Practical uncertainty quantification analysis involving statistically dependent random variables |
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Institution: | 1. Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, India;2. Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt;1. Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon, Hong Kong, China;2. Department of Civil and Environmental Engineering, Michigan State University, East Lansing, Michigan, USA;1. Department of Mathematics, Memari College, Memari, Purba Bardhaman, West Bengal 713146, India;2. Complex Analysis Group, Translational Health Science and Technology Institute, NCR Biotech Science Cluster, Faridabad 121001, India;3. Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata 700032, India;1. BCAM–Basque Center for Applied Mathematics, Alameda de Mazarredo 14, Bilbao E-48009, Basque Country, Spain;2. Ikerbasque–Basque Foundation for Science, Calle de María Díaz de Haro 3, Bilbao E-48013, Basque Country, Spain |
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Abstract: | This article presents a practical refinement of generalized polynomial chaos expansion for uncertainty quantification under dependent input random variables. Unlike the Rodrigues-type formula, which exists for select probability measures, a three-step computational algorithm is put forward to generate a sequence of any approximate measure-consistent multivariate orthonormal polynomials. For uncertainty quantification analysis under dependent random variables, two regression methods, comprising existing standard least-squares and newly developed partitioned diffeomorphic modulation under observable response preserving homotopy (D-MORPH), are proposed to estimate the coefficients of generalized polynomial chaos expansion for the very first time. In contrast to the existing regression devoted so far to the classical polynomial chaos expansion, no tensor-product structure is required or enforced. The partitioned D-MORPH regression is applicable to either an underdetermined or overdetermined system, thus substantially enhancing the ability of the original D-MORPH regression. Numerical results obtained for Gaussian and non-Gaussian probability measures with rectangular or non-rectangular domains point to highly accurate orthonormal polynomials produced by the three-step algorithm. More significantly, the generalized polynomial chaos approximations of mathematical functions and stochastic responses from solid-mechanics problems, in tandem with the partitioned D-MORPH regression, provide excellent estimates of the second-moment properties and reliability from only hundreds of function evaluations or finite element analyses. |
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