Volume 17, Issue 6

In this paper we consider an inverse problem of identification of a right-hand side in a second order parabolic and hyperbolic equations with a parameter. We prove theorems on existence and uniqueness of solutions to these problems.

We propose an algorithm to select parameter subset combinations that can be estimated using an ordinary least-squares (OLS) inverse problem formulation with a given data set. First, the algorithm selects the parameter combinations that correspond to sensitivity matrices with full rank. Second, the algorithm involves uncertainty quantification by using the inverse of the Fisher Information Matrix. Nominal values of parameters are used to construct synthetic data sets, and explore the effects of removing certain parameters from those to be estimated using OLS procedures. We quantify these effects in a score for a vector parameter defined using the norm of the vector of standard errors for components of estimates divided by the estimates. In some cases the method leads to reduction of the standard error for a parameter to less than 1% of the estimate.

This paper proves the existence, uniqueness and stability of solutions of an inverse problem for the 2-D Navier–Stokes equations with the final overdetermination with L 2 initial data and sufficiently large viscosity.

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.

We discuss the relaxation of a class of nonlinear elliptic Cauchy problems with data on a piece S of the boundary surface by means of a variational approach known in the optimal control literature as “equation error method”. By the Cauchy problem is meant any boundary value problem for an unknown function y in a domain χ with the property that the data on S , if combined with the differential equations in χ , allow one to determine all derivatives of y on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy–Kovalevskaya theorem. We also admit overdetermined elliptic systems, in which case the set of those Cauchy data on S for which the Cauchy problem is solvable is very “thin”. For this reason we discuss a variational setting of the Cauchy problem which always possesses a generalised solution.

A new robust algorithm for the pressure/rate deconvolution problem, described by Duhamel's convolution integral, which is a first-kind linear Volterra integral equation, has been developed. A transformation of the convolution integral to a nonlinear one is used to impose explicitly the positivity constraint on the solution. The weighted least-squares method with regularization on the solution by a curvature constraint has been used for computation of the convolution kernel (impulse function or deconvolved pressure) of the system. The algorithm takes into account the errors (or noise) in both the left-hand-side (measured pressures) and flow rate measurements (normally, the time dependent inner boundary condition) of the convolution integral. The solution algorithm also allows one to adjust flow rates and/or the initial reservoir pressure (an initial condition for the solution) during calculations, where both flow rate and the initial pressure may contain some level of uncertainty. For validation of the results of the algorithm, three synthetic examples are presented.