Steadystate elliptic partial differential equations (PDEs) are frequently used to model a diverse range of physical phenomena. The source and boundary data estimation problems for such PDE systems are of prime interest in various engineering disciplines including biomedical engineering, mechanics of materials and earth sciences. Almost all existing solution strategies for such problems can be broadly classified as optimizationbased techniques, which are computationally heavy especially when the problems are formulated on higher dimensional space domains. However, in this dissertation, feedback based state estimation algorithms, known as state observers, are developed to solve such steadystate problems using one of the space variables as timelike. In this regard, first, an iterative observer algorithm is developed that sweeps over regularshaped domains and solves boundary estimation problems for steadystate Laplace equation. It is wellknown that source and boundary estimation problems for the elliptic PDEs are highly sensitive to noise in the data. For this, an optimal iterative observer algorithm, which is a robust counterpart of the iterative observer, is presented to tackle the illposedness due to noise. The iterative observer algorithm and the optimal iterative algorithm are then used to solve source localization and estimation problems for Poisson equation for noisefree and noisy data cases respectively. Next, a divide and conquer approach is developed for threedimensional domains with two congruent parallel surfaces to solve the boundary and the source data estimation problems for the steadystate Laplace and Poisson kind of systems respectively. Theoretical results are shown using a functional analysis framework, and consistent numerical simulation results are presented for several test cases using finite difference discretization schemes.
Date of Award  Jul 19 2017 

Original language  English (US) 

Awarding Institution   Computer, Electrical and Mathematical Sciences and Engineering


Supervisor  Taousmeriem Laleg (Supervisor) 

 observer design
 Inverse problems
 Boundary Estimation
 source localization
 elliptic PDEs
 kalman filter