Barbara Kaltenbacher

Barbara Kaltenbacher

Parameter identification problems typically consist of a model equation, e.g. a (system of) ordinary or partial differential equation(s), and the observation equation. In the conventional reduced setting, the model equation is eliminated via the parameter-to-state map. Alternatively, one might consider both sets of equations (model and observations) as one large system, to which some regularization method is applied. The choice of the formulation (reduced or all-at-once) can make a large difference computationally, depending on which regularization method is used: Whereas almost the same optimality system arises for the reduced and the all-at-once Tikhonov method, the situation is different for iterative methods, especially in the context of nonlinear models. In this paper we will exemplarily provide some convergence results for all-at-once versions of variational, Newton type and gradient based regularization methods. Moreover we will compare the implementation requirements for the respective all-at-one and reduced versions and provide some numerical comparison.

Barbara Kaltenbacher, William Rundell

Reaction-diffusion equations are one of the most common partial differential equations used to model physical phenomenon. They arise as the combination of two physical processes: a driving force $f(u)$ that depends on the state variable $u$ and a diffusive mechanism that spreads this effect over a spatial domain. The canonical form is $u_t - \triangle u = f(u)$. Application areas include chemical processes, heat flow models and population dynamics. The direct or forwards problem for such equations is now very well-developed and understood. However, our interest lies in the inverse problem of recovering the reaction term $f(u)$ not just at the level of determining a few parameters in a known functional form, but recovering the complete functional form itself. To achieve this we set up the usual paradigm for the parabolic equation where $u$ is subject to both given initial and boundary data, then prescribe overposed data consisting of the solution at a later time $T$. For example, in the case of a population model this amounts to census data at a fixed time. Our approach will be two-fold.First we will transform the inverse problem into an equivalent nonlinear mapping from which we seek a fixed point. We will be able to prove important features of this map such as a self-mapping property and give conditions under which it is contractive. Second, we consider the direct map from $f$ through the partial differential operator to the overposed data. We will investigate Newton schemes for this case. In recent decades various anomalous processes have been used to generalize classical Brownian diffusion. Amongst the most popular is one that replaces the usual time derivative by a fractional one of order $α\leq 1$. We will also include this model in our analysis. The final section of the paper shows numerical reconstructions that demonstrate the viability of the suggested approaches.

Bernd Hofmann, Barbara Kaltenbacher, Elena Resmerita

In this paper, we deal with nonlinear ill-posed problems involving monotone operators and consider Lavrentiev's regularization method. This approach, in contrast to Tikhonov's regularization method, does not make use of the adjoint of the derivative. There are plenty of qualitative and quantitative convergence results in the literature, both in Hilbert and Banach spaces. Our aim here is mainly to contribute to convergence rates results in Hilbert spaces based on some types of error estimates derived under various source conditions and to interpret them in some settings. In particular, we propose and investigate new variational source conditions adapted to these Lavrentiev-type techniques. Another focus of this paper is to exploit the concept of approximate source conditions.

Barbara Kaltenbacher, Andrej Klassen

In this paper we provide a convergence analysis of some variational methods alternative to the classical Tikhonov regularization, namely Ivanov regularization (also called method of quasi solutions) with some versions of the discrepancy principle for choosing the regularization parameter, and Morozov regularization (also called method of the residuals). After motivating nonequivalence with Tikhonov regularization by means of an example, we prove well-definedness of the Ivanov and the Morozov method, convergence in the sense of regularization, as well as convergence rates under variational source conditions. Finally, we apply these results to some linear and nonlinear parameter identification problems in elliptic boundary value problems.

Uno Hämarik, Barbara Kaltenbacher, Urve Kangro et al.

We consider ill-posed linear operator equations with operators acting between Banach spaces. For solution approximation, the methods of choice here are projection methods onto finite dimensional subspaces, thus extending existing results from Hilbert space settings. More precisely, general projection methods, the least squares method and the least error method are analyzed. In order to appropriately choose the dimension of the subspace, we consider a priori and a posteriori choices by the discrepancy principle and by the monotone error rule. Analytical considerations and numerical tests are provided for a collocation method applied to a Volterra integral equation in one space dimension.

