New implementation of Elliptic Systems Method for time dependent diffusion tomography with back reflected and transmitted boundary data

摘要

A problem of coefficient recovery from incomplete boundary data in inverse problems is considered. It uses a new implementation of the Elliptic Systems Method (ESM) in time dependent diffusion tomography. The basic formulation of the ESM involves solving a system of coupled fourth-order partial differential equations, with the time variable integrated out using Legendre polynomials. Here, we use C1 Bogner–Fox–Schmit bi-cubic elements over rectangles, with a new treatment of boundary conditions in the common case of incomplete boundary data. This new method is fourth-order accurate for sufficiently smooth functions. The new boundary condition approach allows the use of homogeneous natural boundary conditions on parts of the boundary where no measured data is available. We will focus on a comparison with three previously published examples using back reflected or transmitted data with one or two inclusions. The new implementation gives markedly improved results for inclusion recovery, all of which are achieved without use of additional aids such as weight functions which previously have been thought to be essential, and is shown to be surprisingly robust with respect to noise. We conclude with two examples illustrating the effect of increasing levels of noise.