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In ultrasonics, nonlocal quasilinear wave equations arise when taking into account a class of heat flux laws of Gurtin–Pipkin type within the system of governing equations of sound motion. The present study extends previous work by the authors to incorporate nonlocal acoustic wave equations with quadratic gradient nonlinearities which require a new approach in the energy analysis. More precisely, we investigate the Kuznetsov and Blackstock equations with dissipation of fractional type and identify a minimal set of assumptions on the memory kernel needed for each equation. In particular, we discuss the physically relevant examples of Abel and Mittag–Leffler kernels. We perform the well-posedness analysis uniformly with respect to a small parameter on which the kernels depend and which can be interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the limiting behavior of solutions with respect to this parameter, and how it is influenced by the specific class of memory kernels at hand. Through such a limiting study, we relate the considered nonlocal quasilinear equations to their limiting counterparts and establish the convergence rates of the respective solutions in the energy norm. [ABSTRACT FROM AUTHOR]

A key observation underlying this paper is the fact that the range invariance condition for convergence of regularization methods for nonlinear ill-posed operator equations—such as coefficient identification in partial differential equations (PDEs) from boundary observations—can often be achieved by extending the searched for parameter in the sense of allowing it to depend on additional variables. This clearly counteracts unique identifiability of the parameter, though. The second key idea of this paper is now to restore the original restricted dependency of the parameter by penalization. This is shown to lead to convergence of variational (Tikhonov type) and iterative (Newton-type) regularization methods. We concretize the abstract convergence analysis in a framework typical of parameter identification in PDEs in a reduced and an all-at-once setting. This is further illustrated by three examples of coefficient identification from boundary observations in elliptic and time-dependent PDEs. [ABSTRACT FROM AUTHOR]

This paper considers the Westervelt equation, one of the most widely used models in nonlinear acoustics, and seeks to recover two spatially-dependent parameters of physical importance from time-trace boundary measurements. Specifically, these are the nonlinearity parameter κ (x) often referred to as B / A in the acoustics literature and the wave speed c 0 (x). The determination of the spatial change in these quantities can be used as a means of imaging. We consider identifiability from one or two boundary measurements as relevant in these applications. For a reformulation of the problem in terms of the squared slowness s = 1 / c 0 2 and the combined coefficient η = κ c 0 2 we devise a frozen Newton method and prove its convergence. The effectiveness (and limitations) of this iterative scheme are demonstrated by numerical examples. [ABSTRACT FROM AUTHOR]

We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation describing high‐intensity ultrasound propagation as widely used in medical imaging and therapy. The usual nonlinear term in the standard model using the Westervelt equation in pressure formulation is of the form ppt. However, this should be considered as a low‐order approximation to a more complex physical model where higher order terms will be required. Here we assume a more general case where the form taken is f(p) pt and f is unknown and must be recovered from data measurements. Corresponding to the typical measurement setup, the overposed data consist of time trace observations of the acoustic pressure at a single point or on a one‐dimensional set Σ representing the receiving transducer array at a fixed time. Additionally to an analysis of well‐posedness of the resulting pde, we show injectivity of the linearized forward map from f to the overposed data and use this as motivation for several iterative schemes to recover f. Numerical simulations will also be shown to illustrate the efficiency of the methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider several time-fractional generalizations of the Jordan–Moore–Gibson–Thompson (JMGT) equations in nonlinear acoustics as well as their linear Moore–Gibson–Thompson (MGT) versions. Following the procedure described in Jordan (2014), these time-fractional acoustic equations are derived from four fractional versions of the Maxwell–Cattaneo law in Compte and Metzler (1997). Additionally to providing well-posedness results for each of them, we also study the respective limits as the fractional order tends to one, leading to the classical third order in time (J)MGT equation. [ABSTRACT FROM AUTHOR]

