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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]

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]

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]

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]

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]

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]