arXiv subjects

Sharif Rahman

Publications and source records attributed to Sharif Rahman.

14 recordsLinked to original sources

arXiv2021

A Spline Dimensional Decomposition for Uncertainty Quantification in High Dimensions

This study debuts a new spline dimensional decomposition (SDD) for uncertainty quantification analysis of high-dimensional functions, including those endowed with high nonlinearity and nonsmoothness, if they exist, in a proficient manner. The decomposition creates an hierarchical expansion for an output random variable of interest with respect to measure-consistent orthonormalized basis splines (B-splines) in independent input random variables. A dimensionwise decomposition of a spline space into orthogonal subspaces, each spanned by a reduced set of such orthonormal splines, results in SDD. Exploiting the modulus of smoothness, the SDD approximation is shown to converge in mean-square to the correct limit. The computational complexity of the SDD method is polynomial, as opposed to exponential, thus alleviating the curse of dimensionality to the extent possible. Analytical formulae are proposed to calculate the second-moment properties of a truncated SDD approximation for a general output random variable in terms of the expansion coefficients involved. Numerical results indicate that a low-order SDD approximation of nonsmooth functions calculates the probabilistic characteristics of an output variable with an accuracy matching or surpassing those obtained by high-order approximations from several existing methods. Finally, a 34-dimensional random eigenvalue analysis demonstrates the utility of SDD in solving practical problems.

math.NA↗

arXiv2019

A Spline Chaos Expansion

A spline chaos expansion, referred to as SCE, is introduced for uncertainty quantification analysis. The expansion provides a means for representing an output random variable of interest with respect to multivariate orthonormal basis splines (B-splines) in input random variables. The multivariate B-splines are built from a whitening transformation to generate univariate orthonormal B-splines in each coordinate direction, followed by a tensor-product structure to produce the multivariate version. SCE, as it stems from compactly supported B-splines, tackles locally prominent responses more effectively than the polynomial chaos expansion (PCE). The approximation quality of the expansion is demonstrated in terms of the modulus of smoothness of the output function, leading to the mean-square convergence of SCE to the correct limit. Analytical formulae are proposed to calculate the mean and variance of an SCE approximation for a general output variable in terms of the requisite expansion coefficients. Numerical results indicate that a low-order SCE approximation with an adequate mesh is markedly more accurate than a high-order PCE approximation in estimating the output variances and probability distributions of oscillatory, nonsmooth, and nearly discontinuous functions.

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arXiv2018

Dimension-wise Multivariate Orthogonal Polynomials in General Probability Spaces

This paper puts forward a new generalized polynomial dimensional decomposition (PDD), referred to as GPDD, comprising hierarchically ordered measure-consistent multivariate orthogonal polynomials in dependent random variables. Unlike the existing PDD, which is valid strictly for independent random variables, no tensor-product structure is assumed or required. Important mathematical properties of GPDD are studied by constructing dimension-wise decomposition of polynomial spaces, deriving statistical properties of random orthogonal polynomials, demonstrating completeness of orthogonal polynomials for prescribed assumptions, and proving mean-square convergence to the correct limit, including when there are infinitely many random variables. The GPDD approximation proposed should be effective in solving high-dimensional stochastic problems subject to dependent variables.

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arXiv2018

A Galerkin Isogeometric Method for Karhunen-Loeve Approximation of Random Fields

This paper marks the debut of a Galerkin isogeometric method for solving a Fredholm integral eigenvalue problem, enabling random field discretization by means of the Karhunen-Loeve expansion. The method involves a Galerkin projection onto a finite-dimensional subspace of a Hilbert space, basis splines (B-splines) and non-uniform rational B-splines (NURBS) spanning the subspace, and standard methods of eigensolutions. Compared with the existing Galerkin methods, such as the finite-element and mesh-free methods, the NURBS-based isogeometric method upholds exact geometrical representation of the physical or computational domain and exploits regularity of basis functions delivering globally smooth eigensolutions. Therefore, the introduction of the isogeometric method for random field discretization is not only new; it also offers a few computational advantages over existing methods. In the big picture, the use of NURBS for random field discretization enriches the isogeometric paradigm. As a result, an uncertainty quantification pipeline of the future can be envisioned where geometric modeling, stress analysis, and stochastic simulation are all integrated using the same building blocks of NURBS. Three numerical examples, including a three-dimensional random field discretization problem, illustrate the accuracy and convergence properties of the isogeometric method for obtaining eigensolutions.

