Abstract

In this work, we present stochastic variants of an optimization method for the problem of recovering a spatially varying parameter in a scalar partial differential equation (PDE) from measurement(s) of the state variable. In many inverse problems associated with PDE models, the goal is to utilize observational measurements of the state variable in the estimation of the model parameter(s). In the problem being studied, the hydraulic conductivity of an aquifer (a function of location) is the parameter/coefficient to be recovered and the hydraulic (or piezometric) head is the quantity observed/measured. We discuss the choices of objective functional (such as output least squares and modified output least squares), finite element discretizations, as well as the discrete formulas for the objective functionals, and their gradients. We examine applications of stochastic variations of a gradient-based optimization method for the problem of estimating a coefficient in a scalar PDE used in groundwater modeling. The main focus of this work is to investigate the effects of computational parameters such as step size in gradient-based optimization methods, regularization parameter in the optimization problem, and finite element mesh size in the quality of the recovered parameter. Results of numerical simulations with synthetic data and those with simulation of flow through an aquifer with water channels are presented. We also consider the groundwater flow system modeled by a stochastic PDE model as measurement data used in the solution is often noisy in real-life situations. We propose a stochastic approximation scheme for the problem of estimating hydraulic conductivity from noisy hydraulic head measurements. We present the results of numerical simulations with synthetic data as well as with discrete hydraulic conductivity functions that resemble flow through groundwater channels and discuss variations of the proposed gradient-based scheme.

Publication Date

6-2024

Document Type

Dissertation

Student Type

Graduate

Degree Name

Mathematical Modeling (Ph.D)

Department, Program, or Center

Mathematics and Statistics, School of

College

College of Science

Advisor

Basca Jadamba

Advisor/Committee Member

Chao Peng

Advisor/Committee Member

Akhtar Khan

Campus

RIT – Main Campus

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