Identification of Distributed Parameters in Partial Differential Equations with Incomplete Observed Data
by
M. Sun
The University of Alabama
Coauthors: C. Zheng

Important parameters in partial differential equations vary spatially. Identification of such parameters usually leads to a large scale ill-conditioned optimization problem. Although a globally optimal solution to such an identification problem is generally known to exist, several inherent problems are well documented in the literature, such as nonuniqueness, existence of a large number of local or global minima, and instability. Those problems make it difficult (if not impossible) to find a global solution or even a decent approximation of a global solution. Sometimes, even a global solution can be found by well-designed optimization procedures, the solution may not be physically reasonable. The large dimension of the search space is one of the major causes of those problems. One of commonly used standard strategies in a variety of distributed parameter identification algorithms is to reduce the dimension of the search space by the spatial partition (or zonation). In contrast to existing zonation based identification schemes that require a priori information regarding the shape and/or even the complete configuration of all the parameter zones, we do not impose any one of such assumptions regarding parameter zones because those kinds of zone information may not be easy to get or may be very expansive to get. In contrast to our previous work, we no longer assume that all the different types of zones have been observed at least in one location for each zone type. We only assume that the system state variable and the unknown system parameter are observed at an incomplete number of observation locations and that each parameter zone is connected. There is at least one undetected parameter zone (its location and configuration) but with known parameter value for that zone. Our objective is to find a good parameter distribution that would generate basically the same observation data at the observation locations. We will mainly examine the unknown parameters appearing in the highest order terms of a class of elliptic partial differential equations. The identification algorithm will be described and hypothetical examples will be used for illustrating effectiveness of the method.

Date received: April 1, 1999