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.
Abstract. We study numerical methods for dissipative particle dynamics, a system of stochastic differential equations for simulating particles interacting pairwise according to a soft potential at constant temperature where the total momentum is conserved. We introduce splitting methods and examine the behavior of these methods experimentally. The performance of the methods, particularly temperature control, is compared to the modified velocity Verlet method used in many previous papers.
We consider Monte Carlo methods for simulating solutions to the analogue of the Dirichlet boundary-value problem in which the Laplacian is replaced by the fractional Laplacian and boundary conditions are replaced by conditions on the exterior of the domain. Specifically, we consider the analogue of the so-called 'walk-on-spheres' algorithm. In the diffusive setting, this entails sampling the path of Brownian motion as it uniformly exits a sequence of spheres maximally inscribed in the domain. As this algorithm would otherwise never end, it is truncated when the 'walk-on-spheres' comes within ε > 0 of the boundary. In the setting of the fractional Laplacian, the role of Brownian motion is replaced by an isotropic α-stable process with α ∈ (0, 2). A significant difference to the Brownian setting is that the stable processes will exit spheres by a jump rather than hitting their boundary. This difference ensures that disconnected domains may be considered and that, unlike the diffusive setting, the algorithm ends after an almost surely finite number of steps.issued from x ∈ D, then, by the strong law of large numbers,For practical purposes, since it is impossible to take the limit, one truncates the series of estimates for large n and the central limit theorem gives O(1/n) upper bounds on the variance of the n-term sum, which serves as a numerical error estimate. Although forming the fundamental basis of most Monte Carlo methods for diffusive Dirichlettype problems, (1.3) is an inefficient numerical approach. Least of all, this is because the Monte Carlo simulation of u(x) is independent for each x ∈ D. Moreover, it is unclear how exactly to simulate the path of a Brownian motion on its first exit from D, that is to say, the quantity W τ D . This is because of the fractal properties of Brownian motion, making its path difficult to simulate. This introduces additional numerical errors over and above that of Monte Carlo simulation.A method proposed by (Muller, 1956), for the case that D is convex, sub-samples special points along the path of Brownian motion to the boundary of the domain D. The method does not require a complete simulation of its path and takes advantage of the distributional symmetry of Brownian motion. In order to describe the so-called 'walk-on-spheres', we need to first introduce some notation. We may thus set ρ 0 = x for x ∈ D and define r 1 to be the radius of the largest sphere inscribed in D that is centred at x. This sphere we will call S 1 = {y ∈ R d : |y − ρ 0 | = r 1 }. To avoid special cases, we henceforth assume that the surface area of S 1 ∩ ∂D is zero (this excludes, for example, the case that x = 0 and D is a sphere centred at the origin). Now set ρ 1 ∈ D to be a point uniformly distributed on S 1 and note that, given the assumption in the previous sentence, P x (ρ 1 ∈ ∂D) = 0. Construct the remainder of the sequence (ρ n , n ≥ 1) inductively. Given ρ n−1 , we define the radius, r n , of the largest sphere inscribed in D that is centred at ρ n−1 . Calling this sphere S n , we have that ...
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