A fast and accurate imaging algorithm in optical/diffusion tomography

An n-dimensional (n = 2, 3) inverse problem for the parabolic/diffusion equation u(t) = div(D(x)del u) -a(x)u, u(x, 0) = delta(x - x(0)), x epsilon R-n, t epsilon (0, T) is considered. The problem consists of determining the function a(x) inside of a bounded domain Omega subset of R-n given the values of the solution u(x, t) for a single source location x(0) epsilon delta Omega on a set of detectors {x(i)}(j=1)(m) subset of delta Omega, where delta Omega is the boundary of Omega. A novel numerical method is derived and tested. Numerical tests are conducted for n = 2 and for ranges of parameters which are realistic for applications to early breast cancer diagnosis and the search for mines in murky shallow water using ultrafast laser pulses. The main innovation of this method lies in a new approach for a novel linearized problem (LP). Such a LP is derived and reduced to a well-posed boundary-value problem for a coupled system of elliptic partial differential equations. A principal advantage of this technique is in its speed and accuracy, since it leads to the factorization of well conditioned, sparse matrices with non-zero entries clustered in a narrow band near the diagonal. The authors call this approach the elliptic systems method (ESM). The ESM can be extended to other imaging modalities.