DETERMINATION OF PARAMETERS IN NONLINEAR HYPERBOLIC PDES VIA A MULTIHARMONIC FORMULATION, USED IN PIEZOELECTRIC MATERIAL CHARACTERIZATION

Abstract

In this paper we consider the problem of determining material parameter curves that appear as coefficients in nonlinear partial differential equations of hyperbolic type. In order to demonstrate our ideas of an identification method for this class of problems, we consider the model problem of identifying c in the nonlinear wave equation dtt - (c(dx)dx)x = 0 from boundary measurements. Motivated by the fact that in many applications, this inverse problem is naturally posed in frequency domain rather than in time domain, we work in the Fourier transformed setting. Here, nonlinearity can be accounted for by using a multiharmonic Ansatz for the measured field quantity. The searched for material parameter curves are approximated by polynomials of arbitrary order, which enables a reformulation of the parameter identification problem purely in frequency domain, although the parameter curve is a function of time domain values of the field quantity. Based on this formulation, we develop a reconstruction algorithm by means of the above-mentioned model problem. Regularization of the typically unstable identification problem is here achieved by bandlimiting the data and restricting the number of degrees of freedom in the solution. We outline the extension of the proposed method to more general material parameter identification problems, focusing especially on the piezoelectric PDEs, for which we also give numerical results.