Author: Publisher:
ISBN:
Category :
Languages : en
Pages : 25
Book Description
A Weak Space-time Formulation for the Linear Stochastic Heat Equation
Author: Space-Time Methods
Author: Ulrich LangerPublisher: Walter de Gruyter GmbH & Co KG
ISBN: 3110548488
Category : Mathematics
Languages : en
Pages : 261
Book Description
This volume provides an introduction to modern space-time discretization methods such as finite and boundary elements and isogeometric analysis for time-dependent initial-boundary value problems of parabolic and hyperbolic type. Particular focus is given on stable formulations, error estimates, adaptivity in space and time, efficient solution algorithms, parallelization of the solution pipeline, and applications in science and engineering.
Applied Stochastic Differential Equations
Author: Simo SärkkäPublisher: Cambridge University Press
ISBN: 1316510085
Category : Business & Economics
Languages : en
Pages : 327
Book Description
With this hands-on introduction readers will learn what SDEs are all about and how they should use them in practice.
Numerical Methods for Stochastic Partial Differential Equations with White Noise
Author: Zhongqiang ZhangPublisher: Springer
ISBN: 3319575112
Category : Mathematics
Languages : en
Pages : 391
Book Description
This book covers numerical methods for stochastic partial differential equations with white noise using the framework of Wong-Zakai approximation. The book begins with some motivational and background material in the introductory chapters and is divided into three parts. Part I covers numerical stochastic ordinary differential equations. Here the authors start with numerical methods for SDEs with delay using the Wong-Zakai approximation and finite difference in time. Part II covers temporal white noise. Here the authors consider SPDEs as PDEs driven by white noise, where discretization of white noise (Brownian motion) leads to PDEs with smooth noise, which can then be treated by numerical methods for PDEs. In this part, recursive algorithms based on Wiener chaos expansion and stochastic collocation methods are presented for linear stochastic advection-diffusion-reaction equations. In addition, stochastic Euler equations are exploited as an application of stochastic collocation methods, where a numerical comparison with other integration methods in random space is made. Part III covers spatial white noise. Here the authors discuss numerical methods for nonlinear elliptic equations as well as other equations with additive noise. Numerical methods for SPDEs with multiplicative noise are also discussed using the Wiener chaos expansion method. In addition, some SPDEs driven by non-Gaussian white noise are discussed and some model reduction methods (based on Wick-Malliavin calculus) are presented for generalized polynomial chaos expansion methods. Powerful techniques are provided for solving stochastic partial differential equations. This book can be considered as self-contained. Necessary background knowledge is presented in the appendices. Basic knowledge of probability theory and stochastic calculus is presented in Appendix A. In Appendix B some semi-analytical methods for SPDEs are presented. In Appendix C an introduction to Gauss quadrature is provided. In Appendix D, all the conclusions which are needed for proofs are presented, and in Appendix E a method to compute the convergence rate empirically is included. In addition, the authors provide a thorough review of the topics, both theoretical and computational exercises in the book with practical discussion of the effectiveness of the methods. Supporting Matlab files are made available to help illustrate some of the concepts further. Bibliographic notes are included at the end of each chapter. This book serves as a reference for graduate students and researchers in the mathematical sciences who would like to understand state-of-the-art numerical methods for stochastic partial differential equations with white noise.
A Minicourse on Stochastic Partial Differential Equations
Author: Robert C. DalangPublisher: Springer Science & Business Media
ISBN: 3540859934
Category : Mathematics
Languages : en
Pages : 230
Book Description
This title contains lectures that offer an introduction to modern topics in stochastic partial differential equations and bring together experts whose research is centered on the interface between Gaussian analysis, stochastic analysis, and stochastic PDEs.
On Space-time Quasiconcave Solutions of the Heat Equation
Author: Chuanqiang ChenPublisher:
ISBN: 9781470452438
Category :
Languages : en
Pages :
Book Description
Random Walk and the Heat Equation
Author: Gregory F. LawlerPublisher: American Mathematical Soc.
