Quantification of margins and uncertainties using an active subspace method for approximating bounds

Constantine, Paul G.; Emory, Michael; Palacios, Fredy Orlando Ruiz; Kseib, N; Iaccarino, G.

doi:10.1201/b16387-119

Cited by 4 publications

(7 citation statements)

References 2 publications

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“…In this section, we discuss the classic approach to discovering the active subspace using gradient information [69,67,68,73,70,74,75,90,91]. Recall that we are dealing with a high-dimensional response surface, and that we would like to approximate it as in Eq.…”

Section: Gradient-based Approach To Active Subspace Regressionmentioning

confidence: 99%

“…AS applications focus on computer experiments [67][68][69][70], while PLS has been extensively used to model real experiments with high-dimensional inputs/outputs in the field of chemometrics [79][80][81]. PLS not only projects the input to a lower dimensional space using an orthogonal projection matrix, but, if required, it can do the same to a high-dimensional output.…”

Section: Introductionmentioning

confidence: 99%

“…Mathematically, an AS is described by an orthogonal matrix that projects the original inputs to this low-dimensional manifold. The classic framework for discovering the AS was laid down by Constantine [67][68][69][70]. One builds a positive-definite matrix that depends upon the gradients of the response surface.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Gaussian processes with built-in dimensionality reduction: Applications to high-dimensional uncertainty propagation

Tripathy

Bilionis

Gonzalez

2016

Journal of Computational Physics

164

126

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Uncertainty quantification (UQ) tasks, such as model calibration, uncertainty propagation, and optimization under uncertainty, typically require several thousand evaluations of the underlying computer codes. To cope with the cost of simulations, one replaces the real response surface with a cheap surrogate based, e.g., on polynomial chaos expansions, neural networks, support vector machines, or Gaussian processes (GP). However, the number of simulations required to learn a generic multivariate response grows exponentially as the input dimension increases. This curse of dimensionality can only be addressed, if the response exhibits some special structure that can be discovered and exploited. A wide range of physical responses exhibit a special structure known as an active subspace (AS). An AS is a linear manifold of the stochastic space characterized by maximal response variation. The idea is that one should first identify this low dimensional manifold, project the high-dimensional input onto it, and then link the projection to the output. If the dimensionality of the AS is low enough, then learning the link function is a much easier problem than the original problem of learning a high-dimensional function. The classic approach to discovering the AS requires gradient information, a fact that severely limits its applicability. Furthermore, and partly because of its reliance to gradients, it is not able to handle noisy observations. The latter is an essential trait if one wants to be able to propagate uncertainty through stochastic simulators, e.g., through molecular dynamics codes. In this work, we develop a probabilistic version of AS which is gradient-free and robust to observational noise. Our approach relies on a novel Gaussian process regression with built-in dimensionality reduction. In particular, the AS is represented as an orthogonal projection matrix that serves as yet another covariance function hyper-parameter to be estimated from the data. To train the model, we design a two-step maximum likelihood optimization procedure that ensures the orthogonality of the projection matrix by exploiting recent results on the Stiefel manifold, i.e., the manifold of matrices with orthogonal columns. The additional benefit of our probabilistic formulation, is that it allows us to select the dimensionality of the AS via the Bayesian information criterion. We validate our approach by showing that it can discover the right AS in synthetic examples without gradient information using both noiseless and noisy observations. We demonstrate that our method is able to discover the same AS as the classical approach in a challenging one-hundred-dimensional problem involving an elliptic stochastic partial differential equation with random conductivity. Finally, we use our approach to study the effect of geometric and material uncertainties in the propagation of solitary waves in a one dimensional granular system.

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Section: Gradient-based Approach To Active Subspace Regressionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Gaussian processes with built-in dimensionality reduction: Applications to high-dimensional uncertainty propagation

Tripathy

Bilionis

Gonzalez

2016

Journal of Computational Physics

164

126

View full text Add to dashboard Cite

show abstract

“…In this section, the classic active subspace method which is based on the gradient information [21][22][23][24][25][26][27][28] has been introduced. Let f be the multivariate response surface with D ≫ 1.…”

Section: Active Subspace 221 Classic Active Subspace Approachmentioning

confidence: 99%

High-Dimensional Nonlinear Multi-Fidelity Model with Gradient-Free Active Subspace Method

Lin¹

2020

CICP

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“…Our prior work studies the effects of uncertainties in the RANS turbulence models and how to account for them in a physically meaningful manner [15]. We have also compared the effects of uncertainties in the combustion chemistry to effects of uncertain operating conditions for HyShot II, and we found that the effects of operating conditions were much greater than the effects of the chemistry parameters [10]. The current paper focuses on the HyShot II operating environment vehicle studied by DLR in the HEG shock tunnel [18,19,51].…”

Section: Introductionmentioning

confidence: 97%

Exploiting active subspaces to quantify uncertainty in the numerical simulation of the HyShot II scramjet

Constantine

Emory

Larsson

et al. 2015

Journal of Computational Physics

141

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We present a computational analysis of the reactive flow in a hypersonic scramjet engine with focus on effects of uncertainties in the operating conditions. We employ a novel methodology based on active subspaces to characterize the effects of the input uncertainty on the scramjet performance. The active subspace identifies one-dimensional structure in the map from simulation inputs to quantity of interest that allows us to reparameterize the operating conditions; instead of seven physical parameters, we can use a single derived active variable. This dimension reduction enables otherwise infeasible uncertainty quantification, considering the simulation cost of roughly 9500 CPU-hours per run. For two values of the fuel injection rate, we use a total of 68 simulations to (i) identify the parameters that contribute the most to the variation in the output quantity of interest, (ii) estimate upper and lower bounds on the quantity of interest, (iii) classify sets of operating conditions as safe or unsafe corresponding to a threshold on the output quantity of interest, and (iv) estimate a cumulative distribution function for the quantity of interest.

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Quantification of margins and uncertainties using an active subspace method for approximating bounds

Cited by 4 publications

References 2 publications

Gaussian processes with built-in dimensionality reduction: Applications to high-dimensional uncertainty propagation

Gaussian processes with built-in dimensionality reduction: Applications to high-dimensional uncertainty propagation

High-Dimensional Nonlinear Multi-Fidelity Model with Gradient-Free Active Subspace Method

Exploiting active subspaces to quantify uncertainty in the numerical simulation of the HyShot II scramjet

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