A framework for solving meta inverse problems: experimental design and application to an acoustic source problem

Abstract

The selection of optimal measurement locations in remote sensing or imaging algorithms is of large practical interest in many applications. The target is usually to choose a measurement setup that best resolves some particular quantity of interest. This work describes a general framework for selecting such an optimal setup within a given set Q of possible setups for the formulation and solution of the meta inverse problem. The work shows that it is crucial to incorporate the basic ingredients which are usually part of the inversion process. In particular, it takes care of the nature and the size of the measurement error, the choice of the regularization scheme which is employed for the inverse problem, and the prior knowledge on solutions. The basic idea of the framework is to minimize the errors associated with the reconstruction of a given quantity of interest. Five functional layers which reflect the structure of the meta inverse problem are introduced. Further, with framework adaption, an iterative algorithm is formulated to solve the meta inverse problem at each iterative step in order to obtain improved reconstructions of the inverse problem. Using the initial reconstructions as input for meta inversion, the framework adaption algorithm does not require prior knowledge of the source distribution. The feasibility of the framework adaption algorithm is illustrated by using it to solve the inverse acoustic source problem.