Research Papers

Gradient-Informed Basis Adaptation for Legendre Chaos Expansions

[+] Author and Article Information
Panagiotis A. Tsilifis

Institute of Mathematics,
School of Basic Sciences,
École Polytechnique Fédérale de Lausanne,
Lausanne CH-1015, Switzerland
e-mail: panagiotis.tsilifis@epfl.ch

Manuscript received November 10, 2016; final manuscript received July 5, 2018; published online July 30, 2018. Assoc. Editor: Sez Atamturktur.

J. Verif. Valid. Uncert 3(1), 011005 (Jul 30, 2018) (11 pages) Paper No: VVUQ-16-1030; doi: 10.1115/1.4040802 History: Received November 10, 2016; Revised July 05, 2018

The recently introduced basis adaptation method for homogeneous (Wiener) chaos expansions is explored in a new context where the rotation/projection matrices are computed by discovering the active subspace (AS) where the random input exhibits most of its variability. In the case where a One-dimensional (1D) AS exists, the methodology can be applicable to generalized polynomial chaos expansions (PCE), thus enabling the projection of a high-dimensional input to a single input variable and the efficient estimation of a univariate chaos expansion. Attractive features of this approach, such as the significant computational savings and the high accuracy in computing statistics of interest are investigated.

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Grahic Jump Location
Fig. 1

Left: Plot of the empirical pdf of η=wTξ for w given in Eq. (39). Right: level-5 Clenshaw–Curtis quadrature points on [–1,1] and their mappings on η-space.

Grahic Jump Location
Fig. 2

Left: predictions of the link function using 1D- and 10D-expansions. Right: comparison of the true and the estimated density function bases on 1D-20th-order chaos expansion.

Grahic Jump Location
Fig. 3

Domain of the ammonium transport problem. Ammonium is injected at y = 0.

Grahic Jump Location
Fig. 4

Ten Monte Carlo realizations of the model outputs and their corresponding PC approximations using quadrature level  = 2, 3, 4, and 5, respectively

Grahic Jump Location
Fig. 5

Eigenvalues of the gradient matrix (left) and the values of the dominant eigenvector w (right) for chaos coefficients corresponding to different quadrature levels

Grahic Jump Location
Fig. 6

Estimated coefficients of the chaos expansions using different quadrature rule levels

Grahic Jump Location
Fig. 7

Evaluation of the model output and the 1d PC expansions on 1000 MC samples of η for  = 2, 3, 4, and 5

Grahic Jump Location
Fig. 8

Histogram comparison based on MC samples from the 1D- (left column) and 5D- (right column) chaos expansions with that based on 1000 MC samples drawn from the TOUGH2 simulator (all histograms are normalized such that the bins integrate to 1)



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