Absorbing Boundary Conditions for Solving N-Dimensional Stationary Schrödinger Equations with Unbounded Potentials and Nonlinearities

International audienceWe propose a hierarchy of novel absorbing boundary conditions for the one-dimensional stationary Schrö\-din\-ger equation with general (linear and nonlinear) potential. The accuracy of the new absorbing boundary conditions is investigated numerically for the computation of energies and ground-states for linear and nonlinear Schrödinger equations. It turns out that these absorbing boundary conditions and their variants lead to a higher accuracy than the usual Dirichlet boundary condition. Finally, we give the extension of these ABCs to $N$-dimensional stationary Schrödinger equations

Spectrum analysis of LTI continuous-time systems with constant delays: A literature overview of some recent results

In recent decades, increasingly intensive research attention has been given to dynamical systems containing delays and those affected by the after-effect phenomenon. Such research covers a wide range of human activities and the solutions of related engineering problems often require interdisciplinary cooperation. The knowledge of the spectrum of these so-called time-delay systems (TDSs) is very crucial for the analysis of their dynamical properties, especially stability, periodicity, and dumping effect. A great volume of mathematical methods and techniques to analyze the spectrum of the TDSs have been developed and further applied in the most recent times. Although a broad family of nonlinear, stochastic, sampled-data, time-variant or time-varying-delay systems has been considered, the study of the most fundamental continuous linear time-invariant (LTI) TDSs with fixed delays is still the dominant research direction with ever-increasing new results and novel applications. This paper is primarily aimed at a (systematic) literature overview of recent (mostly published between 2013 to 2017) advances regarding the spectrum analysis of the LTI-TDSs. Specifically, a total of 137 collected articles-which are most closely related to the research area-are eventually reviewed. There are two main objectives of this review paper: First, to provide the reader with a detailed literature survey on the selected recent results on the topic and Second, to suggest possible future research directions to be tackled by scientists and engineers in the field. © 2013 IEEE.MSMT-7778/2014, FEDER, European Regional Development Fund; LO1303, FEDER, European Regional Development Fund; CZ.1.05/2.1.00/19.0376, FEDER, European Regional Development FundEuropean Regional Development Fund through the Project CEBIA-Tech Instrumentation [CZ.1.05/2.1.00/19.0376]; National Sustainability Program Project [LO1303 (MSMT-7778/2014)

On Matrix Nearness Problems: Distance to Delocalization

This paper introduces two new matrix nearness problems that are intended to generalize the distance to instability and the distance to stability. They are named the distance to delocalization and the distance to localization due to their applicability in analyzing the robustness of eigenvalues with respect to arbitrary localization sets (domains) in the complex plane. For the open left-half plane or the unit circle, the distance to the nearest unstable/stable matrix is obtained as a special case. Then, following the theoretical framework of Hermitian functions and the Lyapunov-type localization approach, we present a new Newton-type algorithm for the distance to delocalization (D2D) and study its implementations using both an explicit and an implicit computation of the desired singular values. Since our investigations are motivated by several practical applications, we will illustrate our approach on some of them. Furthermore, in the special case when the distance to delocalization becomes the distance to instability, we will validate our algorithms against the state of the art computational method

Non-Normality In Scalar Delay Differential Equations

Thesis (M.S.) University of Alaska Fairbanks, 2006Analysis of stability for delay differential equations (DDEs) is a tool in a variety of fields such as nonlinear dynamics in physics, biology, and chemistry, engineering and pure mathematics. Stability analysis is based primarily on the eigenvalues of a discretized system. Situations exist in which practical and numerical results may not match expected stability inferred from such approaches. The reasons and mechanisms for this behavior can be related to the eigenvectors associated with the eigenvalues. When the operator associated to a linear (or linearized) DDE is significantly non-normal, the stability analysis must be adapted as demonstrated here. Example DDEs are shown to have solutions which exhibit transient growth not accounted for by eigenvalues alone. Pseudospectra are computed and related to transient growth

Das Spektrum zeitverzögerter Differentialgleichungen: numerische Methoden, Stabilität und Störungstheorie

Three types of problems related to delay-differential equations (DDEs) are treated in this thesis. We first consider the problem of numerically computing the eigenvalues of a DDE. Here, we present an application of a projection method for nonlinear eigenvalue problems (NLEPs). We compare this projection method with other methods, suggested in the literature, and used in software packages. The projection method is computationally superior to all of the other tested method for the presented large-scale examples. We give interpretations of methods based on discretizations in terms of rational approximations. Some notes regarding a special case where the spectrum can be explicitly expressed with a formula containing a matrix version of the are Lambert W function are presented. We clarify its range of applicability, and, by counter-example, show that it does not hold in general. The second part of this thesis is related to exact stability conditions of the DDE. All those combinations of the delays such that there is a purely imaginary eigenvalue (called critical delays) are parameterized. In general, an evaluation of the parameterization map consists of solving a quadratic eigenvalue problem of squared dimension. We show how the computational cost for one evaluation of the map can be reduced by exploiting a relation to a Lyapunov equation. The third and last part of this thesis is about generalizations of perturbation results for NLEPs. A sensitivity formula for the movement of the eigenvalues extends to NLEPs. We introduce a fixed point form for the NLEP, and show that some methods in the literature can be interpreted as set-valued fixed point iterations for which asymptotic convergence can be established. We also show how the Bauer-Fike theorem can be generalized to the NLEP under special conditions.In dieser Arbeit werden drei verschiedene Problemklassen im Bezug zu delay-differential equations (DDEs) behandelt. Als erstes gehen wir auf die Berechnung der Eigenwerte von DDEs ein. In dieser Arbeit wenden wir eine Projektionsmethode für nichtlineare Eigenwertprobleme (NLEPe) an. Wir vergleichen diese mit anderen bereits bekannten Verfahren, wobei die hier vorgestellte Methode bedeutend bessere numerische Eigenschaften für die verwendeten Beispiele hat. Zusätzlich treffen wir Aussagen über Diskretisierungsmethoden zur rationalen Approximation. Desweiteren betrachten wir einen Spezialfall, bei welchem das Spektrum explizit mit Hilfe einer Matrix-Version der Lambert W-Funktion dargestellt werden kann. Für diese Formel bestimmen wir einen möglichen Anwendungsbereich. Im zweiten Teil der Arbeit werden exakte Stabilitätsbedingungen von DDEs betrachtet. Die Menge der Delays, für welche die DDE einen imaginären Eigenwert hat (sogenannte kritische Delays), wird parameterisiert. Im Allgemeinen ist zur Auswertung der Parametrisierungsabbildung das Lösen eines quadratischen Eigenwertproblems nötig, dessen Größe dem Quadrat der Dimension der DDE entspricht. Wir zeigen wie der Rechenaufwand durch Ausnutzung einer Lyapunov-Gleichung reduziert werden kann. Der letzte Teil dieser Arbeit befasst sich mit der Verallgemeinerung der Störungstheorie auf NLEPe. Unter anderem lässt sich eine Sensitivitätsformel auf NLEPe erweitern. Desweiteren wird eine Fixpunktform für NLEPe vorgestellt, und gezeigt dass einige Methoden aus der Literatur als mengenwertige Fixpunktiterationen dargestellt werden können, für welche wir asymptotische Konvergenz feststellen. Wir zeigen zusätzlich, dass das Bauer-Fike Theorem unter bestimmten Bedingungen auf NLEPe verallgemeinert werden kann