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search.filters.applied.f.access: openAccess × Author: Ehrhardt, Matthias × End date: 2009 × Start date: 2000 × Has files: Yes × Type: Text × WGL Institute: WIAS ×

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Now showing 1 - 10 of 15
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    Exact artificial boundary conditions for problems with period structure
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Ehrhardt, Matthias; Zheng, Chunxiong
    Based on the work of Zheng on the artificial boundary condition for the Schrödinger equation with sinusoidal potentials at infinity, an analytical impedance expression is presented for general second order ODE problems with periodic coefficients and its validity is shown to be strongly supported by numerical evidences. This new expression for the kernel of the Dirichlet-to-Neumann mapping of the artificial boundary conditions is then used for computing the bound states of the Schrödinger operator with periodic potentials at infinity. Other potential applications are associated with the exact artificial boundary conditions for some time-dependent problems with periodic structures. As an example, a two-dimensional hyperbolic equation modeling the TM polarization of the electromagnetic field with a periodic dielectric permittivity is considered.
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    A threestepped coordinated level set segmentation method for identifying atherosclerotic plaques on MR-images
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Gloger, Oliver; Ehrhardt, Matthias; Dietrich, Thore; Hellwich, Olaf; Graf, Kristof; Nagel, Eike
    In this work we propose an adapted level set segmentation technique for the recognition of atherosclerotic plaque tissue on magnetic resonance images. The images are 2dimensional crosssectional images and show different profiles from ex-vivo human vessels with high variability in vessel shape. We used a curvature based anisotropic diffusion technique to denoise the magnetic resonance images. The segmentation technique is subdivided into three level set steps. Hence, the result of every phase serves as constructive knowledge for the next level set step. By analyzing and combining carefully all available channel information during the first and second step we are capable to delineate exactly the vessel walls by using and adapting two well-known level set segmentation techniques. The third step controls an enclosing level set which separates the plaque patterns from healthy media tissue. In this step we introduce a local weighting concept to consider intensity information for conspicuous plaque patterns. Furthermore, we propose the introduction of a maximal shrinking distance for the third level set in the vessel wall and compare the results of the local weighting algorithm with and without the concept of the maximal shrinking distance. The incorporation of locally weighted intensity information into the level set method allows the algorithm to automatically distinguish plaque from healthy media tissue. The knowledge of the maximal shrinking distance can improve the segmentation results and enables to delineate tissue areas where plaque is most likely.
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    A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Antoine, Xavier; Arnold, Anton; Besse, Chritophe; Ehrhardt, Matthias; Schädle, Achim
    In this review article we discuss different techniques to solve numerically the time-dependent Schrödinger equation on unbounded domains. We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.
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    Fixed domain transformations and split-step finite difference schemes for nonlinear black-scholes equations for American options
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Ankudinova, Julia; Ehrhardt, Matthias
    Due to transaction costs, illiquid markets, large investors or risks from an unprotected portfolio the assumptions in the classical Black-Scholes model become unrealistic and the model results in strongly or fully nonlinear, possibly degenerate, parabolic diffusion-convection equations, where the stock price, volatility, trend and option price may depend on the time, the stock price or the option price itself. In this chapter we will be concerned with several models from the most relevant class of nonlinear Black-Scholes equations for American options with a volatility depending on different factors, such as the stock price, the time, the option price and its derivatives. We will analytically approach the option price by following the ideas proposed by evcovic and transforming the free boundary problem into a fully nonlinear nonlocal parabolic equation defined on a fixed, but unbounded domain. Finally, we will present the results of a split-step finite difference schemes for various volatility models including the Leland model, the Barles and Soner model and the Risk adjusted pricing methodology model.
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    Fast, stable and accurate method for the Black-Scholes equation of American options
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Ehrhardt, Matthias; Mickens, Ronald E.
