Desmond J. Higham

An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations

A practical and accessible introduction to numerical methods for stochastic differential equations is given. The reader is assumed to be familiar with Eulers method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable; however, no knowledge of advanced probability theory or stochastic processes is assumed. The article is built around

Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations

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Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise

MATLAB programs, and the topics covered include stochastic integration, the Euler--Maruyama method, Milsteins method, strong and weak convergence, linear stability, and the stochastic chain rule.

Connectivity-based parcellation of human cortex using diffusion MRI: Establishing reproducibility, validity and observer independence in BA 44/45 and SMA/pre-SMA.

Traditional finite-time convergence theory for numerical methods applied to stochastic differential equations (SDEs) requires a global Lipschitz assumption on the drift and diffusion coefficients. In practice, many important SDE models satisfy only a local Lipschitz property and, since Brownian paths can make arbitrarily large excursions, the global Lipschitz-based theory is not directly relevant. In this work we prove strong convergence results under less restrictive conditions. First, we give a convergence result for Euler--Maruyama requiring only that the SDE is locally Lipschitz and that the pth moments of the exact and numerical solution are bounded for some p >2. As an application of this general theory we show that an implicit variant of Euler--Maruyama converges if the diffusion coefficient is globally Lipschitz, but the drift coefficient satisfies only a one-sided Lipschitz condition; this is achieved by showing that the implicit method has bounded moments and may be viewed as an Euler--Maruyama approximation to a perturbed SDE of the same form. Second, we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial.

Modeling and Simulating Chemical Reactions

The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces, such as that expounded by Meyn-Tweedie. Application of these Markov chain results leads to straightforward proofs of geometric ergodicity for a variety of SDEs, including problems with degenerate noise and for problems with locally Lipschitz vector fields. Applications where this theory can be usefully applied include damped-driven Hamiltonian problems (the Langevin equation), the Lorenz equation with degenerate noise and gradient systems. The same Markov chain theory is then used to study time-discrete approximations of these SDEs. The two primary ingredients for ergodicity are a minorization condition and a Lyapunov condition. It is shown that the minorization condition is robust under approximation. For globally Lipschitz vector fields this is also true of the Lyapunov condition. However in the locally Lipschitz case the Lyapunov condition fails for explicit methods such as Euler-Maruyama; for pathwise approximations it is, in general, only inherited by specially constructed implicit discretizations. Examples of such discretization based on backward Euler methods are given, and approximation of the Langevin equation studied in some detail.

Numerical methods for nonlinear stochastic differential equations with jumps

The identification of specialized, functional regions of the human cortex is a vital precondition for neuroscience and clinical neurosurgery. Functional imaging modalities are used for their delineation in living subjects, but these methods rely on subject cooperation, and many regions of the human brain cannot be activated specifically. Diffusion tractography is a novel tool to identify such areas in the human brain, utilizing underlying white matter pathways to separate regions of differing specialization. We explore the reproducibility, generalizability and validity of diffusion tractography-based localization in four functional areas across subjects, timepoints and scanners, and validate findings against fMRI and post-mortem cytoarchitectonic data. With reproducibility across modalities, clustering methods, scanners, timepoints, and subjects in the order of 80-90%, we conclude that diffusion tractography represents a useful and objective tool for parcellation of the human cortex into functional regions, enabling studies into individual functional anatomy even when there are no specific activation paradigms available.

Network Properties Revealed through Matrix Functions

Many students are familiar with the idea of modeling chemical reactions in terms of ordinary differential equations. However, these deterministic reaction rate equations are really a certain large-scale limit of a sequence of finer-scale probabilistic models. In studying this hierarchy of models, students can be exposed to a range of modern ideas in applied and computational mathematics. This article introduces some of the basic concepts in an accessible manner and points to some challenges that currently occupy researchers in this area. Short, downloadable MATLAB codes are listed and described.

Geometric De-noising of Protein-Protein Interaction Networks

We present and analyse two implicit methods for Ito stochastic differential equations (SDEs) with Poisson-driven jumps. The first method, SSBE, is a split-step extension of the backward Euler method. The second method, CSSBE, arises from the introduction of a compensated, martingale, form of the Poisson process. We show that both methods are amenable to rigorous analysis when a one-sided Lipschitz condition, rather than a more restrictive global Lipschitz condition, holds for the drift. Our analysis covers strong convergence and nonlinear stability. We prove that both methods give strong convergence when the drift coefficient is one-sided Lipschitz and the diffusion and jump coefficients are globally Lipschitz. On the way to proving these results, we show that a compensated form of the Euler–Maruyama method converges strongly when the SDE coefficients satisfy a local Lipschitz condition and the pth moment of the exact and numerical solution are bounded for some p>2. Under our assumptions, both SSBE and CSSBE give well-defined, unique solutions for sufficiently small stepsizes, and SSBE has the advantage that the restriction is independent of the jump intensity. We also study the ability of the methods to reproduce exponential mean-square stability in the case where the drift has a negative one-sided Lipschitz constant. This work extends the deterministic nonlinear stability theory in numerical analysis. We find that SSBE preserves stability under a stepsize constraint that is independent of the initial data. CSSBE satisfies an even stronger condition, and gives a generalization of B-stability. Finally, we specialize to a linear test problem and show that CSSBE has a natural extension of deterministic A-stability. The difference in stability properties of the SSBE and CSSBE methods emphasizes that the addition of a jump term has a significant effect that cannot be deduced directly from the non-jump literature.

Almost Sure and Moment Exponential Stability in the Numerical Simulation of Stochastic Differential Equations

The emerging field of network science deals with the tasks of modeling, comparing, and summarizing large data sets that describe complex interactions. Because pairwise affinity data can be stored in a two-dimensional array, graph theory and applied linear algebra provide extremely useful tools. Here, we focus on the general concepts of centrality, communicability, and betweenness, each of which quantifies important features in a network. Some recent work in the mathematical physics literature has shown that the exponential of a networks adjacency matrix can be used as the basis for defining and computing specific versions of these measures. We introduce here a general class of measures based on matrix functions, and show that a particular case involving a matrix resolvent arises naturally from graph-theoretic arguments. We also point out connections between these measures and the quantities typically computed when spectral methods are used for data mining tasks such as clustering and ordering. We finish with computational examples showing the new matrix resolvent version applied to real networks.

Backward error and condition of structured linear systems

Understanding complex networks of protein-protein interactions (PPIs) is one of the foremost challenges of the post-genomic era. Due to the recent advances in experimental bio-technology, including yeast-2-hybrid (Y2H), tandem affinity purification (TAP) and other high-throughput methods for protein-protein interaction (PPI) detection, huge amounts of PPI network data are becoming available. Of major concern, however, are the levels of noise and incompleteness. For example, for Y2H screens, it is thought that the false positive rate could be as high as 64%, and the false negative rate may range from 43% to 71%. TAP experiments are believed to have comparable levels of noise. We present a novel technique to assess the confidence levels of interactions in PPI networks obtained from experimental studies. We use it for predicting new interactions and thus for guiding future biological experiments. This technique is the first to utilize currently the best fitting network model for PPI networks, geometric graphs. Our approach achieves specificity of 85% and sensitivity of 90%. We use it to assign confidence scores to physical protein-protein interactions in the human PPI network downloaded from BioGRID. Using our approach, we predict 251 interactions in the human PPI network, a statistically significant fraction of which correspond to protein pairs sharing common GO terms. Moreover, we validate a statistically significant portion of our predicted interactions in the HPRD database and the newer release of BioGRID. The data and Matlab code implementing the methods are freely available from the web site: http://www.kuchaev.com/Denoising.