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Polynomial chaos
About: Polynomial chaos is a research topic. Over the lifetime, 3700 publications have been published within this topic receiving 86289 citations.
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TL;DR: This work presents a reformulation of the Bayesian approach to inverse problems, that seeks to accelerate Bayesian inference by using polynomial chaos expansions to represent random variables, and evaluates the utility of this technique on a transient diffusion problem arising in contaminant source inversion.
484 citations
TL;DR: The method converges in probability as a consequence of sparsity and a concentration of measure phenomenon on the empirical correlation between samples, and it is shown that the method is well suited for truly high-dimensional problems.
479 citations
Book•
01 Jul 1992
TL;DR: In this article, the convergence of the p-th mean of a series of tangent sequences has been studied in the context of Gaussian random series, and the convergence has been shown to be a strong predictor of strong domination of character systems on Abelian groups.
Abstract: 0 Preliminaries.- 0.1 Topology and measures.- 0.2 Tail inequalities.- 0.3 Filtrations and stopping times.- 0.4 Extensions of probability spaces.- 0.5 Bernoulli and canonical Gaussian and ?-stable sequences.- 0.6 Gaussian measures on linear spaces.- 0.7 Modulars on linear spaces.- 0.8 Musielak-Orlicz spaces.- 0.9 Random Musielak-Orlicz spaces.- 0.10 Complements and comments.- Bibliographical notes.- I Random Series.- 1 Basic Inequalities for Random Linear Forms in Independent Random Variables.- 1.1 Levy-Octaviani inequalities.- 1.2 Contraction inequalities.- 1.3 Moment inequalities.- 1.4 Complements and comments.- Best constants in the Levy-Octaviani inequality.- A contraction inequality for mixtures of Gaussian random variables.- Tail inequalities for Bernoulli and Gaussian random linear forms.- A refinement of the moment inequality.- Comparison of moments.- Bibliographical notes.- 2 Convergence of Series of Independent Random Variables.- 2.1 The Ito-Nisio Theorem.- 2.2 Convergence in the p-th mean.- 2.3 Exponential and other moments of random series.- 2.4 Random series in function spaces.- 2.5 An example: construction of the Brownian motion.- 2.6 Karhunen-Loeve representation of Gaussian measures.- 2.7 Complements and comments.- Rosenthal's inequalities.- Strong exponential moments of Gaussian series.- Lattice function spaces.- Convergence of Gaussian series.- Bibliographical notes.- 3 Domination Principles and Comparison of Sums of Independent Random Variables.- 3.1 Weak domination.- 3.2 Strong domination.- 3.3 Hypercontractive domination.- 3.4 Hypercontractivity of Bernoulli and Gaussian series.- 3.5 Sharp estimates of growth of p-th moments.- 3.6 Complements and comments.- More on C-domination.- Superstrong domination.- Domination of character systems on compact Abelian groups.- Random matrices.- Hypercontractivity of real random variables.- More precise estimates on strong exponential moments of Gaussian series.- Growth of p-th moments revisited.- More on strong exponential moments of series of bounded variables.- Bibliographical notes.- 4 Martingales.- 4.1 Doob's inequalities.- 4.2 Convergence of martingales.- 4.3 Tangent and decoupled sequences.- 4.4 Complements and comments.- Bibliographical notes.- 5 Domination Principles for Martingales.- 5.1 Weak domination.- 5.2 Strong domination.- 5.3 Burkholder's method: comparison of subordinated martingales.- 5.4 Comparison of strongly dominated martingales.- 5.5 Gaussian martingales.- 5.6 Classic martingale inequalities.- 5.7 Comparison of the a.s convergence of series of tangent sequences.- 5.8 Complements and comments.- Tangency and ergodic theorems.- Burkholder's method for conditionally Gaussian and conditionally independent martingales.- Necessity of moderate growth of ?.- Comparison of Gaussian martingales revisited.- Comparing H-valued martingales with 2-D martingales.- The principle of conditioning in limit theorems.- Bibliographical notes.- 6 Random Multilinear Forms in Independent Random Variables and Polynomial Chaos.- 6.1 Basic definitions and properties.- 6.2 Maximal inequalities.- 6.3 Contraction inequalities and domination of polynomial chaos.- 6.4 Decoupling inequalities.- 6.5 Comparison of moments of polynomial chaos.- 6.6 Convergence of polynomial chaos.- 6.7 Quadratic chaos.- 6.8 Wiener chaos and Herrnite polynomials.- 6.9 Complements and comments.- Tail and moment comparisons for chaos and its decoupled chaos.- Necessity of the symmetry condition in decoupling inequalities.- Karhunen-Loeve expansion for the Wiener chaos.- ?-stable chaos of degree d ? 2.- Bibliographical notes.- II Stochastic Integrals.- 7 Integration with Respect to General Stochastic Measures.- 7.1 Construction of the integral.- 7.2 Examples of stochastic measures.- 7.3 Complements and comments.- An alternative definition of m-integrability.- Bibliographical notes.- 8 Deterministic Integrands.- 8.1 Discrete stochastic measure.- 8.2 Processes with independent increments and their characteristics.- 8.3 Integration with respect to a general independently scattered measure.- 8.4 Complements and comments.- Stochastic measures with finite p-th moments.- Bibliographical notes.- 9 Predictable Integrands.- 9.1 Integration with respect to processes with independent increments: Decoupling inequalities approach.- 9.2 Brownian integrals.- 9.3 Characteristics of semimartingales.- 9.4 Semimartingale integrals.- 9.5 Complements and comments.- The Bichteler-Dellacherie Theorem.- Semimartingale integrals in Lp.- ?-stable integrals.- Bibliographical notes.- 10 Multiple Stochastic Integrals.- 10.1 Products of stochastic measures.- 10.2 Structure of double integrals.- 10.3 Wiener polynomial chaos revisited.- 10.4 Complements and comments.- Multiple ?-stable integrals.- Bibliographical notes.- A Unconditional and Bounded Multiplier Convergence of Random Series.- A.2 Almost sure convergence.- A.3 Complements and comments.- A hypercontractive view.- Bibliographical notes.- B Vector Measures.- B.1 Extensions of vector measures.- B.2 Boundedness and control measure of stochastic measures.- B.3 Complements and comments.- Bibliographical notes.
469 citations
TL;DR: In this paper, a generalization of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to non-Gaussian probability measures is presented.
Abstract: A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to non-Gaussian probability measures. We present conditions on such measures which imply mean-square convergence of generalized polynomial chaos expansions to the correct limit and complement these with illustrative examples.
469 citations
TL;DR: In this article, an uncertainty quantification scheme based on generalized polynomial chaos (PC) representations is constructed, which is applied to a model problem involving a simplified dynamical system and to the classical problem of Rayleigh-Benard instability.
463 citations