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arXiv2018

A Polynomial Chaos Expansion in Dependent Random Variables

This paper introduces a new generalized polynomial chaos expansion (PCE) comprising measure-consistent multivariate orthonormal polynomials in dependent random variables. Unlike existing PCEs, whether classical or generalized, no tensor-product structure is assumed or required. Important mathematical properties of the generalized PCE are studied by constructing orthogonal decomposition of polynomial spaces, explaining completeness of orthogonal polynomials for prescribed assumptions, exploiting whitening transformation for generating orthonormal polynomial bases, and demonstrating mean-square convergence to the correct limit. Analytical formulae are proposed to calculate the mean and variance of a truncated generalized PCE for a general output variable in terms of the expansion coefficients. An example derived from a stochastic boundary-value problem illustrates the generalized PCE approximation in estimating the statistical properties of an output variable for 12 distinct non-product-type probability measures of input variables.

math.PR↗

arXiv2018

Mathematical Properties of Polynomial Dimensional Decomposition

Many high-dimensional uncertainty quantification problems are solved by polynomial dimensional decomposition (PDD), which represents Fourier-like series expansion in terms of random orthonormal polynomials with increasing dimensions. This study constructs dimension-wise and orthogonal splitting of polynomial spaces, proves completeness of polynomial orthogonal basis for prescribed assumptions, and demonstrates mean-square convergence to the correct limit -- all associated with PDD. A second-moment error analysis reveals that PDD cannot commit larger error than polynomial chaos expansion (PCE) for the appropriately chosen truncation parameters. From the comparison of computational efforts, required to estimate with the same precision the variance of an output function involving exponentially attenuating expansion coefficients, the PDD approximation can be markedly more efficient than the PCE approximation.

math.NA↗

arXiv2017

Wiener-Hermite Polynomial Expansion for Multivariate Gaussian Probability Measures

This paper introduces a new generalized polynomial chaos expansion (PCE) comprising multivariate Hermite orthogonal polynomials in dependent Gaussian random variables. The second-moment properties of Hermite polynomials reveal a weakly orthogonal system when obtained for a general Gaussian probability measure. Still, the exponential integrability of norm allows the Hermite polynomials to constitute a complete set and hence a basis in a Hilbert space. The completeness is vitally important for the convergence of the generalized PCE to the correct limit. The optimality of the generalized PCE and the approximation quality due to truncation are discussed. New analytical formulae are proposed to calculate the mean and variance of a generalized PCE approximation of a general output variable in terms of the expansion coefficients and statistical properties of Hermite polynomials. However, unlike in the classical PCE, calculating the coefficients of the generalized PCE requires solving a coupled system of linear equations. Besides, the variance formula of the generalized PCE contains additional terms due to statistical dependence among Gaussian variables. The additional terms vanish when the Gaussian variables are statistically independent, reverting the generalized PCE to the classical PCE. Numerical examples illustrate the generalized PCE approximation in estimating the statistical properties of various output variables.

math.NA↗

arXiv2016

High-Dimensional Stochastic Design Optimization by Adaptive-Sparse Polynomial Dimensional Decomposition

This paper presents a novel adaptive-sparse polynomial dimensional decomposition (PDD) method for stochastic design optimization of complex systems. The method entails an adaptive-sparse PDD approximation of a high-dimensional stochastic response for statistical moment and reliability analyses; a novel integration of the adaptive-sparse PDD approximation and score functions for estimating the first-order design sensitivities of the statistical moments and failure probability; and standard gradient-based optimization algorithms. New analytical formulae are presented for the design sensitivities that are simultaneously determined along with the moments or the failure probability. Numerical results stemming from mathematical functions indicate that the new method provides more computationally efficient design solutions than the existing methods. Finally, stochastic shape optimization of a jet engine bracket with 79 variables was performed, demonstrating the power of the new method to tackle practical engineering problems.

math.NA↗

arXiv2015

The $f$-Sensitivity Index

This article presents a general multivariate $f$-sensitivity index, rooted in the $f$-divergence between the unconditional and conditional probability measures of a stochastic response, for global sensitivity analysis. Unlike the variance-based Sobol index, the $f$-sensitivity index is applicable to random input following dependent as well as independent probability distributions. Since the class of $f$-divergences supports a wide variety of divergence or distance measures, a plethora of $f$-sensitivity indices are possible, affording diverse choices to sensitivity analysis. Commonly used sensitivity indices or measures, such as mutual information, squared-loss mutual information, and Borgonovo's importance measure, are shown to be special cases of the proposed sensitivity index. New theoretical results, revealing fundamental properties of the $f$-sensitivity index and establishing important inequalities, are presented. Three new approximate methods, depending on how the probability densities of a stochastic response are determined, are proposed to estimate the sensitivity index. Four numerical examples, including a computationally intensive stochastic boundary-value problem, illustrate these methods and explain when one method is more relevant than the others.

