Random-field Solutions to Linear Hyperbolic Stochastic Partial Differential Equations with Variable Coefficients

In this article we show the existence of a random-field solution to linear stochastic partial differential equations whose partial differential operator is hyperbolic and has variable coefficients that may depend on the temporal and spatial argument. The main tools for this, pseudo-differential and Fourier integral operators, come from microlocal analysis. The equations that we treat are second-order and higher-order strictly hyperbolic, and second-order weakly hyperbolic with uniformly bounded coefficients in space. For the latter one we show that a stronger assumption on the correlation measure of the random noise might be needed. Moreover, we show that the well-known case of the stochastic wave equation can be embedded into the theory presented in this article.Comment: 40 pages, final version, Stochastic Processes and their Applications (2017

Fast and Efficient Numerical Methods for an Extended Black-Scholes Model

An efficient linear solver plays an important role while solving partial differential equations (PDEs) and partial integro-differential equations (PIDEs) type mathematical models. In most cases, the efficiency depends on the stability and accuracy of the numerical scheme considered. In this article we consider a PIDE that arises in option pricing theory (financial problems) as well as in various scientific modeling and deal with two different topics. In the first part of the article, we study several iterative techniques (preconditioned) for the PIDE model. A wavelet basis and a Fourier sine basis have been used to design various preconditioners to improve the convergence criteria of iterative solvers. We implement a multigrid (MG) iterative method. In fact, we approximate the problem using a finite difference scheme, then implement a few preconditioned Krylov subspace methods as well as a MG method to speed up the computation. Then, in the second part in this study, we analyze the stability and the accuracy of two different one step schemes to approximate the model.Comment: 29 pages; 10 figure

Information Transmission using the Nonlinear Fourier Transform, Part I: Mathematical Tools

The nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models, is a method for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT decorrelates signal degrees-of-freedom in such models, in much the same way that the Fourier transform does for linear systems. In this three-part series of papers, this observation is exploited for data transmission over integrable channels such as optical fibers, where pulse propagation is governed by the nonlinear Schr\"odinger equation. In this transmission scheme, which can be viewed as a nonlinear analogue of orthogonal frequency-division multiplexing commonly used in linear channels, information is encoded in the nonlinear frequencies and their spectral amplitudes. Unlike most other fiber-optic transmission schemes, this technique deals with both dispersion and nonlinearity directly and unconditionally without the need for dispersion or nonlinearity compensation methods. This first paper explains the mathematical tools that underlie the method.Comment: This version contains minor updates of IEEE Transactions on Information Theory, vol. 60, no. 7, pp. 4312--4328, July 201

Basic theorems of distributions and Fourier transforms

Master of ScienceDepartment of MathematicsMarianne KortenDistribution theory is an important tool in studying partial differential equations. Distributions are linear functionals that act on a space of smooth test functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. There are different possible choices for the space of test functions, leading to different spaces of distributions. In this report, we take a look at some basic theory of distributions and their Fourier transforms. And we also solve some typical exercises at the end