Nonlinear PDE Days

** **

**Klaus
Deckelnick **

**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.`