Abstract

This dissertation presents a computational framework for solving stochastic inverse problems in partial differential equations (PDEs) with random data. The objective is to recover unknown random coefficients in parabolic and elliptic diffusion equations, as well as nearly incompressible linear elasticity models. To achieve this, optimization-based approaches are used to minimize regularized Energy Least Squares (ELS) and Output Least Squares (OLS) objective functionals. Both direct and adjoint methods are used to estimate the gradient and Hessian of the objective functionals with respect to the coefficients. To avoid the expensive computation of the full Hessian matrix, the Newton-CG-like method is also adopted, which relies only on the Hessian-vector products. Finally, we develop Stochastic Galerkin discretizations for both the forward and inverse problems, providing discrete expressions for the objective functionals, their gradients, and their Hessian (and Hessian-vector product), and we finally provide numerical examples. We also present a stochastic auxiliary problem principle for solving variational inequalities. We establish its convergence under suitable assumptions and then apply it to estimate deterministic coefficients in PDEs with random data. The dissertation concludes by proposing future research in uncertainty quantification and the Calderón problem.

Publication Date

8-3-2026

Document Type

Dissertation

Student Type

Graduate

Degree Name

Mathematical Modeling (Ph.D)

Department, Program, or Center

Mathematical Sciences, School of

College

College of Science

Advisor

Akhtar A. Khan

Advisor/Committee Member

Charles Bachmann

Advisor/Committee Member

Basca Jadamba

Campus

RIT – Main Campus

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