Partial Differential Equations (PDEs) are commonly used to model complex systems that arise for example in biology, engineering, chemistry, and elsewhere. The parameters (or coefficients) and the source of PDE models are often unknown and are estimated from available measurements. Despite its importance, solving the estimation problem is mathematically and numerically challenging and especially when the measurements are corrupted by noise, which is often the case. Various methods have been proposed to solve estimation problems in PDEs which can be classified into optimization methods and recursive methods. The optimization methods are usually heavy computationally, especially when the number of unknowns is large. In addition, they are sensitive to the initial guess and stop condition, and they suffer from the lack of robustness to noise. Recursive methods, such as observer-based approaches, are limited by their dependence on some structural properties such as observability and identifiability which might be lost when approximating the PDE numerically. Moreover, most of these methods provide asymptotic estimates which might not be useful for control applications for example. An alternative non-asymptotic approach with less computational burden has been proposed in engineering fields based on the so-called modulating functions. In this dissertation, we propose to mathematically and numerically analyze the modulating functions based approaches. We also propose to extend these approaches to different situations. The contributions of this thesis are as follows. (i) Provide a mathematical analysis of the modulating function-based method (MFBM) which includes: its well-posedness, statistical properties, and estimation errors. (ii) Provide a numerical analysis of the MFBM through some estimation problems, and study the sensitivity of the method to the modulating functions' parameters. (iii) Propose an effective algorithm for selecting the method's design parameters. (iv) Develop a two-dimensional MFBM to estimate space-time dependent unknowns which is illustrated in estimating the source term in the damped wave equation describing the physiological characterization of brain activity. (v) Introduce a moving horizon strategy in the MFBM for on-line estimation and examine its effectiveness on estimating the source term of a first order hyperbolic equation which describes the heat transfer in distributed solar collector systems.
|Date of Award||Oct 8 2017|
|Original language||English (US)|
- Computer, Electrical and Mathematical Science and Engineering
|Supervisor||Meriem Laleg (Supervisor)|
- Inverse problem
- modulating functions