated by the SIP for which the domain is the finite dimensional cube obtained by taking a finite dimensional projection of the countable cube. Chapter 5 begins with an algorithm for approximating the solution of the infinite dimensional SIP, and then proves the algorithm converges to the true solution. Chapter 6 gives a numerical example showing the effects of different decay rates and the relation to truncation to finite dimensions. Chapter 7 reviews popular probabilistic inverse problem methods and proposes a combination of the SIP and statistical models to address problems encountered in practice. | |
| dc.format.medium | born digital | |
| dc.format.medium | doctoral dissertations | |
| dc.identifier | Yang_colostate_0053A_15095.pdf | |
| dc.identifier.uri | https://hdl.handle.net/10217/191489 | |
| dc.language | English | |
| dc.language.iso | eng | |
| dc.publisher | Colorado State University. Libraries | |
| dc.relation.ispartof | 2000-2019 | |
| dc.rights | Copyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright. | |
| dc.subject | generalized contour | |
| dc.subject | measure theory | |
| dc.subject | Hilbert cube | |
| dc.subject | disintegration | |
| dc.title | Infinite dimensional stochastic inverse problems | |
| dc.type | Text | |
| dcterms.rights.dpla | This Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). | |
| thesis.degree.discipline | Statistics | |
| thesis.degree.grantor | Colorado State University | |
| thesis.degree.level | Doctoral | |
| thesis.degree.name | Doctor of Philosophy (Ph.D.) |
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