Heinz W. Engl's research while affiliated with Austrian Academy of Sciences (OeAW) and other places

Preface. 1. Introduction: Examples of Inverse Problems. 2. Ill-Posed Linear Operator Equations. 3. Regularization Operators. 4. Continuous Regularization Methods. 5. Tikhonov Regularization. 6. Iterative Regularization Methods. 7. The Conjugate Gradient Method. 8. Regularization with Differential Operators. 9. Numerical Realization. 10. Tikhonov Regularization of Nonlinear Problems. 11. Iterative Methods for Nonlinear Problems. A. Appendix: A.1. Weighted Polynomial Minimization Problems. A.2. Orthogonal Polynomials. A.3. Christoffel Functions. Bibliography. Index.

Systems biology is a new discipline built upon the premise that an understanding of how cells and organisms carry out their functions cannot be gained by looking at cellular components in isolation. Instead, consideration of the interplay between the parts of systems is indispensable for analyzing, modeling, and predicting systems' behavior. Studying biological processes under this premise, systems biology combines experimental techniques and computational methods in order to construct predictive models. Both in building and utilizing models of biological systems, inverse problems arise at several occasions, for example, (i) when experimental time series and steady state data are used to construct biochemical reaction networks, (ii) when model parameters are identified that capture underlying mechanisms or (iii) when desired qualitative behavior such as bistability or limit cycle oscillations is engineered by proper choices of parameter combinations. In this paper we review principles of the modeling process in systems biology and illustrate the ill-posedness and regularization of parameter identification problems in that context. Furthermore, we discuss the methodology of qualitative inverse problems and demonstrate how sparsity enforcing regularization allows the determination of key reaction mechanisms underlying the qualitative behavior.

We describe a framework for solving nonlinear inverse problems in a random environment. Such problems arise, for instance, in the identification of parameters in a stochastic process or in a differential equation where the parameters themselves are random variables. The corresponding inverse problems can be treated by Tikhonov regularization in a stochastic setup. Both the solution and the data in such inverse problems can be random variables. As an example, the inverse problem considered here concerns the identification of the parameter relating the nucleation rate to the temperature field in a mesoscale model for crystal growth. The derivationof the mesoscale model from a microscale model by geometric averages is outlined in the first sections. We formulate the corresponding inverse problem both for the simply stochastic case, which leads to a deterministic inverse problem, and for the doubly stochastic case yielding a stochasticinverse problem. We apply the stochastic version of the theory of Tikhonov regularization to prove convergence and convergence rates and outline how the stochastic regularization approach can be used to deal with scale-dependent modelling errors.

Ion channels are proteins with a hole down theirmiddle that allow ions to move across otherwise impermeable cellmembranes, thereby controlling many important physiological functions. The transport process of the ions can be described using the Poisson-Nernst-Plank equations, a system of coupled nonlinear partial differential equations. Based on this model we derive a simplified surrogate model that captures the main features of the associated current-voltage curves of the ion channel. This surrogate model is then used to identify individual channel parameters based on current data. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The EM (Expectation-Maximization) algorithm is a convenient tool for approximating maximum likelihood estimators in situations when avail-able data are incomplete, as it is the case for many inverse problems. Our focus here is on the continuous version of the EM algorithm for a Pois-son model, which is known to perform unstably when applied to ill-posed integral equations. We interpret and analyse the EM algorithm as a reg-ularization procedure: We show weak convergence of the iterates to a solution of the equation when exact data are considered. In the case of perturbed data, similar results are established by employing a stopping rule of discrepancy type under boundedness assumptions on the problem data.

The blast furnace process represents a way to produce hot metal from iron ore. In a joint work of Industrial Mathematics Competence Center (IMCC) and Siemens VAI, we have developed a transient mathematical model of the blast furnace with the aim to calculate the form and position of the coke / ore layers and of the cohesive zone. Further, the model should be capable of predicting the amount of reducing agents needed to produce hot metal with some prescribed temperature. In this paper, we present the mathematical model as well as the numerical methods we use and compare the results to the measurements of a real blast furnace.

Given a large, complex ordinary differential equation model of a gene regulatory network, relating its dynamical properties to its network structure is a challenging task. Biologically important questions include: what network components are responsible for the various dynamical behaviors that arise? can the underlying dynamical behavior be essentially attributed to a small number of modules? In this paper, we demonstrate that inverse bifurcation analysis can be used to address such inverse problems. We show that sparsity-promoting regularization strategies, in combination with numerical bifurcation analysis, can be used to identify small sets of ”influential” submodules and parameters within a given network. In addition, hierarchical strategies can be used to generate parameter solutions of increasing cardinality of non-zero entries. We apply the proposed methods to analyze a model of the mammalian G 1/S regulatory module.

Ion channels control many biological processes in cells and consequently a large amount of research is devoted to this topic. Great progress in the understanding of channel function has been made recently using advanced mathematical modeling and simulation. This paper investigates another interesting mathematical topic, namely inverse problems, in connection with ion channels. We concentrate on problems that arise when we try to determine ('identify') one of the structural features of a channel - its permanent charge - from measurements of its function, namely current voltage curves in many solutions. We also try to design channels with desirable properties - for example with particular selectivity properties - using the methods of inverse problems. The use of mathematical methods of identification will help in the design of ecient experiments to determine the properties of ion channels. Closely related mathematical methods will allow the rational design of ion channels useful in many applications, technological and medical. We also discuss certain mathematical issues arising in these inverse problems, such as their ill-posedness and the choice of regularization techniques, as well as challenges in their numerical solution. The L-type calcium channel is studied with the methods of inverse problems to see how mathematics can aid in the analysis of existing ion channels and the design of new ones.