Sharif Rahman

Sharif Rahman

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.

Sharif Rahman

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.

Sharif Rahman

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.

Sharif Rahman

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.

Sharif Rahman

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.

Sharif Rahman

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.

Sharif Rahman, Xuchun Ren, Vaibhav Yadav

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.