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Author: Ehrhardt, Matthias × End date: 2023 × 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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    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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    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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    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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    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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    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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    Numerical simulation of waves in periodic structures
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Ehrhardt, Matthias; Han, Houde; Zheng, Chunxiong
    In this work we present a new numerical technique for solving periodic structure problems. This new approach possesses several advantages. First, it allows for a fast evaluation of the Robin-to-Robin operator for periodic array problems. Secondly, this computational method can also be used for bi-periodic structure problems with local defects. Our strategy is an improvement of the recently developed recursive doubling process by Yuan and Lu. In this paper we consider several problems, such as the exterior elliptic problems with strong coercivity, the time-dependent Schrödinger equation and finally the Helmholtz equation with damping.
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    Evaluation of exact boundary mappings for one-dimensional semiinfinite periodic arrays
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Ehrhardt, Matthias; Sun, Jiguang; Zheng, Chunxiong
    Periodic arrays are structures consisting of geometrically identical subdomains, usually called periodic cells. In this paper, by taking the Helmholtz equation as a model, we consider the definition and evaluation of the exact boundary mappings for general one-dimensional semi-infinite periodic arrays for any real wavenumber. The well-posedness of the Helmholtz equation is established via the limiting absorption principle. An algorithm based on the doubling procedure and extrapolation technique is proposed to derive the exact Sommerfeld-to-Sommerfeld boundary mapping. The advantages of this algorithm are the robustness and simplicity of implementation. But it also suffers from the high computational cost and the resonance wave numbers. To overcome these shortcomings, we propose another algorithm based on a conjecture about the asymptotic behaviour of limiting absorption principle solutions. The price we have to pay is the resolution of two generalized eigenvalue problems, but still the overall computational cost is significantly reduced. Numerical evidences show that this algorithm presents theoretically the same results as the first algorithm. Moreover, some quantitative comparisons between these two algorithms are given.
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    Numerical simulation of quantum waveguides
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Arnold, Anton; Ehrhardt, Matthias; Schulte, Maike
    This chapter is a review of the research of the authors from the last decade and focuses on the mathematical analysis of the Schrödinger model for nano-scale semiconductor devices. We discuss transparent boundary conditions (TBCs) for the time-dependent Schrödinger equation on a two dimensional domain. First we derive the two dimensional discrete TBCs in conjunction with a conservative Crank-Nicolson-type finite difference scheme and a compact nine-point scheme. For this difference equations we derive discrete transparent boundary conditions (DTBCs) in order to get highly accurate solutions for open boundary problems. The presented discrete boundary-valued problem is unconditionally stable and completely reflection-free at the boundary. Then, since the DTBCs for the Schrödinger equation include a convolution w.r.t. time with a weakly decaying kernel, we construct approximate DTBCs with a kernel having the form of a finite sum of exponentials, which can be efficiently evaluated by recursion. In several numerical tests we illustrate the perfect absorption of outgoing waves independent of their impact angle at the boundary, the stability, and efficiency of the proposed method. Finally, we apply inhomogeneous DTBCs to the transient simulation of quantum waveguides with a prescribed electron inflow.
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    A model of an electrochemical flow cell with porous layer
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Ehrhardt, Matthias; Fuhrmann, Jürgen; Linke, Alexander
    In this paper we discuss three different mathematical models for fluid-porous interfaces in a simple channel geometry that appears e.g. in thin-layer channel flow cells. Here the difficulties arise from the possibly different orders of the corresponding differential operators in the different domains. A finite volume discretization of this model allows to calculate the limiting current of the H_2 oxidation in a porous electrode with platinum catalyst particles.