Uncertainty Quantification by Alternative Decompositions of Multivariate Functions
Sharif Rahman
2013
SIAM Journal on Scientific Computing
This article advocates factorized and hybrid dimensional decompositions (FDD/HDD), as alternatives to analysis-of-variance dimensional decomposition (ADD), for second-moment statistical analysis of multivariate functions. New formulas revealing the relationships between component functions of FDD and ADD are proposed. While ADD or FDD is relevant when a function is strongly additive or strongly multiplicative, HDD, whether formed linearly or nonlinearly, requires no specific dimensional
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... ies. Furthermore, FDD and HDD lead to alternative definitions of effective dimension, reported in the current literature only for ADD. New closed-form or analytical expressions are derived for univariate truncations of all three decompositions, followed by mean-squared error analysis of univariate ADD, FDD, and HDD approximations. The analysis finds appropriate conditions when one approximation is better than the other. Numerical results affirm the theoretical finding that HDD is ideally suited to a general function approximation that may otherwise require higher-variate ADD or FDD truncations for rendering acceptable accuracy in stochastic solutions. Introduction. Uncertainty quantification of complex systems entails stochastic computing for a large number of random variables. Although the sampling-based methods can solve any stochastic problem, they generally require numerous deterministic trials and are, therefore, cost prohibitive when each analysis demands expensive finite-element or similar numerical calculations. Existing analytical or approximate methods require additional assumptions, mostly for computational expediency, that begin to deteriorate when the input-output mapping is highly nonlinear and the input variance is arbitrarily large. Furthermore, truly high-dimensional problems are all but impossible to solve using most existing methods, including numerical integration. The root deterrence to practical computability is often related to the high dimension of the multivariate integration or interpolation problem, known as the curse of dimensionality [1]. The dimensional decomposition of a multivariate function [8, 18, 12, 9] addresses the curse of dimensionality to some extent by developing an input-output behavior of complex systems with low effective dimension [2], wherein the degrees of interactions between input variables attenuate rapidly or vanish altogether. A prominent variant of dimensional decomposition is the well-known analysis-ofvariance or ANOVA dimensional decomposition (ADD), first presented by Hoeffding in the 1940s in relation to his seminal work on U -statistics [8] . Since then, ADD has been studied by numerous researchers in disparate fields of mathematics [16, 6, 7] , statistics [10, 3], finance [5] , and basic and applied sciences [11] , including engineering disciplines, mostly for uncertainty quantification [21, 15, 13] . However, ADD constitutes a finite sum of lower-dimensional component functions of a multivariate * ). A3024 ALTERNATIVE DIMENSIONAL DECOMPOSITIONS A3025 function, and is, therefore, predicated on the additive nature of a function decomposition. In contrast, when a response function is dominantly of a multiplicative nature, suitable multiplicative-type decompositions, such as factorized dimensional decomposition (FDD) [20], should be explored. But existing truncations of FDD are limited to only univariate or bivariate approximations, because FDD component functions of three or more variables have yet to be determined. No error analyses exist comparing ADD and FDD, even for respective univariate approximations. Nonetheless, ADD or FDD is relevant as long as the dimensional hierarchy of a stochastic response is also additive or multiplicative. Unfortunately, the dimensional structure of a response function, in general, is not known a priori. Therefore, indiscriminately using ADD or FDD for general stochastic analysis is not desirable. Further complications may arise when a complex system exhibits a response that is dominantly neither additive nor multiplicative. In the latter case, hybrid approaches coupling both additive and multiplicative decompositions, preferably selected optimally, are needed. For such decompositions, it is unknown which truncation parameter should be selected when compared with that for ADD or FDD. Is it possible to solve a stochastic problem by selecting a lower truncation parameter for hybrid decompositions than for ADD or FDD? If the answer is yes, then a significant, positive impact on high-dimensional uncertainty quantification is anticipated. These enhancements, some of which are indispensable, should be pursued without sustaining significant additional cost. The purpose of this paper is threefold. First, a brief exposition of ADD and FDD is given in section 3. A theorem, proven herein, reveals the relationship between all component functions of FDD and ADD, so far available only for univariate and bivariate component functions. Three function classes, comprising purely additive functions, purely multiplicative functions, and their mixtures, are examined to illustrate when and how one decomposition or approximation is better than the other. Second, a new hybrid approach optimally blending ADD and FDD approximations, referred to as hybrid dimensional decomposition (HDD), is presented in section 4 for second-moment analysis. Both linear and nonlinear mixtures of ADD and FDD approximations are supported. Gaining insights from FDD and HDD, alternative definitions of effective dimension are proposed. Third, section 5 reports new explicit formulas for respective univariate approximations derived from ADD, FDD, and HDD. The mean-squared error analyses pertaining to univariate ADD, FDD, and HDD approximations are also described. Numerical results from four elementary yet illuminating examples and a practical engineering problem are reported in sections 3 through 5 as relevant. There are nine new theoretical results stated or proved in this paper: Theorems 3.4, 4.1, and 5.4, Corollaries 3.5 and 4.2, Propositions 5.1 and 5.2, and Lemmas 3.3 and 5.3. Mathematical notations and conclusions are defined or drawn in sections 2 and 6, respectively. Notation. Let N, N 0 , R, and R + 0 represent the sets of positive integer (natural), nonnegative integer, real, and nonnegative real numbers, respectively. For k ∈ N, denote by R k the k-dimensional Euclidean space and by R k×k the set of k × k realvalued matrices. These standard notations will be used throughout the paper. Let (Ω, F , P ) be a complete probability space, where Ω is a sample space, F is a σ-field on Ω, and P : F → [0, 1] is a probability measure. With B N representing the Borel σ-field on R N , N ∈ N, consider an R N -valued random vector X := (X 1 , . . . , X N ) : (Ω, F ) → (R N , B N ) , which describes the statistical uncertainties in all system and input parameters of a high-dimensional stochastic problem. The probability law of X is completely defined by its joint probability density function
doi:10.1137/12089168x
fatcat:umlpyp23xbb6pnghmu25v4de6m