math.NA↗

arXiv2014

A Generalized ANOVA Dimensional Decomposition for Dependent Probability Measures

This article explores the generalized analysis-of-variance or ANOVA dimensional decomposition (ADD) for multivariate functions of dependent random variables. Two notable properties, stemming from weakened annihilating conditions, reveal that the component functions of the generalized ADD have \emph{zero} means and are hierarchically orthogonal. By exploiting these properties, a simple, alternative approach is presented to derive a coupled system of equations that the generalized ADD component functions satisfy. The coupled equations, which subsume as a special case the classical ADD, reproduce the component functions for independent probability measures. To determine the component functions of the generalized ADD, a new constructive method is proposed by employing measure-consistent, multivariate orthogonal polynomials as bases and calculating the expansion coefficients involved from the solution of linear algebraic equations. New generalized formulae are presented for the second-moment characteristics, including triplets of global sensitivity indices, for dependent probability distributions. Furthermore, the generalized ADD leads to extended definitions of effective dimensions, reported in the current literature for the classical ADD. Numerical results demonstrate that the correlation structure of random variables can significantly alter the composition of component functions, producing widely varying global sensitivity indices and, therefore, distinct rankings of random variables. An application to random eigenvalue analysis demonstrates the usefulness of the proposed approximation.

math.NA↗

arXiv2014

Novel Computational Methods for High-Dimensional Stochastic Sensitivity Analysis

This paper presents three new computational methods for calculating design sensitivities of statistical moments and reliability of high-dimensional complex systems subject to random input. The first method represents a novel integration of polynomial dimensional decomposition (PDD) of a multivariate stochastic response function and score functions. Applied to the statistical moments, the method provides mean-square convergent analytical expressions of design sensitivities of the first two moments of a stochastic response. The second and third methods, relevant to probability distribution or reliability analysis, exploit two distinct combinations built on PDD: the PDD-SPA method, entailing the saddlepoint approximation (SPA) and score functions; and the PDD-MCS method, utilizing the embedded Monte Carlo simulation (MCS) of the PDD approximation and score functions. For all three methods developed, the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation. Numerical examples, including a 100-dimensional mathematical problem, indicate that the new methods developed provide not only theoretically convergent or accurate design sensitivities, but also computationally efficient solutions. A practical example involving robust design optimization of a three-hole bracket illustrates the usefulness of the proposed methods.

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arXiv2014

Adaptive-Sparse Polynomial Dimensional Decomposition Methods for High-Dimensional Stochastic Computing

This article presents two novel adaptive-sparse polynomial dimensional decomposition (PDD) methods for solving high-dimensional uncertainty quantification problems in computational science and engineering. The methods entail global sensitivity analysis for retaining important PDD component functions, and a full- or sparse-grid dimension-reduction integration or quasi Monte Carlo simulation for estimating the PDD expansion coefficients. A unified algorithm, endowed with two distinct ranking schemes for grading component functions, was created for their numerical implementation. The fully adaptive-sparse PDD method is comprehensive and rigorous, leading to the second-moment statistics of a stochastic response that converges to the exact solution when the tolerances vanish. A partially adaptive-sparse PDD method, obtained through regulated adaptivity and sparsity, is economical and is, therefore, expected to solve practical problems with numerous variables. Compared with past developments, the adaptive-sparse PDD methods do not require its truncation parameter(s) to be assigned \emph{a priori} or arbitrarily. The numerical results reveal that an adaptive-sparse PDD method achieves a desired level of accuracy with considerably fewer coefficients compared with existing PDD approximations. For a required accuracy in calculating the probabilistic response characteristics, the new bivariate adaptive-sparse PDD method is more efficient than the existing bivariately truncated PDD method by almost an order of magnitude. Finally, stochastic dynamic analysis of a disk brake system was performed, demonstrating the ability of the new methods to tackle practical engineering problems.

math.NA↗

arXiv2013

Approximation Errors in Truncated Dimensional Decompositions

The main theme of this paper is error analysis for approximations derived from two variants of dimensional decomposition of a multivariate function: the referential dimensional decomposition (RDD) and analysis-of-variance dimensional decomposition (ADD). New formulae are presented for the lower and upper bounds of the expected errors committed by bivariately and arbitrarily truncated RDD approximations when the reference point is selected randomly, thereby facilitating a means for weighing RDD against ADD approximations. The formulae reveal that the expected error from the S-variate RDD approximation of a function of N variables, where $0 \le S < N < \infty$, is at least $2^{S+1}$ times greater than the error from the S-variate ADD approximation. Consequently, ADD approximations are exceedingly more precise than RDD approximations. The analysis also finds the RDD approximation to be sub-optimal for an arbitrarily selected reference point, whereas the ADD approximation always results in minimum error. Therefore, the RDD approximation should be used with caution.

math.NA↗

arXiv2013

Uncertainty Quantification by Alternative Decompositions of Multivariate Functions

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 formulae 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 hierarchies. 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.

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