This work deals with the convergence analysis of parabolic perturbations to quasilinear wave equations on smooth bounded domains. In particular, we consider wave equations with nonlinearities of quadratic type, which cover the two classical models of nonlinear acoustics, the Westervelt and Kuznetsov equations. By employing a high-order energy analysis, we obtain convergence of their solutions to the corresponding inviscid equations' solutions in the standard energy norm with a linear rate, assuming small data and a sufficiently short time. The smallness of initial data can, however, be imposed in a lower-order norm than the one needed in the energy analysis. It arises only from ensuring the nondegeneracy of the studied wave equations. In addition, we address the open questions of local well-posedness of the two classical models on bounded domains with a nonnegative, possibly vanishing sound diffusivity. [ABSTRACT FROM AUTHOR]

The family of Jordan–Moore–Gibson–Thompson (JMGT) equations arises in nonlinear acoustics when a relaxed version of the heat flux law is employed within the system of governing equations of sound motion. Motivated by the propagation of sound waves in complex media with anomalous diffusion, we consider here a generalized class of such equations involving two (weakly) singular memory kernels in the principal and non-leading terms. To relate them to the second-order wave equations, we investigate their vanishing relaxation time behavior. The key component of this singular limit analysis are the uniform bounds for the solutions of these nonlinear equations of fractional type with respect to the relaxation time. Their availability turns out to depend not only on the regularity and coercivity properties of the two kernels, but also on their behavior relative to each other and the type of nonlinearity present in the equations. [ABSTRACT FROM AUTHOR]

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 \kappa (x), in what becomes a nonlinear hyperbolic equation with non-local 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 \kappa to the overposed data and from this basis develop and analyse Newton-type schemes for its effective recovery. [ABSTRACT FROM AUTHOR]

We analyze the behavior of third-order-in-time linear and nonlinear sound waves in thermally relaxing fluids and gases as the sound diffusivity vanishes. The nonlinear acoustic propagation is modeled by the Jordan--Moore--Gibson--Thompson equation both in its Westervelttype and in its Kuznetsov-type forms, that is, including general nonlinearities of quadratic type. As it turns out, sufficiently smooth solutions of these equations converge in the energy norm to the solutions of the corresponding inviscid models at a linear rate. Numerical experiments illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

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 systems. [ABSTRACT FROM AUTHOR]

We are interested in the determination of the local nonlinear magnetic material behaviour in electrical steel sheets due to cutting and punching effects. For this purpose, the inverse problem has to be solved, where the objective function, which penalises the difference between the measured and the simulated magnetic flux density, has to be minimised under a constraint defined according to the corresponding partial differential equation model. We use the adjoint method to efficiently obtain the gradients of the objective function with respect to the material parameters. The optimisation algorithm is low-memory Broyden–Fletcher–Goldfarb–Shanno (BFGS), the forward and adjoint formulations are solved using the finite element (FE) method and the ill-posedness is handled via Tikhonov regularisation, in combination with the discrepancy principle. Realistic numerical case studies show promising results. [ABSTRACT FROM AUTHOR]

The damped wave equation with the attenuation proportional to velocity is ubiquitous in science and engineering and a common situation is when the attenuation depends on frequency. The usual way to incorporate this effect is to introduce fractional order derivatives either as a replacement for ut or as modifier through a spatial component with space fractional derivatives. Models for these are very well developed and the effort in this paper is towards the analysis of the inverse problem of recovering critical coefficients or initial states although we also develop constructive methods for these and analyse their degree of ill-conditioning. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The focus of this work is on the analysis of the Westervelt equation modeling nonlinear propagation of high intensity ultrasound, in the practically relevant setting of a truncated computational domain with absorbing boundary conditions. We especially consider the zero and first order nonlinear absorbing boundary conditions devised in [38] in one and two space dimensions. As a matter of fact, the energy identities and estimates presented here were crucial for designing these absorbing boundary conditions in such a way that the desired energy dissipation through the boundary is guaranteed. Under the hypothesis of small initial data, we establish local well-posedness and provide higher order energy estimates, that we expect to be of additional use in boundary control and stabilization. [ABSTRACT FROM AUTHOR]

