arXiv subjects

Barbara Kaltenbacher

Publications and source records attributed to Barbara Kaltenbacher.

At least 19 recordsLinked to original sources

arXiv2026

Linearized uniqueness of space dependent coefficients in a non-autonomous evolution equation from non-local observations

In this paper, we consider identification of space dependent coefficients in a time dependent PDE, which is non-autonomous, due to a bilinear control term. We prove linearized uniqueness from time trace observations, in particular also considering non-local observations in the form of weighted integrals of the state over he spatial domain. The result is obtained in an abstract setting and applied, first of all to the identification of a potential in a diffusion equation for illustration purposes, second to the reconstruction of equilibrium magnetization, relaxation rates and field inhomogeneity as space dependent quantities in model based qunatitative magnetic resonance imaging with the Bloch-Torrey equation, which constitutes the real world application motivating this study.

math.AP↗

arXiv2026

Multi parameter identification in the nonlinear periodic Westervelt equation

Nonlinear ultrasound imaging leverages harmonic wave generation to enhance contrast and spatial resolution beyond the capabilities of conventional linear techniques. This behavior is commonly modeled by the Westervelt equation, which captures finite-amplitude acoustic wave propagation in heterogeneous media. In this work, we investigate an inverse problem for a periodic nonlinear Westervelt equation in $\mathbb{R}^d$, where $d\in\{2,3\}$ with spatially varying coefficients and Robin-type boundary conditions. The objective is to simultaneously reconstruct the sound speed, diffusivity, and nonlinearity parameters from (partial) boundary measurements. We first establish the Fréchet differentiability of the forward solution operator with respect to the unknown parameters, providing a rigorous analytical foundation for parameter identification. To address uniqueness, we introduce a reference-state framework and prove linearized uniqueness of an all-at-once forward operator without requiring the reference states to satisfy the governing equation. Building on these results, we develop an iterative reconstruction scheme based on a frozen Newton-type method, supported by an exact range invariance property. Numerical simulations are presented to illustrate the feasibility and performance of the proposed approach.

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arXiv2026

On the Jordan-Moore-Gibson-Thompson equation of nonlinear acoustics

The JMGT equation was put forward by Pedro Jordan~\cite{jordan2008nonlinear,jordan2014second}, also referring to earlier work by Moore and Gibson~\cite{moore1960propagation}, as well as Thompson~\cite{thompson} to amend the infinite speed of sound paradox of classical models of nonlinear acoustics such as the Westervelt and Kuznetsov's equation. Additionally to its physical significance (and of course related to it), it has given rise to a substantial body of mathematical literature -- possibly even more than the above mentioned classical models. In this paper, we aim to provide a systematic (though inevitably incomplete) overview %and indicate some potential open questions. thereby focusing on well-posedness analysis of initial value and time periodic problems, memory and fractional attenuation as well as singular limits and -- with one example each -- control and inverse problems.

math.AP↗

arXiv2026

On uniqueness of coefficient identification in the Bloch-Torrey equation for magnetic resonance imaging

In this paper we provide some uniqueness results for the (multi-)coefficient identification problem of reconstructing the spatially varying spin density as well as the spin-lattice and spin-spin relaxation times and the local field inhomogeneity in the Bloch-Torrey equation, as relevant in magnetic resonance imaging MRI. To this end, we follow two approaches: (a) Relying on sampling of the k-space and (approximately) explicit reconstruction formulas in the simplified (Bloch) ODE setting, along with perturbation estimates; (b) Relying on infinite speed of propagation due to diffusion. The results on well-posendess and Lipschitz continuous differentiability of the coefficient-to-state map derived for this purpose, are expected to be useful also in the convergence analysis of reconstruction schemes as well in mathematical optimization of the experimental design in MRI.

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arXiv2026

Locally-averaged McCormick relaxations for discretization-regularized inverse problems

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.

math.NA↗

arXiv2026

Identification of space-dependent coefficients in two competing terms of a nonlinear subdiffusion equation

We consider a (sub)diffusion equation with a nonlinearity of the form $pf(u)-qu$, where $p$ and $q$ are space dependent functions. Prominent examples are the Fisher-KPP, the Frank-Kamenetskii-Zeldovich and the Allen-Cahn equations. We devise a fixed point scheme for reconstructing the spatially varying coefficients from interior observations a) at final time under two different excitations b) at two different time instances under a single excitation. Convergence of the scheme as well as local uniqueness of these coefficients is proven. Numerical experiments illustrate the performance of the reconstruction scheme.

math.NA↗

arXiv2026

Decay rates to equilibrium in a nonlinear subdiffusion equation with two counteracting terms

In this paper we prove convergence to a steady state as $t\to\infty$ for solutions to the subdiffusion equation \[ \partial_t^\alpha u - \mathbb{L} u = q(x)u - p(x)f(u) + r \] with the exponential ($\alpha=1$) or power law ($\alpha\in[0,1)$) rates under mild conditions on the coefficients $p$, $q$, the nonlinearity $f$, the source $r$, and the elliptic operator $\mathbb{L}$.

