arXiv subjects

Barbara Kaltenbacher

Publications and source records attributed to Barbara Kaltenbacher.

At least 55 records · Page 3Linked to original sources

arXiv2021

Fractional time stepping and adjoint based gradient computation in an inverse problem for a fractionally damped wave equation

In this paper we consider the inverse problem of identifying the initial data in a fractionally damped wave equation from time trace measurements on a surface, as relevant in photoacoustic or thermoacoustic tomography. We derive and analyze a time stepping method for the numerical solution of the corresponding forward problem. Moreover, to efficiently obtain reconstructions by minimizing a Tikhonov regularization functional (or alternatively, by computing the MAP estimator in a Bayesian approach), we develop an adjoint based scheme for gradient computation. Numerical reconstructions in two space dimensions illustrate the performance of the devised methods.

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arXiv2021

A model reference adaptive system approach for nonlinear online parameter identification

Dynamical systems, for instance in model predictive control, often contain unknown parameters, which must be determined during system operation. Online or on-the-fly parameter identification methods are therefore necessary. The challenge of online methods is that one must continuously estimate parameters as experimental data becomes available. The existing techniques in the context of time-dependent partial differential equations exclude the case where the system depends nonlinearly on the parameters.Based on a model reference adaptive system approach, we present an online parameter identification method for nonlinear infinite-dimensional evolutionary system.

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arXiv2021

On an inverse problem of nonlinear imaging with fractional damping

This paper considers the attenuated Westervelt equation in pressure formulation. The attenuation is by various models proposed in the literature and characterised by the inclusion of non-local operators that give power law damping as opposed to the exponential of classical models. The goal is the inverse problem of recovering a spatially dependent coefficient in the equation, the parameter of nonlinearity $κ(x)$, in what becomes a nonlinear hyperbolic equation with nonlocal terms. The overposed measured data is a time trace taken on a subset of the domain or its boundary. We shall show injectivity of the linearised map from $κ$ to the overposed data used to recover it and from this basis develop and analyse Newton-type schemes for its effective recovery.

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arXiv2021

On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements

We consider an undetermined coefficient inverse problem for a non-\\linear partial differential equation occurring in high intensity ultrasound propagation as used in acoustic tomography. In particular, we investigate the recovery of the nonlinearity coefficient commonly labeled as $B/A$ in the literature which is part of a space dependent coefficient $κ$ in the Westervelt equation governing nonlinear acoustics. Corresponding to the typical measurement setup, the overposed data consists of time trace measurements on some zero or one dimensional set $Σ$ representing the receiving transducer array. After an analysis of the map from $κ$ to the overposed data, we show injectivity of its linearisation and use this as motivation for several iterative schemes to recover $κ$. Numerical simulations will also be shown to illustrate the efficiency of the methods.

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arXiv2021

On uniqueness and reconstruction of a nonlinear diffusion term in a parabolic equation

The problem of recovering coefficients in a diffusion equation is one of the basic inverse problems. Perhaps the most important term is the one that couples the length and time scales and is often referred to as {\it the\/} diffusion coefficient $a$ in $u_t - \nabla(a\nabla u) = f$. In this paper we seek the unknown $a$ assuming that $a=a(u)$ depends only on the value of the solution at a given point. Such diffusion models are the basic of a wide range of physical phenomena such as nonlinear heat conduction, chemical mixing and population dynamics. We shall look at two types of overposed data in order to effect recovery of $a(u)$: the value of a time trace $u(x_0,t)$ for some fixed point $x_0$ on the boundary of the region $Ω$; or the value of $u$ on an interior curve $Σ$ lying within $Ω$. As examples, these might represent a temperature measurement on the boundary or a census of the population in some subset of $Ω$ taken at a fixed time $T>0$. In the latter case we shall show a uniqueness result that leads to a constructive method for recovery of $a$. Indeed, for both types of measured data we shall show reconstructions based on the iterative algorithms developed in the paper.

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arXiv2021

On the simultaneous recovery of the conductivity and the nonlinear reaction term in a parabolic equation

This paper considers the inverse problem of recovering both the unknown, spatially-dependent conductivity $a(x)$ and the nonlinear reaction term $f(u)$ in a reaction-diffusion equation from overposed data. These measurements can consist of: the value of two different solution measurements taken at a later time $T$; time-trace profiles from two solutions; or both final time and time-trace measurements from a single forwards solve data run. We prove both uniqueness results and the convergence of iteration schemes designed to recover these coefficients. The last section of the paper shows numerical reconstructions based on these algorithms.

