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.





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