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An Introduction to Computational Stochastic PDEs

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.

Table of Contents

  1. Cover
  2. Half-title page
  3. Series page
  4. Title
  5. Copyright
  6. Contents
  7. Preface
  8. PART ONE DETERMINISTIC DIFFERENTIAL EQUATIONS
    1. 1 Linear Analysis
      1. 1.1 Banach spaces Cr and Lp
      2. 1.2 Hilbert spaces L2 and Hr
      3. 1.3 Linear operators and spectral theory
      4. 1.4 Fourier analysis
      5. 1.5 Notes
      6. Exercises
    2. 2 Galerkin Approximation and Finite Elements
      1. 2.1 Two-point boundary-value problems
      2. 2.2 Variational formulation of elliptic PDEs
      3. 2.3 The Galerkin finite element method for elliptic PDEs
      4. 2.4 Notes
      5. Exercises
    3. 3 Time-dependent Differential Equations
      1. 3.1 Initial-value problems for ODEs
      2. 3.2 Semigroups of linear operators
      3. 3.3 Semilinear evolution equations
      4. 3.4 Method of lines and finite differences for semilinear PDEs
      5. 3.5 Galerkin methods for semilinear PDEs
      6. 3.6 Finite elements for reaction–diffusion equations
      7. 3.7 Non-smooth error analysis
      8. 3.8 Notes
      9. Exercises
  9. PART TWO STOCHASTIC PROCESSES AND RANDOM FIELDS
    1. 4 Probability Theory
      1. 4.1 Probability spaces and random variables
      2. 4.2 Least-squares approximation and conditional expectation
      3. 4.3 Convergence of random variables
      4. 4.4 Random number generation
      5. 4.5 Notes
      6. Exercises
    2. 5 Stochastic Processes
      1. 5.1 Introduction and Brownian motion
      2. 5.2 Gaussian processes and the covariance function
      3. 5.3 Brownian bridge, fractional Brownian motion, and white noise
      4. 5.4 The Karhunen-Loève expansion
      5. 5.5 Regularity of stochastic processes
      6. 5.6 Notes
      7. Exercises
    3. 6 Stationary Gaussian Processes
      1. 6.1 Real-valued stationary processes
      2. 6.2 Complex-valued random variables and stochastic processes
      3. 6.3 Stochastic integrals
      4. 6.4 Sampling by quadrature
      5. 6.5 Sampling by circulant embedding
      6. 6.6 Notes
      7. Exercises
    4. 7 Random Fields
      1. 7.1 Second-order random fields
      2. 7.2 Circulant embedding in two dimensions
      3. 7.3 Turning bands method
      4. 7.4 Karhunen-Loève expansion of random fields
      5. 7.5 Sample path continuity for Gaussian random fields
      6. 7.6 Notes
      7. Exercises
  10. PART THREE STOCHASTIC DIFFERENTIAL EQUATIONS
    1. 8 Stochastic Ordinary Differential Equations
      1. 8.1 Examples of SODEs
      2. 8.2 Itô integral
      3. 8.3 Itô SODEs
      4. 8.4 Numerical methods for Itô SODEs
      5. 8.5 Strong approximation
      6. 8.6 Weak approximation
      7. 8.7 Stratonovich integrals and SODEs
      8. 8.8 Notes
      9. Exercises
    2. 9 Elliptic PDEs with Random Data
      1. 9.1 Variational formulation on D
      2. 9.2 Monte Carlo FEM
      3. 9.3 Variational formulation on D × Ω
      4. 9.4 Variational formulation on D × Γ
      5. 9.5 Stochastic Galerkin FEM on D × Γ
      6. 9.6 Stochastic collocation FEM on D × Γ
      7. 9.7 Notes
      8. Exercises
    3. 10 Semilinear Stochastic PDEs
      1. 10.1 Examples of semilinear SPDEs
      2. 10.2 Q-Wiener process
      3. 10.3 Itô stochastic integrals
      4. 10.4 Semilinear evolution equations in a Hilbert space
      5. 10.5 Finite difference method
      6. 10.6 Galerkin and semi-implicit Euler approximation
      7. 10.7 Spectral Galerkin method
      8. 10.8 Galerkin finite element method
      9. 10.9 Notes
      10. Exercises
  11. Appendix A
  12. References
  13. Index