Barbara Kaltenbacher, William Rundell

This paper considers the inverse problem of recovering both the unknown, spatially-dependent conductivity $a(x)$ and the potential $q(x)$ in a parabolic equation from overposed data consisting of the value of solution profiles taken at a later time $T$. We show both uniqueness results and the convergence of an iteration scheme designed to recover these coefficients. We also allow a more general setting, in particular when the usual time derivative is replaced by one of fractional order and when the potential term is coupled with a known nonlinearity $f$ of the form $q(x)f(u)$.

Christian Clason, Barbara Kaltenbacher, Daniel Wachsmuth

So-called functional error estimators provide a valuable tool for reliably estimating the discretization error for a sum of two convex functions. We apply this concept to Tikhonov regularization for the solution of inverse problems for partial differential equations, not only for quadratic Hilbert space regularization terms but also for nonsmooth Banach space penalties. Examples include the measure-space norm (i.e., sparsity regularization) or the indicator function of an $L^\infty$ ball (i.e., Ivanov regularization). The error estimators can be written in terms of residuals in the optimality system that can then be estimated by conventional techniques, thus leading to explicit estimators. This is illustrated by means of an elliptic inverse source problem with the above-mentioned penalties, and numerical results are provided for the case of sparsity regularization.

Christian Clason, Barbara Kaltenbacher, Elena Resmerita

This survey reviews variational and iterative methods for reconstructing non-negative solutions of ill-posed problems in infinite-dimensional spaces. We focus on two classes of methods: variational methods based on entropy-minimization or constraints, and iterative methods involving projections or non-negativity-preserving multiplicative updates. We summarize known results and point out some open problems.

Kristian Bredies, Barbara Kaltenbacher, Elena Resmerita

This work deals with a regularization method enforcing solution sparsity of linear ill-posed problems by appropriate discretization in the image space. Namely, we formulate the so called least error method in an $\ell^1$ setting and perform the convergence analysis by choosing the discretization level according to an a priori rule, as well as two a posteriori rules, via the discrepancy principle and the monotone error rule, respectively. Depending on the setting, linear or sublinear convergence rates in the $\ell^1$-norm are obtained under a source condition yielding sparsity of the solution. A part of the study is devoted to analyzing the structure of the approximate solutions and of the involved source elements.

Barbara Kaltenbacher, Igor Shevchenko

The focus of this work is on the construction of a family of nonlinear absorbing boundary conditions for the Westervelt equation in one and two space dimensions. The principal ingredient used in the design of such conditions is pseudo-differential calculus. This approach enables to develop high order boundary conditions in a consistent way which are typically more accurate than their low order analogs. Under the hypothesis of small initial data, we establish local well-posedness for the Westervelt equation with the absorbing boundary conditions. The performed numerical experiments illustrate the efficiency of the proposed boundary conditions for different regimes of wave propagation.

Barbara Kaltenbacher, Ivan Tomba

This paper is a close follow-up of Kaltenbacher and Tomba 2013 and Jin 2012, where Newton-Landweber iterations have been shown to converge either (unconditionally) without rates or (under an additional regularity assumption) with rates. The choice of the parameters in the method were different in each of these two cases. We now found a unified and more general strategy for choosing these parameters that enables both convergence and convergence rates. Moreover, as opposed to the previous one, this choice yields strong convergence as the noise level tends to zero, also in the case of no additional regularity. Additionally, the resulting method appears to be more efficient than the one from Kaltenbacher and Tomba 2013, as our numerical tests show.

Romana Boiger, Jan Hasenauer, Sabrina Hross et al.

