Nonlinear PDE Days Karlsruhe January 20

Nonlinear PDE Days Karlsruhe January 20 – 21, 2011

Klaus Deckelnick

University of Magdeburg, Germany

Numerical analysis of an inverse problem for the eikonal equation

We are concerned with the inverse problem of determining the speed function in an eikonal equation using

observations of the arrival time on a fixed surface. This is formulated as an optimisation

problem for a quadratic functional with the state equation being the eikonal equation coupled to the

so-called Soner boundary condition. The state equation is discretised by a suitable finite difference

scheme for which we obtain existence, uniqueness and an error bound. We set up an approximate optimisation

problem and show that a subsequence of the discrete mimina converges to a solution of the continuous

optimisation problem as the mesh size goes to zero. The derivative of the discrete functional is calculated with the help

of an adjoint equation which can be solved efficiently by using fast marching techniques. Finally we

describe some numerical results.