Domain Decomposition Methods for Inverse Problems

Inverse problems related to the estimation of coeecients of partial diierential equations are illposed. Practical applications often use the t-to-data output-least-square's method to recover the coeecients. In this work, we develop parallel nonoverlapping domain decomposition algorithms to estimate the diiusion coeecient associated with elliptic diierential equations. In order to realize the domain decomposition methods, we combine the function decomposition approach of Tai95a] and the augmented Lagrangian techniques of IK90, KT97b]. The output-least-square's method minimizes the output error over the whole domain. When decomposing the domain into nonoverlapping subdomains the output error over the whole domain equals the sum of the output errors in the subdomains. Thus, by borrowing ideas from Tai95a], parallel methods can be used to nd the minimizer. In this approach the partial diierential equation arises as a constraint in the optimization problem whose proper treatment is essential. In this work we propose to incorporate it by an augmented Lagrangian technique. 1.1 Introduction In this work we develop parallel domain decomposition algorithms for parameter estimation problems associated with elliptic diierential equations. In order to realize these methods we combine the function decomposition approach of Tai95b] and the augmented Lagrangian techniques of IK90, KT97b].