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 ∞ {L^{\infty}} as a preimage space. The theoretical findings are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

The problem of optimal energy harvesting for a piezoelectric element driven by mechanical vibrations is stated in terms of an ODE system with hysteresis under the time derivative coupling a mechanical oscillator with an electric circuit with or without inductance. In the piezoelectric constitutive law, both the self-similar piezoelectric butterfly character of the hysteresis curves and feedback effects are taken into account in a thermodynamically consistent way. The physical parameters of the harvester are chosen to be the control variable, and the goal is to maximize the harvested energy for a given mechanical load and a given time interval. If hysteresis is modeled by the Preisach operator, the system is shown to be well-posed with continuous data dependence. For the special case of the play operator, we derive first-order necessary optimality conditions and an explicit form of the gradient of the total harvested energy functional in terms of solutions to the adjoint system. [ABSTRACT FROM AUTHOR]

This work is concerned with the study of fundamental models from nonlinear acoustics. In Part I, a hierarchy of nonlinear damped wave equations arising in the description of sound propagation in thermoviscous fluids is deduced. In particular, a rigorous justification of two classical models, the Kuznetsov and Westervelt equations, retained as limiting systems for vanishing thermal conductivity and consistent initial data, is given. Numerical comparisons that confirm and complement the theoretical results are provided in Part II. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of identifying parameters in stochastic differential equations. For this purpose, we transform the originally stochastic and nonlinear state equation to a deterministic linear partial differential equation for the transition probability density. We provide an appropriate likelihood cost function for parameter fitting, and derive an adjoint based approach for the computation of its gradient. [ABSTRACT FROM AUTHOR]

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 overly 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 not only the reduced and all-at-once formulations but also the well-known and highly versatile variational approach (also called the 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 explore 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. [ABSTRACT FROM AUTHOR]

We are interested in shape sensitivity analysis for an optimization problem arising in medical applications of high intensity focused ultrasound. The goal is to find the optimal shape of a focusing acoustic lens so that the desired acoustic pressure at a kidney stone is achieved. Coupling of the silicone acoustic lens and nonlinearly acoustic fluid region is modeled by the Westervelt equation with nonlinear strong damping and piecewise constant coefficients. We follow the variational approach to calculating the shape derivative of the cost functional which does not require computing the shape derivative of the state variable; however assumptions of certain spatial regularity of the primal and the adjoint state are needed to obtain the derivative, in particular for its strong form according to the Delfour-Hadamard-Zolésio structure theorem. [ABSTRACT FROM AUTHOR]

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. In this manuscript, we evaluate adjoint sensitivity analysis for parameter estimation in large scale biochemical reaction networks. We present the approach for time-discrete measurement and compare it to state-of-the-art methods used in systems and computational biology. Our comparison reveals a significantly improved computational efficiency and a superior scalability of adjoint sensitivity analysis. The computational complexity is effectively independent of the number of parameters, enabling the analysis of large- and genome-scale models. Our study of a comprehensive kinetic model of ErbB signaling shows that parameter estimation using adjoint sensitivity analysis requires a fraction of the computation time of established methods. The proposed method will facilitate mechanistic modeling of genome-scale cellular processes, as required in the age of omics. [ABSTRACT FROM AUTHOR]

In this paper we prove well-posedness and exponential stability for a general model of nonlinear acoustics. The key steps of the proof are a multiplicative splitting of the differential operator into a heat and a strongly damped wave part plus some lower order remainder, as well as derivation of an appropriate spectral bound. [ABSTRACT FROM AUTHOR]

We show higher interior regularity for the Westervelt equation with strong nonlinear damping term ofq-Laplace type. Secondly, we investigate an interface coupling problem for these models, which arise, e.g. in the context of medical applications of high intensity focused ultrasound in the treatment of kidney stones. We show that the solution to the coupled problem exhibits piecewiseregularity in space, provided that the gradient of the acoustic pressure is essentially bounded in space and time on the whole domain. This result is of importance in numerical approximations of the present problem as well as in gradient based algorithms for finding the optimal shape of a focusing acoustic lens in lithotripsy. [ABSTRACT FROM PUBLISHER]

