| In this dissertation we consider input-output systems modeled by parabolic partial differential equations of the form {dollar}{dollar}hsb{lcub}t{rcub}({lcub}bf x{rcub},t)=Asb{lcub}rm a(x){rcub}h({lcub}bf x{rcub},t),eqno(0.1){dollar}{dollar}where {dollar}Asb{lcub}rm a(x){rcub}{dollar} is a linear parabolic differential operator depending on some unknown spatial parameter a(x). Given a such a physical system together with observational data and appropriate initial and boundary conditions, one estimates the parameter a(x) from the available information. Different methods of resolving the inverse problem lead to different mathematical models describing the physical system.; The question we address is how to compare different mathematical models that describe the same physical system. In particular, we argue that the quality of a model should be judged by how well the observations are reproduced over a class of inputs. Frequency domain methods are used to define and measure distances between models as well as between models and the physical system. We define a family of norms on the transfer function and show how these norms can be estimated using only input-output data. We also examine which types of input functions are best suited to be used in the estimation of the norms.; We develop the necessary theory to formulate the problem (0.1) in a semigroup setting. The theory is then demonstrated on an example.; Numerical experiments support the theory. Both the one-dimensional and the two-dimensional time-dependent flow equations serve as examples. |
