Local search methods for the solution of implicit inverse problems

In this paper, we propose two local search algorithms based on the Tabu Search and the Simulated Annealing methods, respectively, for the solution of implicit inverse problems. In general, the proposed methods work as follows: given a noisy observation and a right-hand side function of a partial differential equation, for each model component a sub-domain is built about it and the corresponding part of observed components in such sub-domain is projected onto a space generated by a pre-defined set of local basis functions. For the projection, the singular value decomposition is applied to the data set in order to discard singular vectors corresponding to small singular values in pursuance of reducing the impact of noise on the projected data. The right-hand side function is then utilized in order to estimate the quality of the projection onto such sub-space. Since different basis functions provide different spaces onto which the data can be projected, the well-known Tabu Search and Simulated Annealing methods are utilized in order to enrich the search space of the optimal set of basis functions. This is the optimal combination of basis functions which minimizes the error during the projection step. After this, the local solutions are mapped back onto the global domain from which the global solution of the inverse problem is approximated. A strength of our proposed method is that no assumption is needed over the measurements to be assimilated. Experimental tests are performed making use of a parabolic partial differential equation and different noise levels for the data error. The results reveal that the use of the proposed implementations can provide accurate estimates in a root-mean-square error sense of the reference solution with even large data errors.