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 nonconvex cost function under a norm constraint, where nonconvexity is caused by nonlinearity of the inverse problem. Minimization is done by iterative approximation, using (nonconvex) 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 []. We here present a convergence analysis of the overall method as well as numerical experiments. [ABSTRACT FROM AUTHOR]

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 regularizat ion 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-once and reduced versions and provide some numerical comparison. [ABSTRACT FROM AUTHOR]

In this work, we investigate the inverse problem of determining the kernel functions that best describe the mechanical behavior of a complex medium modeled by a general nonlocal viscoelastic wave equation. To this end, we minimize a tracking-type data misfit function under this PDE constraint. We perform the well-posedness analysis of the state and adjoint problems and, using these results, rigorously derive the first-order sensitivities. Numerical experiments in a three-dimensional setting illustrate the method. • Kernel recovery method for general nonlocal viscoelasticity. • Minimization of a tracking-type data misfit function governed by non-local viscoelastic wave equations. • Rigorous derivation of first-order sensitivities. • Numerical experiments in three-dimensional settings. [ABSTRACT FROM AUTHOR]

We propose a hysteretic model for electromechanical coupling in piezoelectric materials, with the strain and the electric field as inputs and the stress and the polarization as outputs. This constitutive law satisfies the thermodynamic principles and exhibits good agreement with experimental measurements. Moreover, when it is coupled with the mechanical and electromagnetic balance equations, the resulting PDE system is well-posed under the hypothesis that hysteretic effects take place only in one preferred direction. We prove the existence and uniqueness of its global weak solutions for each initial data with prescribed regularity. One of the tools is a new Lipschitz continuity theorem for the inverse Preisach operator with time dependent coefficients. [ABSTRACT FROM AUTHOR]

We consider the Westervelt equation in an unbounded domain and propose nonlinear absorbing boundary conditions for its efficient and robust numerical simulations. We use the theory of pseudo- and para-differential operators as well as asymptotic expansions to derive local in space and time absorbing boundary conditions of low to high orders in a consistent way. We show that the pseudo- and para-differential theories lead to essentially the same absorbing boundary conditions in terms of computational efficiency and numerical accuracy, whereas the asymptotic expansions result in exactly the same boundary conditions as the ones obtained with the para-differential approach. Moreover, we demonstrate that the use of pseudo- and para-differential operators leads to the same boundary conditions if the nonlinear function to be linearized vanishes at zero. The numerical studies demonstrate both the efficiency and effectiveness of the developed boundary conditions for different regimes of wave propagation in a wide range of excitation frequencies and angles of incidence. [ABSTRACT FROM AUTHOR]

Efficient time integration methods based on operator splitting are introduced for theWestervelt equation, a nonlinear damped wave equation that arises in nonlinear acoustics as a mathematical model for the propagation of sound waves in high intensity ultrasound applications. A global error estimate is deduced for the first-order Lie-Trotter splitting method, confirming that the splitting method remains stable, and that the nonstiff convergence order is retained in situations where the problem data are sufficiently regular. Fundamental ingredients in the stability and error analysis are regularity results for the Westervelt equation and related linear evolution equations of hyperbolic and parabolic type. Numerical examples illustrate and complement the theoretical investigations. [ABSTRACT FROM AUTHOR]

Abstract: A particular feature of certain microelectromechanical systems (MEMS) is the appearance of a so-called “pull-in” instability, corresponding to a singularity in the underlying PDE model. We here consider a transient MEMS model and its optimal control via the dielectric properties of the membrane and/or the applied voltage. In contrast to the static case, the control problem suffers from low dimensionality of the control compared to the state and hence requires different techniques for establishing first order optimality conditions. For this purpose, we here use a relaxation approach combined with a localization technique. [Copyright &y& Elsevier]

