Modified Landweber iterations in a multilevel algorithm applied to inverse problems in piezoelectricity

Abstract

In piezoelectric applications, especially when the devices are used as actuators, the piezoelectric materials are driven under large signals which cause a nonlinear behavior. One way to model the nonlinearities is by functional dependencies of the material parameters on the electric field strength or the mechanical strain, respectively. The focus lies in the inverse problem, namely the identification of the parameter curves by appropriate measurements of charge signals over time. The problem is assumed to be ill-posed, since in general measured data are contaminated with noise. The solution process requires regularizing methods where modified Landweber iterations are in the focus. Implementations of modified Landweber iterations, namely the steepest descent and minimal error method can be shown to perform much faster than classical Landweber iterations due to the flexible handling of the relaxation parameter.

In our application, parameter curve identification in nonlinear piezoelectricity, the sought-for quantities require to be discretized. Therefore, an iterative multilevel algorithm as proposed by Scherzer [Numer.Math.: 579–600, 1998] is investigated where the iterations begin with coarse discretizations of the parameter curves profitting from the inherent regularization property of coarse discretization. At an advanced state of the iterations the algorithm switches according to an inner discrepancy principle to finer levels of discretization. By this, a sufficient smooth resolution of the sought-for quantities can be achieved. Convergence results and the regularizing property of such an iterative multilevel algorithm are proven. Numerical identification results are presented at the end of this article.