Abstract

The primary objective of this thesis is to develop a fast and efficient computational framework for the nonlinear inverse problem of identifying a variable coefficient in a system of partial differential equation modelling the response of an incompressible elastic object under some known body forces and boundary traction. The main novelty of this contribution is to use, for the first time, of the so-called heavy ball with friction method for inverse problems. The heavy ball with friction dynamical system is a nonlinear oscillator with damping. The key idea is to pose the inverse problem as an optimization problem, derive its optimality system, and then seek the solution through a trajectory of a dynamical system. In this work, we will study four different optimization formulations for the nonlinear inverse problem and thoroughly compare their convergence and numerical performance. Since we use a second-order method, we also investigate a general second-order hybrid and a second-order adjoint method for an efficient computation of the hessian of the output least-squares formulation. The stability of the dynamical system approach with respect to the contamination in the data is thoroughly investigate in the context of a simpler elliptic partial differential equation. The mixed finite element approach is used to discretize the direct as well as the inverse problems.

Library of Congress Subject Headings

Elastography--Mathematics; Inverse problems (Differential equations); Differential equations, Partial; Mathematical optimization

Publication Date

9-21-2015

Document Type

Thesis

Student Type

Graduate

Degree Name

Applied and Computational Mathematics (MS)

Department, Program, or Center

School of Mathematical Sciences (COS)

Advisor

Akhtar Khan

Advisor/Committee Member

Baasansuren Jadamba

Advisor/Committee Member

Patricia Clark

Comments

Physical copy available from RIT's Wallace Library at QA378.5 .E58 2015

Campus

RIT – Main Campus

Plan Codes

ACMTH-MS

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