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For the first time, a class of implicit Runge-Kutta time discretisation methods is studied for nonlinear damped wave equations arising in nonlinear acoustics. The analysis in particular applies to the Westervelt, Jordan-Moore-Gibson-Thompson, and Blackstock-Crighton-Brunnhuber-Jordan-Kuznetsov equations. Under appropriate regularity, consistency, and smallness requirements on the time-continuous solutions, global error bounds are obtained from energy estimates for the time-discrete solutions. Existence and uniqueness of time-discrete solutions as well as their convergence are proven under weaker conditions on the initial data, based on energy estimates that are established in a continuous setting and then transferred to the time discretisations. [ABSTRACT FROM AUTHOR]

In this paper we consider the iteratively regularized Gauss-Newton method (IRGNM) in its classical Tikhonov version as well as two further—Ivanov type and Morozov type—versions. In these two alternative versions, regularization is achieved by imposing bounds on the solution or by minimizing some regularization functional under a constraint on the data misfit, respectively. 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∞ source. For the latter, we also provide numerical results with the Ivanov type IRGNM. [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 paper we analyze a second order method for regularizing nonlinear inverse problems, which comes from adapting Halley's method to the ill-posed situation. The method is related to the iteratively regularized Gauss-Newton method and enhanced by the use of second derivatives of the forward operator. For parameter identification problems in PDEs, these second derivatives often require not much additional computational cost as compared to first order derivatives. We prove convergence and convergence rates under certain conditions on the forward operator in Hilbert spaces. Numerical tests for a problem of identifying a distributed coefficient in an elliptic PDE illustrate the performance of the method. [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]

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]

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]

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]

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]