Wave propagation from lateral Cauchy data using a boundary element method

Abstract

The wave equation with lateral Cauchy data is considered in three-dimensional annular domains. It is a linear inverse ill-posed problem, where data in the form of the function values and normal derivative is given on the outer boundary surface. To reconstruct the missing data on the inner surface a recent boundary integral approach is applied and extended to three-dimensions. A transformation in time (semi-discretization) is employed via the Laguerre transform resulting in a Cauchy problem for a system consisting of a sequence of inhomogeneous elliptic equations. The solution to this system is represented as a sequence of single-layer potentials invoking a fundamental sequence of the elliptic equations, with the benefit of avoiding the use of any volume potentials or domain discretization. Matching the data against the single-layer representation of the solution gives a system of boundary integral equations to be solved for a sequence of densities. A Galerkin boundary element discretization is applied to this latter system, triangulating the boundary surfaces and using piecewise constant approximating functions. Tikhonov regularization is used when solving the sequence of discrete linear equations. Numerical examples show the feasibility of the proposed approach rendering accurate and stable approximations of the missing lateral data with little computational effort.