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Wavelet projection methods for solving pseudodifferential inverse problems

E. P. Serrano

Centro de Matemática Aplicada, Universidad de San Martín, San Martín, Argentina

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M. I. Troparevsky

Departamento de Matemática, Facultad de Ingeniería, Universidad de Buenos Aires, CABA, Buenos Aires, Argentina

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, and

M. A. Fabio

Centro de Matemática Aplicada, Universidad de San Martín, San Martín, Argentina

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https://doi.org/10.1142/S0219691314500258Cited by:1 (Source: Crossref)

Abstract

We consider the Inverse Problem (IP) associated to an equation of the form Af = g, where A is a pseudodifferential operator with symbol. It consists in finding a solution f for given data g. When the operator A is not strongly invertible and the data is perturbed with noise, the IP may be ill-posed and the solution must be approximate carefully. For the present application we regard a particular orthonormal wavelet basis and perform a wavelet projection method to construct a solution to the Forward Problem (FP). The approximate solution to the IP is achieved based on the decomposition of the perturbed data calculating the elementary solutions that are nearly the preimages of the wavelets. Based on properties of both, the basis and the operator, and taking into account the energy of the data, we can handle the error that arises from the partial knowledge of the data and from the non-exact inversion of each element of the wavelet basis. We estimate the error of the approximation and discuss the advantages of the proposed scheme.

Keywords:

AMSC: 22E46, 53C35, 57S20

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