On the inverse problem of vibro-acoustography.

The aim of this paper is to put the problem of vibroacoustic imaging into the mathematical framework of inverse problems (more precisely, coefficient identification in PDEs) and regularization. We present a model in frequency domain, prove uniqueness of recovery of the spatially varying nonlinearity parameter from measurements of the acoustic pressure at multiple frequencies, and derive Newton as well as gradient based reconstruction methods.

Iterative regularization for constrained minimization formulations of nonlinear inverse problems.

In this paper we study the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type methods. We carry out a convergence analysis in the sense of regularization methods and discuss applicability to the problem of identifying the spatially varying diffusivity in an elliptic PDE from different sets of observations. Among these is a novel hybrid imaging technology known as impedance acoustic tomography, for which we provide numerical experiments.

Convergence and adaptive discretization of the IRGNM Tikhonov and the IRGNM Ivanov method under a tangential cone condition in Banach space.

In this paper we consider the iteratively regularized Gauss-Newton method (IRGNM) in its classical Tikhonov version as well as two further-Ivanov type and Morozov type-versions. In these two alternative versions, regularization is achieved by imposing bounds on the solution or by minimizing some regularization functional under a constraint on the data misfit, respectively. We do so in a general Banach space setting and under a tangential cone condition, while convergence (without source conditions, thus without rates) has so far only been proven under stronger restrictions on the nonlinearity of the operator and/or on the spaces.

Scalable Parameter Estimation for Genome-Scale Biochemical Reaction Networks.

Mechanistic mathematical modeling of biochemical reaction networks using ordinary differential equation (ODE) models has improved our understanding of small- and medium-scale biological processes. While the same should in principle hold for large- and genome-scale processes, the computational methods for the analysis of ODE models which describe hundreds or thousands of biochemical species and reactions are missing so far. While individual simulations are feasible, the inference of the model parameters from experimental data is computationally too intensive.

A modified and stable version of a perfectly matched layer technique for the 3-d second order wave equation in time domain with an application to aeroacoustics.

We consider the second order wave equation in an unbounded domain and propose an advanced perfectly matched layer (PML) technique for its efficient and reliable simulation. In doing so, we concentrate on the time domain case and use the finite-element (FE) method for the space discretization. Our un-split-PML formulation requires four auxiliary variables within the PML region in three space dimensions.

FEM-Based determination of real and complex elastic, dielectric, and piezoelectric moduli in piezoceramic materials.

We propose an enhanced iterative scheme for the precise reconstruction of piezoelectric material parameters from electric impedance and mechanical displacement measurements. It is based on finite-element simulations of the full three-dimensional piezoelectric equations, combined with an inexact Newton or nonlinear Landweber iterative inversion scheme. We apply our method to two piezoelectric materials and test its performance.