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Numerical analysis of an inverse problem for the eikonal equation
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Deckelnick, Klaus, Elliott, Charles M. and Styles, Vanessa. (2011) Numerical analysis of an inverse problem for the eikonal equation. Numerische Mathematik, Volume 119 (Number 2). pp. 245269. ISSN 0029599X

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Official URL: http://dx.doi.org/10.1007/s002110110386z
Abstract
We are concerned with the inverse problem for an eikonal equation of determining the speed function 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 socalled 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.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Numerical analysis, Eikonal equation, Inverse problems (Differential equations) 
Journal or Publication Title:  Numerische Mathematik 
Publisher:  Springer 
ISSN:  0029599X 
Official Date:  2011 
Volume:  Volume 119 
Number:  Number 2 
Page Range:  pp. 245269 
Identification Number:  10.1007/s002110110386z 
Status:  Peer Reviewed 
Publication Status:  Published 
Access rights to Published version:  Restricted or Subscription Access 
References:  [1] R. Abgrall. Numerical discretitzation of boundary conditions for first order HamiltonJacobi equations. SIAM J. Numer. Anal., 41:22332261, 2003. [3] M. Bardi and I. CapuzzoDolcetta. Optimal control and viscosity solutions of 
URI:  http://wrap.warwick.ac.uk/id/eprint/39075 
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