Gradient-Informed Basis Adaptation for Legendre Chaos Expansions

Tsilifis, Panagiotis

doi:10.1115/1.4040802

Cited by 7 publications

(7 citation statements)

References 60 publications

(74 reference statements)

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“…We use n = 300 terms in Eq. (33), which leads to an input random vector X of dimension 300. We are interested in approximating the displacement of the oscillator at t = 8 s, u (8 s).…”

Section: Nonlinear Oscillatormentioning

confidence: 99%

“…Although AS can lead to vast dimensionality reduction, in high-dimensional problems with black box numerical models, the additional computational cost from the numerical evaluation of the gradients might be prohibitive. The AS method has been combined with PCEs in [33], wherein the covariance of the gradient vector is computed based on a low-order PCE.…”

Section: Introductionmentioning

confidence: 99%

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PLS-based adaptation for efficient PCE representation in high dimensions

Papaioannou

Ehre

Straub

2019

Journal of Computational Physics

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Uncertainty quantification of engineering systems modeled by computationally intensive numerical models remains a challenging task, despite the increase in computer power. Efficient uncertainty propagation of such models can be performed by use of surrogate models, such as polynomial chaos expansions (PCE). A major drawback of standard PCE is that its predictive ability decreases with increase of the problem dimension for a fixed computational budget. This is related to the fact that the number of terms in the expansion increases fast with the input variable dimension. To address this issue, Tipireddy and Ghanem (2014) introduced a sparse PCE representation based on a transformation of the coordinate system in Gaussian input variable spaces. In this contribution, we propose to identify the projection operator underlying this transformation and approximate the coefficients of the resulting PCE through partial least squares (PLS) analysis. The proposed PCE-driven PLS algorithm identifies the directions with the largest predictive significance in the PCE representation based on a set of samples from the input random variables and corresponding response variable. This approach does not require gradient evaluations, which makes it efficient for high dimensional problems with black-box numerical models. We assess the proposed approach with three numerical examples in high-dimensional input spaces, comparing its performance with low-rank tensor approximations. These examples demonstrate that the PLS-based PCE method provides accurate representations even for strongly non-linear problems.

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“…We use n = 300 terms in Eq. (33), which leads to an input random vector X of dimension 300. We are interested in approximating the displacement of the oscillator at t = 8 s, u (8 s).…”

Section: Nonlinear Oscillatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

PLS-based adaptation for efficient PCE representation in high dimensions

Papaioannou

Ehre

Straub

2019

Journal of Computational Physics

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“…As was emphasized in the original basis adaptation method [52], our approach applies specifically to the Hermite Chaos with Gaussian input variables, as the distribution of the projected variables would otherwise be arbitrary, resulting in non optimal polynomial representation, due to the loss of the orthogonality property. A recent attempt to adapt the basis of non-Hermite Chaos [57], has been restricted to projections on 1-dimensional bases, as those can be easily mapped to uniformly distributed inputs and the Legendre Chaos can then be employed. This however required an a priori indication that such a 1-dimensional adaptation exists, which was validated using a gradient-based criterion to compute the rotation, that again relies on pseudo-spectral approaches and we therefore restrain from such exploitations here.…”

Section: Introductionmentioning

confidence: 99%

Compressive sensing adaptation for polynomial chaos expansions

Tsilifis¹,

Huan²,

Safta³

et al. 2019

Journal of Computational Physics

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“…The rotation in question is adapted to specific quantities of interest and provides a reduction of full polynomial expansions through a rotation of the independent variables. This formulation enables the concentration of the probabilistic content into a very low-dimensional structure that can be easily explored using low-level quadrature rules, active subspaces [25,26], 1 -minimization [27] and even analytical mappings [28], allowing the segregation and reordering of uncertainties. In this work, we attempt a Bayesian approach to the basis adaptation framework for Homogeneous (Hermite) Chaos that views both the chaos coefficients and the input rotation matrix as random variables defined on probability spaces and we employ inference techniques for characterization of their probability measures.…”

Section: Introductionmentioning

confidence: 99%

Bayesian adaptation of chaos representations using variational inference and sampling on geodesics

Tsilifis

Ghanem

2018

Proc. R. Soc. A.

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A novel approach is presented for constructing polynomial chaos representations of scalar quantities of interest (QoI) that extends previously developed methods for adaptation in Homogeneous Chaos spaces. In this work, we develop a Bayesian formulation of the problem that characterizes the posterior distributions of the series coefficients and the adaptation rotation matrix acting on the Gaussian input variables. The adaptation matrix is thus construed as a new parameter of the map from input to QoI, estimated through Bayesian inference. For the computation of the coefficients' posterior distribution, we use a variational inference approach that approximates the posterior with a member of the same exponential family as the prior, such that it minimizes a Kullback–Leibler criterion. On the other hand, the posterior distribution of the rotation matrix is explored by employing a Geodesic Monte Carlo sampling approach, consisting of a variation of the Hamiltonian Monte Carlo algorithm for embedded manifolds, in our case, the Stiefel manifold of orthonormal matrices. The performance of our method is demonstrated through a series of numerical examples, including the problem of multiphase flow in heterogeneous porous media.

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Gradient-Informed Basis Adaptation for Legendre Chaos Expansions

Cited by 7 publications

References 60 publications

PLS-based adaptation for efficient PCE representation in high dimensions

PLS-based adaptation for efficient PCE representation in high dimensions

Compressive sensing adaptation for polynomial chaos expansions

Bayesian adaptation of chaos representations using variational inference and sampling on geodesics

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