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On the method of calculating the modulus of continuity of the inverse operator and its modifications with application to non-linear problems of geoelectrics

M. I. Shimelevich

Russian State Geological Prospecting University, Moscow

Abstract: The article considers a priori estimates of the ambiguity (error) of approximate solutions of conditionally correct nonlinear inverse problems based on the modulus of continuity of the inverse operator and its modifications. It is shown that in the class of piecewise constant solutions defined on a given parametrization grid, the modulus of continuity of the inverse operator and its modifications monotonously increase with increasing mesh dimension. A method is proposed for constructing an optimal parameterization grid that has a maximum dimension provided that the modulus of continuity of the inverse operator does not exceed a given value. A numerical algorithm for calculating the modulus of continuity of the inverse operator and its modifications using Monte Carlo algorithms is presented; questions of convergence of the algorithm are investigated. The proposed method is also applicable for calculating classical posterior error estimates. Numerical examples are given for nonlinear inverse problems of geoelectrics.

Keywords: inverse problem; modulus of continuity of an operator; a priori and a posteriori estimates; Monte Carlo; geoelectrics.

UDC: 519.6

Received: 29.06.2020

DOI: 10.26089/NumMet.v21r430