On trajectories of complex-valued interior transmission eigenvalues

This paper investigates properties of complex-valued eigenvalue trajectories for the interior transmission problem parametrized by the index of refraction for homogeneous media. Our theoretical analysis for the unit disk shows that the only intersection points with the real axis, as well as the unique trajectorial limit points as the refractive index tends to infinity, are Dirichlet eigenvalues of the Laplacian. Complementing numerical experiments even give rise to an underlying one-to-one correspondence between Dirichlet eigenvalues of the Laplacian and complex-valued interior transmission eigenvalue trajectories. We also examine other scatterers than the disk for which similar numerical observations can be made. We summarize our results in a conjecture for general simply-connected scatterers

On the iterative regularization of non-linear illposed problems in L∞

Parameter identification tasks for partial differential equations are non-linear illposed problems where the parameters are typically assumed to be in $L^{\infty}$ . This Banach space is non-smooth, non-reflexive and non-separable and requires therefore a more sophisticated regularization treatment than the more regular $L^p$-spaces with $1 < p < \infty$. We propose a novel inexact Newton-like iterative solver where the Newton update is an approximate minimizer of a smooth Tikhonov functional over a finite-dimensional space whose dimension increases as the iteration progresses. In this way, all iterates stay bounded and the regularizer, delivered by a discrepancy principle, converges weakly-$\star$? to a solution when the noise level decreases to zero. Our theoretical results are demonstrated by numerical experiments based on the acoustic wave equation in one spatial dimension. This model problem satisfies all assumptions from our theoretical analysis

On the iterative regularization of non-linear illposed problems in L∞L^{\infty }

Parameter identification tasks for partial differential equations are non-linear illposed problems where the parameters are typically assumed to be in L∞. This Banach space is non-smooth, non-reflexive and non-separable and requires therefore a more sophisticated regularization treatment than the more regular Lp-spaces with 1<p<∞. We propose a novel inexact Newton-like iterative solver where the Newton update is an approximate minimizer of a smoothed Tikhonov functional over a finite-dimensional space whose dimension increases as the iteration progresses. In this way, all iterates stay bounded in L∞ and the regularizer, delivered by a discrepancy principle, converges weakly-⋆ to a solution when the noise level decreases to zero. Our theoretical results are demonstrated by numerical experiments based on the acoustic wave equation in one spatial dimension. This model problem satisfies all assumptions from our theoretical analysis

Die Fundamentallösungsmethode zur Berechnung innerer Transmissionseigenwerte

This thesis deals with a novel approach for analyzing and computing interior transmission eigenvalues of (piecewise) homogeneous media in two dimensions. It is based on approximating boundary data of respective eigenfunctions by the method of fundamental solutions. However, since a straightforward implementation would solely exploit ill-conditioned matrices and thus evoke spurious results, a stabilization scheme is incorporated. The combined method is then studied with a distinction between isotropic and anisotropic materials, and complemented by novel approximation theory each. Numerical validations complete the investigations for different wave type scenariosDiese Arbeit befasst sich mit einem neuen Zugang für die Analyse und Berechnung innerer Transmissionseigenwerte (stückweise) homogener Medien in zwei Dimensionen. Dieser basiert auf einer Randdatenapproximation zugehöriger Eigenfunktionen mithilfe der Fundamentallösungsmethode. Da eine einfache Implementierung lediglich schlechtkonditionierte Matrizen ausnutzt und daher verfälschte Ergebnisse liefert, wird zusätzlich ein Stabilisierungsschema integriert. Die kombinierte Methode wird sowohl für isotrope als auch anisotrope Materialen untersucht und in beiden Fällen durch entsprechende Theorien komplementiert. Numerische Ergebnisse runden die Arbeit für verschiedene Wellentypen ab

The method of fundamental solutions for computing interior transmission eigenvalues

This thesis deals with a novel approach for analyzing and computing interior transmission eigenvalues of (piecewise) homogeneous media in two dimensions. It is based on approximating boundary data of respective eigenfunctions by the method of fundamental solutions. However, since a straightforward implementation would solely exploit ill-conditioned matrices and thus evoke spurious results, a stabilization scheme is incorporated. The combined method is then studied with a distinction between isotropic and anisotropic materials, and complemented by novel approximation theory each. Numerical validations complete the investigations for different wave type scenarios

Computing interior transmission eigenvalues for homogeneous and anisotropic media

The method of fundamental solutions is investigated in a stabilized version for the computation of interior transmission eigenvalues in two dimensions for homogeneous and anisotropic media without voids. This approach has already proven to be very competitive in practice for the isotropic framework among regular scattering shapes and keeps predominating through its simplicity as being mesh- and integration free. We give a new approximation analysis, present various numerical results and show that the eigenvalue spectrum for isotropic scatterers is generally different from the corresponding anisotropic borderline cases