math.AP↗

arXiv2026

Existence, uniqueness, and numerical solutions of the nonlinear periodic Westervelt equation

In this paper, we study the nonlinear periodic Westervelt equation with excitations located within a bounded domain in $\mathbb{R}^d$, where $d \in \{2,3\}$, subject to Robin boundary conditions. This problem is of particular interest for advancing imaging techniques that exploit nonlinearity of the acoustic propagation. We establish the existence and uniqueness of solutions in both the linear and the nonlinear setting, thereby allowing for spatially varying coefficients as relevant in quantitative imaging. Derivation of a multiharmonic formulation enables us to show the generation of higher harmonics (that is, responses at multiples of the fundamental frequency) due nonlinear wave propagation. An iterative scheme for solving the resulting system is proposed that relies on successive resolution of these higher harmonics, and its convergence under smallness conditions on the excitation is proven. Furthermore, we investigate the numerical solution of the resulting system of Helmholtz equations, employing a conforming finite element method for its discretization. Through an implementation of the proposed methodology, we illustrate how acoustic waves propagate in nonlinear media. This study aims to enhance our understanding of ultrasound propagation dynamics, which is essential for obtaining high-quality images from limited in vivo and boundary measurements.

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arXiv2025

Imaging nonlinearity coefficient and sound speed with the JMGT equation in frequency domain

In this paper we prove uniqueness and stability of reconstruction of two coefficients (sound speed and nonlinearity parameter) in the Jordan-Moore-Gibson-Thompson JMGT equation of nonlinear acoustics, relying on observations resulting from only two sources. A key tool for this purpose is a multiharmonic expansion of the PDE solution, which reflects the physical phenomenon of higher harmonics appearing due to nonlinearity and allows us to work in frequency domain. Based on this result, we derive a regularization property of reconstruction with JMGT as the relexation time tends to zero (in the spirit of a quasi reversibility method) for reconstruction from the classical Westervelt equation.

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arXiv2025

Optimization of a Nonlinear Acoustics -- Structure Interaction Model

In this paper, we consider a control/shape optimization problem of a nonlinear acoustics-structure interaction model of PDEs, whereby acoustic wave propagation in a chamber is governed by the Westervelt equation, and the motion of the elastic part of the boundary is governed by a 4th order Kirchoff equation. We consider a quadratic objective functional capturing the tracking of prescribed desired states, with three types of controls: 1) An excitation control represented by prescribed Neumann data for the pressure on the excitation part of the boundary 2) A mechanical control represented by a forcing function in the Kirchoff equations and 3) Shape of the excitation part of the boundary represented by a graph function. Our main result is the existence of solutions to the minimization problem, and the characterization of the optimal states through an adjoint system of PDEs derived from the first-order optimality conditions.

math.OC↗

arXiv2025

Acoustic nonlinearity parameter tomography with the Jordan-Moore-Gibson-Thompson equation in frequency domain

This paper aims to combine the advantages of the Jordan-Moore-Gibson-Thompson JMGT equation as an advanced model in nonlinear acoustics with a frequency domain formulation of the forward and inverse problem of acoustic nonlinearity parameter tomography, enabling the multiplication of information by nonlinearity. Our main result is local uniqueness of the space dependent nonlinearity parameter from boundary measurements, which we achieve by linearized uniqueness with an Implicit Function type perturbation argument in appropriately chosen topologies. Moreover, we shortly dwell on the application of a regularized Newton type method for reconstructing the nonlinearity coefficient, whose convergence can be established by means of the linear uniqueness result.

math.AP↗

arXiv2025

On the optimal choice of the illumination function in photoacoustic tomography

This work studies the inverse problem of photoacoustic tomography (more precisely, the acoustic subproblem) as the identification of a space-dependent source parameter. The model consists of a wave equation involving a time-fractional damping term to account for power law frequency dependence of the attenuation, as relevant in ultrasonics. We solve the inverse problem in a Bayesian framework using a Maximum A Posteriori (MAP) estimate, and for this purpose derive an explicit expression for the adjoint operator. On top of this, we consider optimization of the choice of the laser excitation function, which is the time-dependent part of the source in this model, to enhance the reconstruction result. The method employs the $A$-optimality criterion for Bayesian optimal experimental design with Gaussian prior and Gaussian noise. To efficiently approximate the cost functional, we introduce an approximation scheme based on projection onto finite-dimensional subspaces. Finally, we present numerical results that illustrate the theory.

math.NA↗

arXiv2025

Reconstruction of space-dependence and nonlinearity of a reaction term in a subdiffusion equation

In this paper we study the simultaneous reconstruction of two coefficients in a reaction-subdiffusion equation, namely a nonlinearity and a space dependent factor. The fact that these are coupled in a multiplicative matter makes the reconstruction particularly challenging. Several situations of overposed data are considered: boundary observations over a time interval, interior observations at final time, as well as a combination thereof. We devise fixed point schemes and also describe application of a frozen Newton method. In the final time data case we prove convergence of the fixed point scheme as well as uniqueness of both coefficients. Numerical experiments illustrate performance of the reconstruction methods, in particular dependence on the differentiation order in the subdiffusion equation.