• Photoacoustic tomography PAT with fractional attenuation, as relevant in ultrasonics. • Two key computational aspects: time stepping; adjoint based gradient computation. • Both tasks require special treatment of the fractional damping term. • Relevant more generally in inverse problems for fractionally damped wave equations. 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. [ABSTRACT FROM AUTHOR]

Abstract: 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. For a reduced version (rPML), we present a long time stability proof based on an energy analysis. The numerical case studies and an application example demonstrate the good performance and long time stability of our formulation for treating open domain problems. [Copyright &y& Elsevier]

In this paper, we provide convergence and convergence rate results for a Newton-type method with a modified version of Landweber iteration as an inner iteration in a Banach space setting. Numerical experiments illustrate the performance of the method. [ABSTRACT FROM AUTHOR]

Abstract: We consider optimal control of nonlinear partial differential equations involving potentially singular solution-dependent terms. Singularity can be prevented by either restricting controls to a closed admissible set for which well-posedness of the equation can be guaranteed, or by explicitly enforcing pointwise bounds on the state. By means of an elliptic model problem, we contrast the requirements for deriving the existence of solutions and first order optimality conditions for both the control-constrained and the state-constrained formulation. Our analysis as well as numerical tests illustrate that control constraints lead to severe restrictions on the attainable states, which is not the case for state constraints. [Copyright &y& Elsevier]

We consider a third order in time equation which arises, e.g. as a model for wave propagation in viscous thermally relaxing fluids. This equation displays, even in the linear version, a variety of dynamical behaviors for its solution that depend on the physical parameters in the equation. These range from non-existence and instability to exponential stability (in time) as was shown for the constant coefficient case in Ref. 23. In case of vanishing diffusivity of the sound, there is a lack of generation of a semigroup associated with the linear dynamics. If diffusivity of the sound is positive, the linear dynamics is described by a strongly continuous hyperbolic-like evolution. This evolution is exponentially stable provided sufficiently large viscous damping is accounted for in the model. In this paper, we consider the full nonlinear model referred to as Jordan-Moore-Gibson-Thompson equation. This model can be seen as a "hyperbolic" version of Kuznetsov's equation, where the linearization of the latter corresponds to an analytic semigroup. This is no longer valid for the presently considered third-order model whose linearization is associated with a group structure. In order to carry out the analysis of the nonlinear model, we first consider time and space-dependent viscosity which then leads to evolution rather than semigroup generators. Decay rates for both "natural" and "higher" level energies are derived. Relevant physical parameters that are responsible for spectral behavior (continuous and point spectrum) are identified. The theoretical estimates proved in the paper are confirmed by numerical simulations. The derived energy estimates are then used in order to establish global well-posedness and exponential decay for the solutions to the nonlinear equation. [ABSTRACT FROM AUTHOR]

In this paper we investigate the regularizing properties of discretization in preimage space for linear and nonlinear ill-posed operator equations with noisy data. We propose to choose the discretization level, that acts as a regularization parameter in this context, by a discrepancy principle. While general convergence has been shown not to hold, we provide convergence results under appropriate conditions on the exact solution. [ABSTRACT FROM AUTHOR]

In this paper, we perform a sensitivity analysis for shape optimization problems arising in models we suggest for a lithotripter. More precisely, we use two models based on high intensity ultrasound focusing by an acoustic lens, where through changing the shape of the lens we try to achieve a favourable focusing. The models are based on acoustic wave equations with piecewise constant coefficients, a linear and a nonlinear one, where for the nonlinear one we use the Westervelt equation. The sensitivity analysis is performed using an adjoint approach. [ABSTRACT FROM AUTHOR]

In this paper we consider the thermoacoustic inverse problem of identifying the oscillatory heat release from pressure measurements. We consider the spatially onedimensional and time harmonic case. Three different regularization methods for the stable solution of this ill-posed inverse problem are proposed: Lavrent'ev's method, regularization by discretization, and a method based on an explicit formula combined with regularized numerical differentiation. For these methods, the results of numerical experiments are documented. [ABSTRACT FROM AUTHOR]

