Search

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 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]

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]

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]

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]

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]

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]

• 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: 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]

This paper considers the inverse problem of recovering both the unknown, spatially-dependent conductivity a (x) and the potential q (x) in a parabolic equation from overposed data consisting of the value of solution profiles taken at a later time T. We show both uniqueness results and the convergence of an iteration scheme designed to recover these coefficients. We also allow a more general setting, in particular when the usual time derivative is replaced by one of fractional order and when the potential term is coupled with a known nonlinearity f of the form q (x) f (u). [ABSTRACT FROM AUTHOR]