Quantifying Uncertainty For an Elliptic Inverse Problem with Finite Data

Abstract

The field of inverse problems is an area of applied mathematics that is of great importance in several scientific and industrial applications. Since an inverse problem is typically based on non-linear and ill-posed models it is often a very difficult problem to solve.

In this thesis we consider a model inverse problem in which we seek to reconstruct a coefficient in an elliptic partial differential equation from finite data. Firstly, we provide a general framework for solving such problem in Chapter 1. Then, the inverse problem is transformed into a system of ODEs by the method of characteristics. The emphasis is on the following three aspects, where new results are obtained: a) Uniqueness of the solution of our model problem follows under some assumptions (given in Chapter 2), involving some basic concepts and using the method of characteristics. A result concerning uniqueness of the unknown parameter in the reduced model problem is proved, when the dependent variable u is completely known. b) Error estimates are derived in two dimensions using radial basis function (RBF) methods (Chapter 3, 4). An RBF inequality related to our model problem is reviewed in Chapter 3. Also an error estimate between the exact value and the approximate value of the unknown coefficient is given in Chapter 4. This inequality shows us that our parameter uncertainty is bounded by a norm in a suitable space, as well as properties of the domain and information gathered from observational data. To prove this inequality, we also derive a PDE estimate, a Sobolev embedding estimate and a RBF interpolation estimate in Chapter 4. c) Numerical methods and reconstruction algorithms are presented (Chapter 5). We provide two different algorithms to reconstruct the unknown parameter a and multiple numerical simulations are presented in Chapter 5. Numerical results show that the error estimate ||ã-a||L∞ and mesh norm h have a linear relationship in the log-log plane. We compare the approximation results in 3 cases: a = 1 on the boundary ∂Ω, a = 1 on a non-characteristic curve Λ and asymptotic boundary condition. Numerical results show that more a priori information gives us better approximations.