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arXiv2021

Iterative regularization for constrained minimization formulations of nonlinear inverse problems

In this paper we the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type. 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 techology known as impedance acoustic tomography, for which we provide numerical experiments.

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arXiv2021

All-at-once formulation meets the Bayesian approach: A study of two prototypical linear inverse problems

In this work, the Bayesian approach to inverse problems is formulated in an all-at-once setting. The advantages of the all-at-once formulation are known to include the avoidance of a parameter-to-state map as well as numerical improvements, especially when considering nonlinear problems. In the Bayesian approach, prior knowledge is taken into account with the help of a prior distribution. In addition, the error in the observation equation is formulated by means of a distribution. This method naturally results in a whole posterior distribution for the unknown target, not just point estimates. This allows for further statistical analysis including the computation of credible intervals. We combine the Bayesian setting with the all-at-once formulation, resulting in a novel approach for investigating inverse problems. With this combination we are able to chose a prior not only for the parameter, but also for the state variable, which directly influences the parameter. Furthermore, errors not only in the observation equation, but additionally, in the model can be taken into account. %The aim of this approach is not only to accomplish reasonable reconstructions of the unknown parameter but also to maximize the information gained from measurements through combining it with prior knowledge, obtained either from certain expertise or former investigation in the model. We analyze this approach with the help of two linear standard examples, namely the inverse source problem for the Poisson equation and the backwards heat equation, i.e. a stationary and a time dependent problem. Appropriate function spaces and derivation of adjoint operators are investigated. To assess the degree of ill-posedness, we analyze the singular values of the corresponding all-at-once forward operators. %as well as the convergence of the method. Finally, joint priors are designed and numerically tested.

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arXiv2020

Some application examples of minimization based formulations of inverse problems and their regularization

In this paper we extend a recent idea of formulating and regularizing inverse problems as minimization problems, so without using a forward operator, thus avoiding explicit evaluation of a parameter-to-state map. We do so by rephrasing three application examples in this minimization form, namely (a) electrical impedance tomography with the complete electrode model (b) identification of a nonlinear magnetic permeability from magnetic flux measurements (c) localization of sound sources from microphone array measurements. To establish convergence of the proposed regularization approach for these problems, we first of all extend the existing theory. In particular, we take advantage of the fact that observations are finite dimensional here, so that inversion of the noisy data can to some extent be done separately, using a right inverse of the observation operator. This new approach is actually applicable to a wide range of real world problems.

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arXiv2020

The Inverse Problem of Reconstructing Reaction-Diffusion Systems

This paper considers the inverse problem of recovering state-dependent source terms in a reaction-diffusion system from overposed data consisting of the values of the state variables either at a fixed finite time (census-type data) or a time trace of their values at a fixed point on the boundary of the spatial domain. We show both uniqueness results and the convergence of an iteration scheme designed to recover these sources. This leads to a reconstructive method and we shall demonstrate its effectiveness by several illustrative examples.

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arXiv2019

On the Jordan-Moore-Gibson-Thompson equation: well-posedness with quadratic gradient nonlinearity and singular limit for vanishing relaxation time

In this paper, we consider the Jordan-Moore-Gibson-Thompson equation, a third order in time wave equation describing the nonlinear propagation of sound that avoids the infinite signal speed paradox of classical second order in time strongly damped models of nonlinear acoustics, such as the Westervelt and the Kuznetsov equation. We show well-posedness in an acoustic velocity potential formulation with and without gradient nonlinearity, corresponding to the Kuznetsov and the Westervelt nonlinearities, respectively. Moreover, we consider the limit as the parameter of the third order time derivative that plays the role of a relaxation time tends to zero, which again leads to the classical Kuznetsov and Westervelt models. To this end, we establish appropriate energy estimates for the linearized equations and employ fixed-point arguments for well-posedness of the nonlinear equations. The theoretical results are illustrated by numerical experiments.

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arXiv2019

All-at-once versus reduced iterative methods for time dependent inverse problems

In this paper we investigate all-at-once versus reduced regularization of dynamic inverse problems on finite time intervals $(0,T)$. In doing so, we concentrate on iterative methods and nonlinear problems, since they have already been shown to exhibit considerable differences in their reduced and all-at-once versions, whereas Tikhonov regularization is basically the same in both settings. More precisely, we consider Landweber iteration, the iteratively regularized Gauss-Newton method, and the Landweber-Kaczmarz method, the latter relying on cyclic iteration over a subdivision of the problem into subsequent subintervals of $(0,T)$. Part of the paper is devoted to providing an appropriate function space setting as well as establishing the required differentiability results needed for well-definedness and convergence of the methods under consideration. Based on this, we formulate and compare the above mentioned iterative methods in their all-at-once and their reduced version. Finally, we provide some convergence results in the framework of Hilbert space regularization theory and illustrate the convergence conditions by an example of an inverse source problem for a nonlinear diffusion equation.