ISBN: 0821848291
Category : Mathematics
Languages : en
Pages : 170
Book Description
The heat equation can be derived by averaging over a very large number of particles. Traditionally, the resulting PDE is studied as a deterministic equation, an approach that has brought many significant results and a deep understanding of the equation and its solutions. By studying the heat equation and considering the individual random particles, however, one gains further intuition into the problem. While this is now standard for many researchers, this approach is generally not presented at the undergraduate level. In this book, Lawler introduces the heat equations and the closely related notion of harmonic functions from a probabilistic perspective. The theme of the first two chapters of the book is the relationship between random walks and the heat equation. This first chapter discusses the discrete case, random walk and the heat equation on the integer lattice; and the second chapter discusses the continuous case, Brownian motion and the usual heat equation. Relationships are shown between the two. For example, solving the heat equation in the discrete setting becomes a problem of diagonalization of symmetric matrices, which becomes a problem in Fourier series in the continuous case. Random walk and Brownian motion are introduced and developed from first principles. The latter two chapters discuss different topics: martingales and fractal dimension, with the chapters tied together by one example, a random Cantor set. The idea of this book is to merge probabilistic and deterministic approaches to heat flow. It is also intended as a bridge from undergraduate analysis to graduate and research perspectives. The book is suitable for advanced undergraduates, particularly those considering graduate work in mathematics or related areas.
Topics in Stochastic Analysis and Riemannian Foliations
Author: Qi FengPublisher:
ISBN:
Category : Electronic dissertations
Languages : en
Pages : 192
Book Description
This dissertation contains three research directions. In the first direction, we use rough paths theory to study stochastic differential equations and SPDEs. We first prove convergence and the rate of convergence of the Taylor expansion for the solutions of differential equations driven by $p$-rough paths with $p>2$. The main results are the Castell expansion and the tail estimate for the remainder terms. Our results apply to differential equations driven by continuous centered Gaussian process with finite $2D~\rho-$variation and fBm with $H>1/4$. We then give a new and simple method to get a priori bounds on rough partial differential equations. The technique is based on a weak formulation of the equation and a rough version of Gronwall's lemma. The method is presented on a linear stochastic heat equation. In the second direction, we study stochastic analysis on the horizontal paths space of totally geodesic Riemannian foliations. We first develop Malliavin calculus on the horizontal path space and then prove the quasi-invariance of horizontal Wiener measure. We further prove a Log-Sobolev inequality, the improved Log-Sobolev inequality and the equivalence of two-sided uniform Ricci curvature bounds to functional inequalities. We also obtain concentration and tail estimates. In the third direction, we study Ricci flow on totally geodesic Riemannian foliations. Under the transverse Ricci flow, we prove two types of differential Harnack inequalities for the positive solutions of the heat equation. We also get a time dependent generalized curvature dimension inequality. As consequences, we get parabolic Harnack inequalities and heat kernel upper bounds.
Singular Random Dynamics
Author: Massimiliano GubinelliPublisher: Springer Nature
ISBN: 3030295451
Category : Mathematics
Languages : en
Pages : 316
Book Description
Written by leading experts in an emerging field, this book offers a unique view of the theory of stochastic partial differential equations, with lectures on the stationary KPZ equation, fully nonlinear SPDEs, and random data wave equations. This subject has recently attracted a great deal of attention, partly as a consequence of Martin Hairer's contributions and in particular his creation of a theory of regularity structures for SPDEs, for which he was awarded the Fields Medal in 2014. The text comprises three lectures covering: the theory of stochastic Hamilton–Jacobi equations, one of the most intriguing and rich new chapters of this subject; singular SPDEs, which are at the cutting edge of innovation in the field following the breakthroughs of regularity structures and related theories, with the KPZ equation as a central example; and the study of dispersive equations with random initial conditions, which gives new insights into classical problems and at the same time provides a surprising parallel to the theory of singular SPDEs, viewed from many different perspectives. These notes are aimed at graduate students and researchers who want to familiarize themselves with this new field, which lies at the interface between analysis and probability.
An Introduction to Computational Stochastic PDEs
Author: Gabriel J. LordPublisher: Cambridge University Press
ISBN: 1139915770
Category : Mathematics
Languages : en
Pages : 516
Book Description
This book gives a comprehensive introduction to numerical methods and analysis of stochastic processes, random fields and stochastic differential equations, and offers graduate students and researchers powerful tools for understanding uncertainty quantification for risk analysis. Coverage includes traditional stochastic ODEs with white noise forcing, strong and weak approximation, and the multi-level Monte Carlo method. Later chapters apply the theory of random fields to the numerical solution of elliptic PDEs with correlated random data, discuss the Monte Carlo method, and introduce stochastic Galerkin finite-element methods. Finally, stochastic parabolic PDEs are developed. Assuming little previous exposure to probability and statistics, theory is developed in tandem with state-of-the-art computational methods through worked examples, exercises, theorems and proofs. The set of MATLAB® codes included (and downloadable) allows readers to perform computations themselves and solve the test problems discussed. Practical examples are drawn from finance, mathematical biology, neuroscience, fluid flow modelling and materials science.