    We propose a simple model for the behaviour of long-time investors on stock markets, consisting of three particles, which represent the current price of the stock, and the opinion of the buyers, or sellers resp., about the right trading price. As time evolves both groups of traders update their opinions with respect to the current price. The update speed is controled by a parameter $\gamma$, the price process is described by a geometric Brownian motion. The stability of the market is governed by the difference of the buyers' opinion and the sellers' opinion. We prove that the distance
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    Implementing exact absorbing boundary condition for the linear one-dimensional Schrödinger problem with variable potential by Titchmarsh-Weyl theory
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Ehrhardt, Matthias; Zheng, Chunxiong
    A new approach for simulating the solution of the time-dependent Schrödinger equation with a general variable potential will be proposed. The key idea is to approximate the Titchmarsh-Weyl m-function (exact Dirichlet-to-Neumann operator) by a rational function with respect to a suitable spectral parameter. With the proposed method we can overcome the usual high-frequency restriction for absorbing boundary conditions of general variable potential problems. We end up with a fast computational algorithm for absorbing boundary conditions that are accurate for the full frequency band
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    Fast numerical methods for waves in periodic media
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Ehrhardt, Matthias; Zheng, Chunxiong
    Periodic media problems widely exist in many modern application areas like semiconductor nanostructures (e.g. quantum dots and nanocrystals), semi-conductor superlattices, photonic crystals (PC) structures, meta materials or Bragg gratings of surface plasmon polariton (SPP) waveguides, etc. Often these application problems are modeled by partial differential equations with periodic coefficients and/or periodic geometries. In order to numerically solve these periodic structure problems efficiently one usually confines the spatial domain to a bounded computational domain (i.e. in a neighborhood of the region of physical interest). Hereby, the usual strategy is to introduce so-called artificial boundaries and impose suitable boundary conditions. For wave-like equations, the ideal boundary conditions should not only lead to w ell-posed problems, but also mimic the perfect absorption of waves traveling out of the computational domain through the artificial boundaries ...
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    Discrete transparent boundary conditions for the Schrödinger equation on circular domains
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Arnold, Anton; Ehrhardt, Matthias; Schulte, Maike; Sofronov, Ivan
    We propose transparent boundary conditions (TBCs) for the time-dependent Schrödinger equation on a circular computational domain. First we derive the two-dimensional discrete TBCs in conjunction with a conservative Crank-Nicolson finite difference scheme. The presented discrete initial boundary-value problem is unconditionally stable and completely reflection-free at the boundary. Then, since the discrete TBCs for the Schrödinger equation with a spatially dependent potential include a convolution w.r.t. time with a weakly decaying kernel, we construct approximate discrete TBCs with a kernel having the form of a finite sum of exponentials, which can be efficiently evaluated by recursion. In numerical tests we finally illustrate the accuracy, stability, and efficiency of the proposed method. As a by-product we also present a new formulation of discrete TBCs for the 1D Schrödinger equation, with convolution coefficients that have better decay properties than those from the literature.
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    Comparison of the continuous, semi-discrete and fully-discrete Transparent Boundary Conditions (TBC) for the parabolic wave equation 1. Theory
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Šumichrast, L'ubomír; Ehrhardt, Matthias
    For the simulation of the propagation of optical waves in open wave guiding structures of integrated optics the parabolic approximation of the scalar wave equation is commonly used. This approach is commonly termed the beam propagation method (BPM). It is of paramount importance to have well-performing transparent boundary conditions applied on the boundaries of the finite computational window, to enable the superfluous portion of the propagating wave to radiate away from the wave guiding structure. Three different formulations (continuous, semi-discrete and fully-discrete) of the non-local transparent boundary conditions are described and compared here.
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    Mathematical modeling of channel-porous layer interfaces in PEM fuel cells
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Ehrhardt, Matthias; Fuhrmann, J.; Holzbecher, E.; Linke, A.
    In proton exchange membrane (PEM) fuel cells, the transport of the fuel to the active zones, and the removal of the reaction products are realized using a combination of channels and porous diffusion layers. In order to improve existing mathematical and numerical models of PEM fuel cells, a deeper understanding of the coupling of the flow processes in the channels and diffusion layers is necessary. After discussing different mathematical models for PEM fuel cells, the work will focus on the description of the coupling of the free flow in the channel region with the filtration velocity in the porous diffusion layer as well as interface conditions between them. The difficulty in finding effective coupling conditions at the interface between the channel flow and the membrane lies in the fact that often the orders of the corresponding differential operators are different, e.g., when using stationary (Navier-)Stokes and Darcy's equation. Alternatively, using the Brinkman model for the porous media this difficulty does not occur. We will review different interface conditions, including the well-known Beavers-Joseph-Saffman boundary condition and its recent improvement by Le Bars and Worster.