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arXiv2019

Regularization of a backwards parabolic equation by fractional operators

The backwards diffusion equation is one of the classical ill-posed inverse problems, related to a wide range of applications, and has been extensively studied over the last 50 years. One of the first methods was that of {\it quasireversibility\/} whereby the parabolic operator is replaced by a differential operator for which the backwards problem in time is well posed. This is in fact the direction we will take but will do so with a nonlocal operator; an equation of fractional order in time for which the backwards problem is known to be ``almost well posed.'' We shall look at various possible options and strategies but our conclusion for the best of these will exploit the linearity of the problem to break the inversion into distinct frequency bands and to use a different fractional order for each. The fractional exponents will be chosen using the discrepancy principle under the assumption we have an estimate of the noise level in the data. An analysis of the method is provided as are some illustrative numerical examples.

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arXiv2019

The Ivanov regularized Gauss-Newton method in Banach space with an a posteriori choice of the regularization radius

In this paper we consider the iteratively regularized Gauss-Newton method, where regularization is achieved by Ivanov regularization, i.e., by imposing a priori constraints on the solution. We propose an a posteriori choice of the regularization radius, based on an inexact Newton / discrepancy principle approach, prove convergence and convergence rates under a variational source condition as the noise level tends to zero, and provide an analysis of the discretization error. Our results are valid in general, possibly nonreflexive Banach spaces, including, e.g., $L^\infty$ as a preimage space. The theoretical findings are illustrated by numerical experiments.

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arXiv2019

Minimization based formulations of inverse problems and their regularization

The conventional way of formulating inverse problems such as identification of a (possibly infinite dimensional) parameter, is via some forward operator, which is the concatenation of the observation operator with the parameter-to-state-map for the underlying model. Recently, all-at-once formulations have been considered as an alternative to this reduced formulation, avoiding the use of a parameter-to-state map, which would sometimes lead to too restrictive conditions. Here the model and the observation are considered simultaneously as one large system with the state and the parameter as unknowns. A still more general formulation of inverse problems, containing both the reduced and the all-at-once formulation, but also the well-known and highly versatile so-called variational approach (also called Kohn-Vogelius functional approach) as special cases, is to formulate the inverse problem as a minimization problem -- instead of an equation -- for the state and parameter. Regularization can be incorporated via imposing constraints and/or adding regularization terms to the objective. In this paper, after giving a motivation by formulating the electrical impedance tomography (EIT) problem by means of the classical Kohn-Vogelius functional, we will dwell on the regularization aspects for such variational formulations in an abstract setting. Indeed, combination of regularization by constraints and by penalization leads to new methods that are applicable without solving forward problems. In particular, for the EIT problem we will consider a method employing box constraints in a very natural manner to incorporate the discrepancy principle for regularization parameter choice as well as a priori information on the searched for conductivity.

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arXiv2019

Parameter identification for the Landau-Lifshitz-Gilbert equation in Magnetic Particle Imaging

Magnetic particle imaging (MPI) is a tracer-based technique for medical imaging where the tracer consists of ironoxide nanoparticles. The key idea is to measure the particle response to a temporally changing external magnetic field to compute the spatial concentration of the tracer inside the object. A decent mathematical model demands for a data-driven computation of the system function which does not only describe the measurement geometry but also encodes the interaction of the particles with the external magnetic field. The physical model of this interaction is given by the Landau-Lifshitz-Gilbert (LLG) equation. The determination of the system function can be seen as an inverse problem of its own which can be interpreted as a calibration problem for MPI. In this contribution the calibration problem is formulated as an inverse parameter identification problem for the LLG equation. We give a detailed analysis of the direct as well as the inverse problem in an all-at-once as well as in a reduced setting. The analytical results yield a deeper understanding of inverse problems connected to the LLG equation and provide a starting point for the development of robust numerical solution methods in MPI.

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arXiv2019

Recovery of multiple coefficients in a reaction-diffusion equation

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 $q(x)f(u)$.

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arXiv2019

On the identification of a nonlinear term in a reaction-diffusion equation

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.

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