Identification of spatially-varying parameters in distributed parameter systems given an observation of the state is as a rule an ill-posed problem in the sense of Hadamard. Even in case when the solution is unique, it does not depend continuously on the data. The identification problem that motivated this work arises in the description of petroleum reservoirs and subsurface aquifers; it consists of identifying the spatially-varying parameter α(x,y) in the diffusion equation ut = (αux)x + (αuy)y + f given an observation of u at a discrete set of spatial locations.
The question of uniqueness of α (identifiability problem) is first investigated. The analysis is restricted to the one-dimensional version of the above equation i.e. to ut = (αux)x + f and an observation of u at a single point. The identifiability problem is formulated as an inverse Sturm-Liouville problem for (αy')' + λy = 0. It is proved that the eigenvalues and the normalizing constants determine the above Sturm-Liouville operator uniquely. Identifiability and non-identifiability results are obtained for three special cases.
The problem of constructing stable approximate solutions to identification problems in distributed parameter systems is next investigated. The concept of regularization, widely used in solving linear Fredholm integral equations, is developed for the solution of such problems. A general regularization identification theory is presented and applied to the identification of parabolic systems. Two alternative numerical approaches for the minimization of the smoothing functional are investigated: (i) classical Banach space gradient methods and (ii) discretized minimization methods. The latter use finite-dimensional convergent approximations in Sobolev spaces and are based on an appropriate convergence theorem. The performance of the regularization identification method is evaluated by numerical experiments on the identification of spatially-varying diffusivity α in the diffusion equation.