Partial differential equation (PDE) models are widely used in engineering and natural sciences to describe spatio-temporal processes. The parameters of the considered processes are often unknown and have to be estimated from experimental data. Due to partial observations and measurement noise, these parameter estimates are subject to uncertainty. This uncertainty can be assessed using profile likelihoods, a reliable but computationally intensive approach. In this paper, we introduce an integration based approach for the profile likelihood calculation for inverse problems with PDE constraints. While existing approaches rely on repeated optimization, the proposed approach exploits a dynamical system evolving along the likelihood profile. We derive the dynamical system for the reduced and the full estimation problem and study its properties. To evaluate the proposed method, we compare it with state-of-the-art algorithms for a simple reaction-diffusion model for a cellular patterning process. We observe a good accuracy of the method as well as a significant speed up as compared to established methods. Integration based profile calculation facilitates rigorous uncertainty analysis for computationally demanding parameter estimation problems with PDE constraints.

Barbara Kaltenbacher

The use of second order information on the forward operator often comes at a very moderate additional computational price in the context of parameter identification probems for differential equation models. On the other hand the use of general (non-Hilbert) Banach spaces has recently found much interest due to its usefulness in many applications. This motivates us to extend the second order method previously considered by the author in a Hilbert space setting, (see also Hettlich and Rundell 2000) to a Banach space setting and analyze its convergence. We here show rates results for a particular source condition and different exponents in the formulation of Tikhonov regularization in each step. This includes a complementary result on the (first order) iteratively regularized Gauss-Newton method (IRGNM) in case of a one-homogeneous data misfit term, which corresponds to exact penalization. The results clearly show the possible advantages of using second order information, which get most pronounced in this exact penalization case. Numerical simulations for a coefficient identification problem in an elliptic PDE illustrate the theoretical findings.

Barbara Kaltenbacher, Paul Manns

In this paper, by means of a standard model problem, we devise an approach to computing approximate dual bounds for use in global optimization of coefficient identification in partial differential equations (PDEs) by, e.g., (spatial) branch-and-bound methods. Linearization is achieved by a McCormick relaxation (that is, replacing the bilinear PDE term by a linear one and adding inequality constraints), combined with local averaging to reduce the number of inequalities. Optimization-based bound tightening allows us to tighten the relaxation and thus reduce the induced error. Combining this with a quantification of the discretization error and the propagated noise, we prove that the resulting discretization regularizes the inverse problem, thus leading to an overall convergent scheme. Numerical experiments illustrate the theoretical findings.

Barbara Kaltenbacher, Mario Luiz Previatti de Souza

In this paper we consider the Iteratively Regularized Gauss-Newton Method (IRGNM) in its classical Tikhonov version and in an Ivanov type version, where regularization is achieved by imposing bounds on the solution. 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. Moreover, we provide a convergence result for the discretized problem with an appropriate control on the error and show how to provide the required error bounds by goal oriented weighted dual residual estimators. The results are illustrated for an inverse source problem for a nonlinear elliptic boundary value problem, for the cases of a measure valued and of an $L^\infty$ source. For the latter, we also provide numerical results with the Ivanov type IRGNM.

Barbara Kaltenbacher, Franz Rendl, Elena Resmerita

In this paper we present a method for the regularized solution of nonlinear inverse problems, based on Ivanov regularization (also called method of quasi solutions or constrained least squares regularization). This leads to the minimization of a non-convex cost function under a norm constraint, where non-convexity is caused by nonlinearity of the inverse problem. Minimization is done by iterative approximation, using (non-convex) quadratic Taylor expansions of the cost function. This leads to repeated solution of quadratic trust region subproblems with possibly indefinite Hessian. Thus the key step of the method consists in application of an efficient method for solving such quadratic subproblems, developed by Rendl and Wolkowicz [10]. We here present a convergence analysis of the overall method as well as numerical experiments.

Romana Boiger, Barbara Kaltenbacher

Online parameter identification is of importance, e.g., for model predictive control. Since the parameters have to be identified simultaneously to the process of the modeled system, dynamical update laws are used for state and parameter estimates. Most of the existing methods for infinite dimensional systems either impose strong assumptions on the model or cannot handle partial observations. Therefore we propose and analyze an online parameter identification method that is less restrictive concerning the underlying model and allows for partial observations and noisy data. The performance of our approach is illustrated by some numerical experiments.