In this paper we show boundary observability and boundary stabilizability by linear feedbacks for a class of nonlinear wave equations including the undamped Westervelt model used in nonlinear acoustics. We prove local existence for undamped generalized Westervelt equations with homogeneous Dirichlet boundary conditions as well as global existence and exponential decay with absorbing type boundary conditions. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with optimal Neumann boundary control for the Westervelt and the Kuznetsov equations, which are equations of nonlinear acoustics. Specifically, functionals of tracking type with applications in noninvasive ultrasonic medical treatments are considered. Existence of optimal controls is established and first order necessary optimality conditions are derived. Stability of the minimizer with respect to perturbations in the data as well as convergence of the controls when the regularization parameter tends to zero is shown. [Copyright &y& Elsevier]

The parabolic equation of this paper is a nonlinear one with the unknown coefficient depending on the derivative of the solution. A uniqueness result is proven by the method of Carleman estimates. The applicability of this result is illustrated for parameter identification problems in population dynamics and magnetics. For the latter application, we provide numerical results using a reconstruction method based on a multiharmonic formulation of the problem. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of identifying a coefficient function in the principal part of a nonlinear hyperbolic PDE from overdetermined boundary data of the PDE solution. Assuming that the coefficient function can be written as a linear combination of finitely many ansatz functions, we derive a stability and uniqueness result that is based on an identifiability condition in terms of the initial data as well as on the smallness of the time interval. In doing so, we distinguish between the two- and higher dimensional case where functionals on the boundary can be measured and the one-dimensional situation with only point measurements at one boundary point. Our results also give perspectives to the case of an infinite dimensional coefficient function. [ABSTRACT FROM AUTHOR]

Abstract: Due to the principle of regularization by restricting the number of degrees of freedom, truncating the Cholesky factorization of a symmetric positive definite matrix can be expected to have a stabilizing effect. Based on this idea, we consider four different approaches for regularizing ill-posed linear operator equations. Convergence in the noise free case as well as—with an appropriate a priori truncation rule—in the situation of noisy data is analyzed. Moreover, we propose an a posteriori truncation rule and characterize its convergence. Numerical tests illustrate the theoretical results. Both analysis and computations suggest one of the four variants to be favorable to the others. [Copyright &y& Elsevier]

In this paper, we propose a multilevel method for solving nonlinear ill-posed operator equations By minimizing the distance to some initial guess under the constraint of a discretized version of the operator equation for different levels of discretization, we define a sequence of regularized approximations to the exact solution, which is shown to be stable and convergent for arbitrary initial guess and can be computed via a multilevel procedure that altogether yields a globally convergent method. Moreover, this approach enables one to relax restrictions on the nonlinearity of the forward operator, as were used in previous work on regularization methods for nonlinear ill-posed problems. [ABSTRACT FROM AUTHOR]

We introduce a class of stabilizing Newton-Kaczmarz methods for nonlinear ill-posed problems and analyze their convergence and regularization behavior. As usual for iterative methods for solving nonlinear ill-posed problems, conditions on the nonlinearity (or the derivatives) have to be imposed in order to obtain convergence. As we shall discuss in general and in some specific examples, the nonlinearity conditions obtained for the Newton-Kaczmarz methods are less restrictive than those for previously existing iteration methods and can be verified for several practical applications. We also discuss the discretization and efficient numerical solution of the linear problems arising in each step of a Newton-Kaczmarz method, and we carry out numerical experiments for two model problems. [ABSTRACT FROM AUTHOR]

This paper is concerned with nonlinear ill-posed operator equations F(a) = y (e.g., parameter identification problems) and their approximate solution by a Newton-type method that is regularized by projecting the linearized equation in each Newton step onto a finite-dimensional space (e.g., a finite element space) and by stopping the Newton iteration at an appropriate index. We prove convergence as the iteration index n goes to infinity in the noise-free case and convergence as the data noise level δ goes to zero in the case of noisy data, as well as convergence rates under certain additional conditions. [ABSTRACT FROM AUTHOR]