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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 models arising in simulations of fluid flows. Spectral stochastic methods are probabilistic in nature, and are consequently rooted in the 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.

Mathematics. --- Computational Science and Engineering. --- Fluid- and Aerodynamics. --- Numerical and Computational Physics. --- Partial Differential Equations. --- Discrete Mathematics in Computer Science. --- Computational complexity. --- Differential equations, partial. --- Computer science. --- Mathématiques --- Complexité de calcul (Informatique) --- Informatique --- Fluid dynamics --- Mathematical models. --- Fluid dynamics -- Mathematical models. --- Engineering & Applied Sciences --- Civil & Environmental Engineering --- Mathematics --- Physical Sciences & Mathematics --- Applied Mathematics --- Civil Engineering --- Mathematics - General --- Mathematical models --- Spectral theory (Mathematics) --- Physics. --- Computer mathematics. --- Fluids. --- Spectroscopy. --- Microscopy. --- Spectroscopy and Microscopy. --- Computational Mathematics and Numerical Analysis. --- Mathematical Modeling and Industrial Mathematics. --- Analysis, Microscopic --- Light microscopy --- Micrographic analysis --- Microscope and microscopy --- Microscopic analysis --- Optical microscopy --- Optics --- Analysis, Spectrum --- Spectra --- Spectrochemical analysis --- Spectrochemistry --- Spectroscopy --- Chemistry, Analytic --- Interferometry --- Radiation --- Wave-motion, Theory of --- Absorption spectra --- Light --- Spectroscope --- Hydraulics --- Mechanics --- Physics --- Hydrostatics --- Permeability --- Models, Mathematical --- Simulation methods --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Qualitative --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Monograph