Spectral methods for uncertainty quantification. With applications to computational fluid dynamics.

*(English)*Zbl 1193.76003
Dordrecht: Springer (ISBN 978-90-481-3519-6/hbk; 978-94-007-3192-9/pbk; 978-90-481-3520-2/ebook). xvi, 536 p. (2010).

This book presents applications of spectral methods to problems of uncertainty propagation and quantification in model-based computations, focusing on the computational and algorithmic features of these methods most useful in dealing with models based on partial differential equations, in particular, with models arising in simulations of fluid flows. Spectral stochastic methods are probabilistic in nature, and are consequently rooted in rich mathematical foundations associated with probability and measure spaces. A brief discussion is provided of only those theoretical aspects needed to set the stage for subsequent applications. These are demonstrated through detailed treatments of elementary problems, as well as in more elaborate examples involving vortex-dominated flows and compressible flows at low Mach numbers. Some recent developments are also outlined in the book, including iterative techniques (such as stochastic multigrids and Newton schemes), intrusive and non-intrusive formalisms, spectral representations using mixed and discontinuous bases, multi-resolution approximations, and adaptive techniques. Readers are assumed to be familiar with elementary methods for numerical solution of time-dependent partial differential equations; prior experience with spectral approximation is helpful but not essential.

Reviewer: Wilhelm Heinrichs (Essen)