#### Abstract

Numerous mathematical models in applied and industrial mathematics take the form of a partial differential equation involving certain variable coefficients. These coefficients are known and they often describe some physical properties of the model. The direct problem in this context is to solve the partial differential equation. By contrast, an inverse problem asks for the identification of the variable coefficient when a certain measurement of a solution of the partial differential equation is available. A commonly used approach to inverse problems is to solve an optimization problem whose solution is an approximation of the sought coefficient. Such optimization problems are typically solved by discrete iterative schemes. It turns out that most known iterative schemes have their continuous counterparts given in terms of dynamical systems. However, such differential equations are usually solved by specific differential equation solvers. The primary objective of this thesis is to test the feasibility of differential equations based solvers for solving elliptic inverse problems. We will use differential equation solvers such as Euler's Method, Trapezoidal Method, Runge-Kutta Method and Adams-Bashforth Method. In addition, these solvers will also be compared to built-in MATLAB ODE solvers. The performance and accuracy of these methods to solve inverse problems will be thoroughly discussed.

#### Library of Congress Subject Headings

Inverse problems (Differential equations)

#### Publication Date

8-11-2014

#### Document Type

Thesis

#### Student Type

Graduate

#### Department, Program, or Center

School of Mathematical Sciences (COS)

#### Advisor

Akhtar Khan

#### Advisor/Committee Member

Baasansuren Jadamba

#### Advisor/Committee Member

Patricia Clark

#### Recommended Citation

Teravainen, Corinne, "Continuous Methods for Elliptic Inverse Problems" (2014). Thesis. Rochester Institute of Technology. Accessed from

https://scholarworks.rit.edu/theses/8338

#### Campus

RIT – Main Campus

#### Plan Codes

ACMTH-MS

## Comments

Physical copy available from RIT's Wallace Library at QA377 .T47 2014