Applications And Adaptations Of A Globally Convergent Numerical Method In Inverse Problems

Abstract

In our terminology "globally convergent numerical method" means a numerical method whose convergence to a good approximation of the correct solution is independent of the initial approximation in inverse problems. A numerical imaging algorithm has been proposed to solve a coecient inverse problem for an elliptic equation and then the algorithm is validated with the data generated by computer simulation. Previouswork in this eld was focused on the steady-state optical problem with multiple source positions moving along a straight line as well as the frequency domain problem with sweeping frequency. This work includes the steady-state thermal tomography problem with multiple source positions moving along a straight line as well as the time-dependent optical tomography problem using only two fixed source positions. Aconvergence analysis shows that this method converges globally assuming the smallness of the asymptotic solution (the so-called tail function). A heuristic approach for approximating the "new tail-function" has been utilized and verified in numerical experiments, so has the global convergence. Numerical experiments in the 2D timedependentoptical and steady-state thermal property reconstruction are presented.