Nonsmooth Nonlinearities in Applications from Hydrology

Kavanagh, Kathleen Rose. Nonsmooth Nonlinearities in Applications From Hydrology. (Under the direction of C.T. Kelley) This work has two parts; simulation of unsaturated flow and optimization of remediation problems. For the unsaturated flow simulation, we propose an adaptive time stepping scheme based on error control for Richards’ equation, a model for flow in unsaturated porous media. The motivation for this work is a ground and surface water simulator being developed by the U.S. Engineering Research Development Center called the ADaptive Hydrology Model. ADH uses unstructured, adaptive finite elements. ADH advances in time implicitly, solving the nonlinear equations with an inexact—Newton method with a two-level domain decomposition preconditioner. The nonlinearity in Richards’ Equation can be non-Lipschitz and nonsmooth. Standard theory for temporal integration may not apply for certain physical parameters. We consider a method for error estimation and control for temporal adaption. In the optimization section, we investigate a suite of test problems from the literature that are intended for benchmarking purposes and comparison of optimization algorithms. The objective functions can be nonsmooth, nonconvex, or have several minima that may trap standard gradient based methods. We apply the implicit filtering algorithm to some such problems. NONSMOOTH NONLINEARITIES IN APPLICATIONS FROM HYDROLOGY by Kathleen R. Kavanagh a dissertation submitted to the graduate faculty of north carolina state university in partial fulfillment of the requirements for the degree of doctor of philosophy applied mathematics, computational mathematics concentration raleigh, north carolina July 2003