Über die Autorin bzw. den Autor

Dongbin Xiu is associate professor of mathematics at Purdue University.

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"Short and comprehensive, this book is appropriate for novices of polynomial chaos. Many diverse fields are adopting this method, and this book can be used for first-year graduate studies as well as senior undergraduate courses. The book includes important new developments, such as non-Gaussian processes and stochastic collocation methods."--George Karniadakis, Brown University

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"Short and comprehensive, this book is appropriate for novices of polynomial chaos. Many diverse fields are adopting this method, and this book can be used for first-year graduate studies as well as senior undergraduate courses. The book includes important new developments, such as non-Gaussian processes and stochastic collocation methods."--George Karniadakis, Brown University

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Numerical Methods for Stochastic Computations

PRINCETON UNIVERSITY PRESS

Contents

Preface..................................................................................xiChapter 1 Introduction...................................................................1Chapter 2 Basic Concepts of Probability Theory...........................................9Chapter 3 Survey of Orthogonal Polynomials and Approximation Theory......................25Chapter 4 Formulation of Stochastic Systems..............................................44Chapter 5 Generalized Polynomial Chaos...................................................57Chapter 6 Stochastic Galerkin Method.....................................................68Chapter 7 Stochastic Collocation Method..................................................78Chapter 8 Miscellaneous Topics and Applications..........................................89Appendix A Some Important Orthogonal Polynomials in the Askey Scheme.....................105Appendix B The Truncated Gaussian Model G([alpha], [beta])...............................113References...............................................................................117Index....................................................................................127

Chapter One

The goal of this chapter is to introduce the idea behind stochastic computing in the context of uncertainty quantification (UQ). Without using extensive discussions (of which there are many), we will use a simple example of a viscous Burgers' equation to illustrate the impact of input uncertainty on the behavior of a physical system and the need to incorporate uncertainty from the beginning of the simulation and not as an afterthought.

1.1 STOCHASTIC MODELING AND UNCERTAINTY QUANTIFICATION

Scientific computing has become the main tool in many fields for understanding the physics of complex systems when experimental studies can be lengthy, expensive, inflexible, and difficulty to repeat. The ultimate goal of numerical simulations is to predict physical events or the behaviors of engineered systems. To this end, extensive efforts have been devoted to the development of efficient algorithms whose numerical errors are under control and understood. This has been the primary goal of numerical analysis, which remains an active research branch. What has been considered much less in classical numerical analysis is understanding the impact of errors, or uncertainty, in data such as parameter values and initial and boundary conditions.

The goal of UQ is to investigate the impact of such errors in data and subsequently to provide more reliable predictions for practical problems. This topic has received an increasing amount of attention in past years, especially in the context of complex systems where mathematical models can serve only as simplified and reduced representations of the true physics. Although many models have been successful in revealing quantitative connections between predictions and observations, their usage is constrained by our ability to assign accurate numerical values to various parameters in the governing equations. Uncertainty represents such variability in data and is ubiquitous because of our incomplete knowledge of the underlying physics and/or inevitable measurement errors. Hence in order to fully understand simulation results and subsequently to predict the true physics, it is imperative to incorporate uncertainty from the beginning of the simulations and not as an afterthought.

1.1.1 Burgers' Equation: An Illustrative Example

Let us consider a viscous Burgers' equation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where u is the solution field and v > 0 is the viscosity. This is a well-known nonlinear partial differential equation (PDE) for which extensive results exist. The presence of viscosity smooths out the shock discontinuity that would develop otherwise. Thus, the solution has a transition layer, which is a region of rapid variation and extends over a distance of O(v) as v [down arrow] 0. The location of the transition layer z, defined as the zero of the solution profile u(t, z) = 0, is at zero when the solution reaches steady state. If a small amount of (positive) uncertainty exists in the value of the left boundary condition (possibly due to some bias measurement or estimation errors), i.e., u(-1) = 1 + [delta], where 0 < [delta] << 1, then the location of the transition can change significantly. For example, if d is a uniformly distributed random variable in the range of (0, 0.1), then the average steady-state solution with v = 0.05 is the solid line in figure 1.1. It is clear that a small uncertainty of 10 percent can cause significant changes in the final steady-state solution whose average location is approximately at z [approximately equal to] 0.8, resulting in a O(1) difference from the solution with an idealized boundary condition containing no uncertainty. (Details of the computations can be found in [123].)

The Burgers' equation example demonstrates that for some problems, especially nonlinear ones, a small uncertainty in data may cause nonnegligible changes in the system output. Such changes cannot be captured by increasing resolution of the classical numerical algorithms if the uncertainty is not incorporated at the beginning of the computations.

1.1.2 Overview of Techniques

The importance of understanding uncertainty has been realized by many for a long time in disciplines such as civil engineering, hydrology, control, etc. Consequently many methods have been devised to tackle this issue. Because of the "uncertain" nature of the uncertainty, the most dominant approach is to treat data uncertainty as random variables or random processes and recast the original deterministic systems as stochastic systems.

We remark that these types of stochastic systems are different from classical stochastic differential equations (SDEs) where the random inputs are idealized processes such as Wiener processes, Poisson processes, etc., and tools such as stochastic calculus have been developed extensively and are still under active research. (See, for example, [36, 55, 57, 85].)

1.1.2.1 Monte Carlo- and Sampling-Based Methods

One of the most commonly used methods is Monte Carlo sampling (MCS) or one of its variants. In MCS, one generates (independent) realizations of random inputs based on their prescribed probability distribution. For each realization the data are fixed and the problem becomes deterministic. Upon solving the deterministic realizations of the problem, one collects an ensemble of solutions, i.e., realizations of the random solutions. From this ensemble, statistical information can be extracted, e.g., mean and variance. Although MCS is straightforward to apply as it only requires repetitive executions of deterministic simulations, typically a large number of executions are needed, for the solution statistics converge relatively slowly. For example, the mean value typically converges as 1/[square root of K], where K is the number of realizations (see, for example, [30]). The need for a large number of realizations for accurate results can incur an excessive computational burden, especially for systems that are already computationally intensive in their deterministic settings.

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