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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]

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]