Multiwave imaging in an enclosure with variable wave speed

TL;DR

This paper develops methods for reconstructing initial wave conditions in an enclosed region with variable wave speed using boundary measurements, accounting for physical boundaries and reflections, with constructive proofs and numerical algorithms.

Contribution

It introduces new boundary observability techniques for wave reconstruction in enclosed domains with impedance boundaries, accommodating variable wave speeds and partial boundary data.

Findings

Explicit solvable equation for initial condition derived

Convergent Neumann series reconstruction proposed

Methods applicable under geometrical conditions with variable wave speed

Abstract

In this paper we consider the mathematical model of thermo- and photo-acoustic tomography for the recovery of the initial condition of a wave field from knowledge of its boundary values. Unlike the free-space setting, we consider the wave problem in a region enclosed by a surface where an impedance boundary condition is imposed. This condition models the presence of physical boundaries such as interfaces or acoustic mirrors which reflect some of the wave energy back into the enclosed domain. By recognizing that the inverse problem is equivalent to a statement of boundary observability, we use control operators to prove the unique and stable recovery of the initial wave profile from knowledge of boundary measurements. Since our proof is constructive, we explicitly derive a solvable equation for the unknown initial condition. This equation can be solved numerically using the conjugate gradient method. We also propose an alternative approach based on the stabilization of waves. This leads to an exponentially and uniformly convergent Neumann series reconstruction when the impedance coefficient is not identically zero. In both cases, if well-known geometrical conditions are satisfied, our approaches are naturally suited for variable wave speed and for measurements on a subset of the boundary.