A computational framework for the solution of infinite-dimensional Bayesian statistical inverse problems with application to global seismic inversion

Quantifying uncertainties in large-scale forward and inverse PDE simulations has emerged as a central challenge facing the field of computational science and engineering. The promise of modeling and simulation for prediction, design, and control cannot be fully realized unless uncertainties in models are rigorously quantified, since this uncertainty can potentially overwhelm the computed result. While statistical inverse problems can be solved today for smaller models with a handful of uncertain parameters, this task is computationally intractable using contemporary algorithms for complex systems characterized by large-scale simulations and high-dimensional parameter spaces. In this dissertation, I address issues regarding the theoretical formulation, numerical approximation, and algorithms for solution of infinite-dimensional Bayesian statistical inverse problems, and apply the entire framework to a problem in global seismic wave propagation. Classical (deterministic) approaches to solving inverse problems attempt to recover the “best-fit” parameters that match given observation data, as measured in a particular metric. In the statistical inverse problem, we go one step further to return not only a point estimate of the best medium properties, but also a complete statistical description of the uncertain parameters. The result is a posterior probability distribution that describes our state of knowledge after learning from the available data, and provides a complete description of parameter uncertainty. In this dissertation, a computational framework for such problems is described that wraps around the existing forward solvers, as long as they are appropriately equipped, for a given physical problem. Then a collection of tools, insights and numerical methods may be applied to solve the problem, and interrogate the resulting posterior distribution, which describes our final state of knowledge. We demonstrate the framework with numerical examples, including inference of a heterogeneous compressional wavespeed field for a problem in global seismic wave propagation with 10⁶ parameters.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Computational science, engineering, and mathematics

Grantor

The University of Texas at Austin

Year dc:date.issued

2015

Author dc:creator

• Martin, James Robert, Ph. D.

Advisor dc:contributor.advisor

• Ghattas, Omar N.

Committee members dc:contributor.committeemember

• Biros, George
• Demkowicz, Leszek
• Fomel, Sergey
• Marzouk, Youssef
• Moser, Robert

• Bayesian inference
• Infinite-dimensional inverse problems
• Uncertainty quantification
• Low-rank approximation
• Optimality
• Scalable algorithms
• High performance computing
• Markov chain Monte Carlo
• Stochastic Newton
• Seismic wave propagation

Language dc:language.iso

en

Handle dc:identifier.uri

http://hdl.handle.net/2152/31374

OAI identifier oai:identifier

oai:repositories.lib.utexas.edu:2152/31374