Randomized learning of the second-moment matrix of a smooth function

Eftekhari, Armin; Wakin, Michael B.; Li, Ping; Constantine, Paul G.

doi:10.3934/fods.2019015

Cited by 3 publications

(3 citation statements)

References 45 publications

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“…Second, computation of the full Jacobian matrix for each sample is not necessary. By leveraging ideas from randomized numerical linear algebra [33,34], one can construct accurate estimates of the leading eigenspaces of CX and CY using only matrix–vector products.…”

Section: Discussionmentioning

confidence: 99%

A low-rank ensemble Kalman filter for elliptic observations

et al. 2022

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We propose a regularization method for ensemble Kalman filtering (EnKF) with elliptic observation operators. Commonly used EnKF regularization methods suppress state correlations at long distances. For observations described by elliptic partial differential equations, such as the pressure Poisson equation (PPE) in incompressible fluid flows, distance localization should be used cautiously, as we cannot disentangle slowly decaying physical interactions from spurious long-range correlations. This is particularly true for the PPE, in which distant vortex elements couple nonlinearly to induce pressure. Instead, these inverse problems have a low effective dimension: low-dimensional projections of the observations strongly inform a low-dimensional subspace of the state space. We derive a low-rank factorization of the Kalman gain based on the spectrum of the Jacobian of the observation operator. The identified eigenvectors generalize the source and target modes of the multipole expansion, independently of the underlying spatial distribution of the problem. Given rapid spectral decay, inference can be performed in the low-dimensional subspace spanned by the dominant eigenvectors. This low-rank EnKF is assessed on dynamical systems with Poisson observation operators, where we seek to estimate the positions and strengths of point singularities over time from potential or pressure observations. We also comment on the broader applicability of this approach to elliptic inverse problems outside the context of filtering.

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Section: Discussionmentioning

confidence: 99%

A low-rank ensemble Kalman filter for elliptic observations

et al. 2022

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“…If λ m + 1 , …, λ n are small enough, the original response function can be approximated as

g (boldx) = g ({boldWW}^{normalT} x) = g ({boldW}_{1} {W_{1}}^{normalT} x + {boldW}_{2} {W_{2}}^{normalT} x) \approx g ({boldW}_{1} {W_{1}}^{normalT} x) = g ({boldW}_{1} u) = f (boldu),

where

u = {boldW}_{1}^{normalT} x \in {normalℝ}^{m}

is the new rotated and reduced coordinate system. The error of this approximation can be found in Reference .…”

Section: Active Subspace Methodsmentioning

confidence: 99%

“…Lam et al proposed a multifidelity (MF) method to estimate the AS matrix, and they provided the error bar of this estimator. Eftekhari et al presented an efficient algorithm to estimate the AS matrix using only randomly samples.…”

Section: Introductionmentioning

confidence: 99%

Hierarchical surrogate model with dimensionality reduction technique for high‐dimensional uncertainty propagation

Cheng

Lü

2019

Numerical Meth Engineering

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Summary In this article, hierarchical surrogate model combined with dimensionality reduction technique is investigated for uncertainty propagation of high‐dimensional problems. In the proposed method, a low‐fidelity sparse polynomial chaos expansion model is first constructed to capture the global trend of model response and exploit a low‐dimensional active subspace (AS). Then a high‐fidelity (HF) stochastic Kriging model is built on the reduced space by mapping the original high‐dimensional input onto the identified AS. The effective dimensionality of the AS is estimated by maximum likelihood estimation technique. Finally, an accurate HF surrogate model is obtained for uncertainty propagation of high‐dimensional stochastic problems. The proposed method is validated by two challenging high‐dimensional stochastic examples, and the results demonstrate that our method is effective for high‐dimensional uncertainty propagation.

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