COMPLEX SYSTEMS
Solving Inverse Problems by Optimal Control Techniques
In the last few years, we have developed and implemented a general method to find approximate solutions of inverse problems of identification type by optimal control techniques. Given partial and/or noisy observations of the solution of a partial differential equation (PDE) problem, the goal is to identify one or several coefficients in the PDE, initial conditions, or boundary conditions. Regarding the coefficient to be identified as an adjustable control, one varies this control until the corresponding solution data best matches the given observations. The approximate coefficient is completely and explicitly characterized through the solution of an optimality system. As the approximation parameter tends to zero, the corresponding optimal controls converge to the sought for coefficient. Compared to classical regularization methods (e.g. Tikhonov regularization coupled with optimization schemes), our approach presents several advantages, namely: (i) a systematic, general, and rigorous procedure; (ii) explicit expressions for the approximations of the solution; and (iii) a convenient numerical solution of these approximations.
We have successfully applied this approach to several cases involving wave equations. In one situation, the shape of part of the spatial boundary was identified along with a reflective coefficient in the boundary condition. In some cases, the possibility of error or noise in the observations was also treated. In our most recent paper, numerical examples were computed to illustrate the successful implementation of the approximation technique. This work has is contained in four journal publications and several conference and invited papers.
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