Local Misfit Approximation in Memetic Solving of Ill-Posed Inverse Problems

The approximation of the objective function is a well known method of speeding up optimization process, especially if the objective evaluation is costly. This is the case of inverse parametric problems formulated as global optimization ones, in which we recover partial differential equation parameters by minimizing the misfit between its measured and simulated solutions. Typically, the approximation used to build the surrogate objective is rough but globally applicable in the whole admissible domain. The authors try to carry out a different task of detailed misfit approximation in the regions of low sensitivity (plateaus). The proposed complex method consists of independent \(C^0\) Lagrange approximation of the misfit and its gradient, based on the nodes obtained during the dedicated memetic process, and the subsequent projection of the obtained components (single or both) on the space of B-splines. The resulting approximation is globally \(C^1\), which allows us to use fast gradient-based local optimization methods. Another goal attained in this way is the estimation of the shape of plateau as an appropriate level set of the approximated objective. The proposed strategy can be applied for solving ill-conditioned real world inverse problems, e.g., appearing in the oil deposit investigation. We show the results of preliminary tests of the method on two benchmarks featuring convex and non-convex U-shaped plateaus.