Article · Journal of Inverse and Ill-Posed Problems · October 1, 2004
Justification of optimization methods for inverse integro-differential hyperbolic problems
A. T. Ayapbergenova, Sergey Igorevich Kabanikhin, Alfredo Lorenzi
Abstract
As is well-known, optimization is the most widely used method for solving numerically inverse problems in applications. The main theoretical difficulty consists in justifying the existence of the cost (misfit) function in an appropriate space. Indeed, in order to prove the existence of the trace of the solution to the direct problem one should suppose that the desired coefficients are sufficiently (and, sometimes, even too much!) smooth. In the present paper we overcome (in some sense) this gap treating our inverse problem as a nonlinear system of integral equations in L2. More exactly, first we deal with the problem of determining a pair of functions u : ∆(l) → R and h : [0, l]→ R, where
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