The Boundary Element Method in Geophysical Survey by Balgaisha Mukanova & Igor Modin

Author:Balgaisha Mukanova & Igor Modin
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(3.52)

This functional represents the difference between the data Λσ that we obtain if the given function σ is substituted into the direct problem, and the measured data ψ. Then, the inverse problem is reduced to the minimization problem of the functional (3.52). This problem again raises questions concerning the existence of a minimum of this functional in the set of possible functions σ ∈ Σ. However, in many practically important cases, the problem of minimizing J(σ) can be solved by applying the Tikhonov regularization method (Ivanov 1962; Tikhonov and Arsenin 1977). First, we should restrict the operator Λ to some compact subset S of Σ (Tikhonov and Arsenin 1977). In numerical simulations, this subset is bounded and belongs to some finite dimensional space; the minimization problem can thus be reduced to a problem in finite dimensional space. Moreover, a quasi-solution to the inverse problem is sought as a minimizer of the regularized misfit functional, which has the form

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