Adaptive model order reduction for the Jacobian calculation in inverse multi-frequency problem for Maxwell's equations

This work develops a model order reduction method for a numerical solution of an inverse multi-frequency eddy current problem using a rational interpolation of the transfer function in the complex plane. We use the Pade interpolation in the complex frequency plane; this allows us to speed up the calculation of the frequency-dependent Jacobian in the inversion procedure without loosing accuracy. Interpolating frequencies are chosen adaptively to reduce the maximal approximation error. We use the error indicator that is equivalent to a seminorm of the residual. The efficiency of the developed approach is demonstrated by applying it to the inverse magnetotelluric problem, which is a geophysical electromagnetic remote sensing method used in mineral, geothermal, and groundwater exploration. In this application, the transfer function values are needed for shifts in a purely imaginary interval. Thus we consider the interpolating shifts in the same interval as well as in a purely real interval, containing the spectrum of the operator. Numerical tests show an excellent performance of the proposed methods characterized by a significant reduction of computational time without loss of accuracy of the calculated Jacobian.