Numerical analysis of an inverse problem for the eikonal equation
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. 245-269. ISSN 0029-599X
WRAP_Elliott_finalVMS-June2011Inverse_eikonal.pdf - Accepted Version
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Official URL: http://dx.doi.org/10.1007/s00211-011-0386-z
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 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.
|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|
|Page Range:||pp. 245-269|
|Access rights to Published version:||Restricted or Subscription Access|
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