Upcoming Event: PhD Dissertation Defense

On Derivative-Informed Uncertainty Quantification

Joshua Chen, Ph.D. Candidate, Oden Institute

Abstract

The estimation of various statistical quantities is important in various aspects of engineering, finance, and science. These quantities are often either central statistics (mean, mode, median, etc.), or measures of uncertainty (covariance, higher moments, (conditional-)  value-at-risk, entropy, etc.). These quantities aid in the understanding of the status of a system and often inform decision-making under various risk-tolerances on the value of these quantities. However, when the underlying systems which produce these quantities of interest (QoIs) are complex, e.g. modeled by systems of partial differential equations (PDEs) describing physical laws, the estimation of these QoIs can become incredibly computationally demanding, i.e. requiring days or months of computational resources on supercomputers. 

When the benefit of performing such computations out-weighs its driving purpose, we call this computationally intractable. To address intractable computations associated with quantifying uncertainty, especially for systems governed by computationally-complex PDEs, we develop numerical approximations, study some theoretical aspects, and test them on challenging test-problems. All of these numerical approximations, at their core, involve the careful usage of derivatives of quantities dependent on PDEs.

Two primary challenges in uncertainty quantification are addressed. Firstly, in Chapter 1, we address the estimation of challenging-to-compute integrals with respect to some underlying probability measure. This arises often in forward uncertainty quantification, in which one needs to estimate the impact of uncertainty of properties of a system (random variables or random functions) on other (usually directly measured) aspects of a system (other random variables or random functions). Secondly, we address the estimation of posterior probability distributions. In Chapter 2, we address the estimation of a single posterior for a Bayesian inverse problem, speeding up the estimation so that any number of posteriors can be estimated rapidly. In Chapter 3, we extend this approach to estimating every posterior of a Bayesian inverse problem all-at-once, a more challenging problem.

 

Biography

 Joshua Chen received is Bachelor's degree from Virginia Tech in Structural Engineering. His was awarded a Master's of Science in Computational Mechanics and a Master's of Science in Computational Design from Carnegie Mellon University. Most recently, he completed a Masters of Science in Computational Science, Engineering, and Mathematics from the Oden Institute.