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In the present work, a novel stochastic method has been developed and investigated in order to face the time-reduced inverse scattering problem, governed by the Helmholtz equation, outside connected or disconnected obstacles supporting boundary conditions of Dirichlet type. On the basis of the stochastic analysis, a series of efficient and alternative stochastic representations of the scattering field have been constructed. These novel representations constitute conceptually the probabilistic analogue of the well known deterministic integral representations involving the famous Green’s functions, and so merit special importance. Their advantage lies in their intrinsic probabilistic nature, allowing to solve the direct and inverse scattering problem in the realm of local methods, which are strongly preferable in comparison with the traditional global ones. The aforementioned locality reflects the ability to handle the scattering field only in small bounded portions of the scattering medium by monitoring suitable stochastic processes, confined in narrow sub-regions where data are available. Especially in the realm of the inverse scattering problem, two different schemes are proposed facing reconstruction from the far field and near field data, respectively. The crucial characteristic of the inversion is that the reconstruction is fulfilled through stochastic experiments, taking place in the interior of conical regions whose base belong to the data region, while their vertices detect appropriately the supporting surfaces of the sought scatterers.

A Conditioned Probabilistic Method for the Solution of the Inverse Acoustic Scattering Problem