This entry-level graduate textbook is devoted to the numerical methods of physical dynamic systems modeling under stochastic uncertainty. It is focused on the methods based on the generalized polynomial chaos (gPC) methodology.
Roughly speaking, the author deals with partial differential equations with coefficients, boundary and/or initial conditions depending on a finite set \(Z\) of random variables. (When a random process is involved it is approximated by e.g. a finite number of the Karhunen-Loève expansion terms.) The approximate solution to the equation is represented as \( u_N(x,t,Z)=\sum \hat u_i(x,t)\Phi_i(Z) \), where \(\{\Phi_i\}\) is a set of polynomials of degree up to \(N\), orthogonal with respect to the distribution of \(Z\), i.e. \(E\Phi_i(Z)\Phi_k(Z)=0\) if \(i\not=k\).
The nonrandom coefficients \(\hat u_i\) are fitted to the equation by some numerical methods (such as the Galerkin method). Then the stochastic properties of the solution can be analyzed by Monte-Carlo methods with sampling from \(Z\). Another way is to use stochastic collocation methods in which \(u_N(x,t,Z)\) is fitted numerically at points \(Z=Z_j\), \(j=1,2,\dots,K\) and then the obtained solution is interpolated numerically over the domain of \(Z\).
Precision and computational efficiency of these methods are analyzed in the book both analytically and numerically on some (rather simple but instructive) examples. The author discusses also the problems of unknown parameters estimation, prediction by the ensemble Kalman filtering and analysis of uncertainty with random domain.
The book contains a short introduction to Probability and Stochastic Processes, Orthogonal Polynomials and Approximation Theory.