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arXiv2025

Forward and Inverse Problems in Nonlinear Acoustics

The importance of ultrasound is well established in the imaging of human tissue. In order to enhance image quality by exploiting nonlinear effects, recently techniques such as harmonic imaging and nonlinearity parameter tomography have been put forward. As soon as the pressure amplitude exceeds a certain bound, the classical linear wave equation loses its validity and more general nonlinear versions have to be used. Another characteristic property of ultrasound propagation in human tissue is frequency power law attenuation, leading to fractional derivative damping models in time domain. In this contribution we will first of all dwell on modeling nonlinearity on the one hand and fractional damping on the other hand. Moreover we will give an idea on the challenges in the analysis of the resulting PDEs and discuss some parameter asymptotics. Finally, we address a relevant inverse problems in this context, the above mentioned task of nonlinearity parameter imaging, which leads to a coefficient identification problem for a quasilinear wave equation.

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arXiv2024

Adaptive Reduced Basis Trust Region Methods for Parameter Identification Problems

In this contribution, we are concerned with model order reduction in the context of iterative regularization methods for the solution of inverse problems arising from parameter identification in elliptic partial differential equations. Such methods typically require a large number of forward solutions, which makes the use of the reduced basis method attractive to reduce computational complexity. However, the considered inverse problems are typically ill-posed due to their infinite-dimensional parameter space. Moreover, the infinite-dimensional parameter space makes it impossible to build and certify classical reduced-order models efficiently in a so-called "offline phase". We thus propose a new algorithm that adaptively builds a reduced parameter space in the online phase. The enrichment of the reduced parameter space is naturally inherited from the Tikhonov regularization within an iteratively regularized Gauß-Newton method. Finally, the adaptive parameter space reduction is combined with a certified reduced basis state space reduction within an adaptive error-aware trust region framework. Numerical experiments are presented to show the efficiency of the combined parameter and state space reduction for inverse parameter identification problems with distributed reaction or diffusion coefficients.

math.NA↗

arXiv2024

Well-posedness of the time-periodic Jordan-Moore-Gibson-Thompson equation

Motivated by applications of nonlinear ultrasonics under continuous wave excitation, we study the Jordan-Moore-Gibson-Thompson (JMGT) equation -- a third order in time quasilinear PDE -- under time periodicity conditions. Here the coefficient of the third order time derivative is the so-called relaxation time and a thorough understanding of the limiting behaviour for vanishing relaxation time is essential to link these JMGT equations to classical second order models in nonlinear acoustics, As compared to the meanwhile well understood initial value problem for JMGT, the periodic setting poses substantial challenges due to a loss of temporal regularity, while the analysis still requires an $L^\infty$ control of solutions in space and time in order to maintain stability or equivalently, to avoid degeneracy of the second time derivative coefficient. We provide a full well-posedness analysis with and without gradient nonlinearity, as relevant for modelling non-cumulative nonlinear effects, under practically relevant mixed boundary conditions. The source-to-state map is thus well-defined and we additionally show it to be Lipschitz continuously differentiable, a result that is useful for inverse problems applications such as acoustic nonlinearity tomography. The energy bounds derived for the well-posedness analysis of periodic JMGT equations also allow to fully justify the singular limit for vanishing relaxation time.

math.AP↗

arXiv2024

Existence of solutions to k-Wave models of nonlinear ultrasound propagation in biological tissue

We investigate models for nonlinear ultrasound propagation in soft biological tissue based on the one that serves as the core for the software package k-Wave. The systems are solved for the acoustic particle velocity, mass density, and acoustic pressure and involve a fractional absorption operator. We first consider a system that incorporates additional viscosity in the equation for momentum conservation. By constructing a Galerkin approximation procedure, we prove the local existence of its solutions. In view of inverse problems arising from imaging tasks, the theory allows for the variable background mass density, speed of sound, and the nonlinearity parameter in the systems. Secondly, under stronger conditions on the data, we take the vanishing viscosity limit of the problem, thereby rigorously establishing the existence of solutions for the limiting system as well.

math.AP↗

arXiv2024

A first order in time wave equation modeling nonlinear acoustics

In this paper we focus on a small amplitude approximation of a Navier-Stokes-Fourier system modeling nonlinear acoustics. Omitting all third and higher order terms with respect to certain small parameters, we obtain a first order in time system containing linear and quadratic pressure and velocity terms. Subsequently, the well-posedness of the derived system is shown using the classical method of Galerkin approximation in combination with a fixed point argument. We first prove the well-posedness of a linearized equation using energy estimates and then the well-posedness of the nonlinear system using a Newton-Kantorovich type argument. Based on this, we also obtain global in time well-posedness for small enough data and exponential decay. This is in line with semigroup results for a linear part of the system that we provide as well.

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