Multifidelity Dimension Reduction via Active Subspaces

Lam, Rémi; Zahm, Olivier; Marzouk, Youssef; Willcox, Karen

doi:10.1137/18m1214123

Cited by 35 publications

(33 citation statements)

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“…We demonstrate the randomized error estimator for two test cases which are derived from the benchmark problem introduced in [36]. The source code is freely available in Matlab at the address 3 so that all numerical results presented in this section are entirely reproducible.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Randomized residual‐based error estimators for the proper generalized decomposition approximation of parametrized problems

Smetana

Zahm

2020

Numerical Meth Engineering

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This paper introduces a novel error estimator for the Proper Generalized Decomposition (PGD) approximation of parametrized equations. The estimator is intrinsically random: It builds on concentration inequalities of Gaussian maps and an adjoint problem with random right-hand side, which we approximate using the PGD. The effectivity of this randomized error estimator can be arbitrarily close to unity with high probability, allowing the estimation of the error with respect to any user-defined norm as well as the error in some quantity of interest. The performance of the error estimator is demonstrated and compared with some existing error estimators for the PGD for a parametrized time-harmonic elastodynamics problem and the parametrized equations of linear elasticity with a high-dimensional parameter space.in general only as costly as the computation of the PGD approximation or even significantly less expensive, which makes our error estimator strategy attractive from a computational viewpoint. To build this estimator, we first estimate the norm of the error with a Monte Carlo estimator using Gaussian random vectors whose covariance is chosen according to the desired error measure, e.g., user-defined norms or quantity of interest. Then, we introduce a dual problem with random right-hand side the solution of which allows us to rewrite the error estimator in terms of the residual of the original equation. Approximating the random dual problems with the PGD yields a fast-to-evaluate error estimator that has low marginal cost.Next, we relate our error estimator to other stopping criteria that have been proposed for the PGD solution. In [2] the authors suggest letting an error estimator in some linear QoI steer the enrichment of the PGD approximation. In order to estimate the error they employ an adjoint problem with the linear QoI as a right-hand side and approximate this adjoint problem with the PGD as well. It is important to note that, as observed in [2], this type of error estimator seems to require in general a more accurate solution of the dual than the primal problem. In contrast, the framework we present in this paper often allows using a dual PGD approximation with significantly less terms than the primal approximation. An adaptive strategy for the PGD approximation of the dual problem for the error estimator proposed in [2] was suggested in [23] in the context of computational homogenization. In [1] the approach proposed in [2] is extended to problems in nonlinear solids by considering a suitable linearization of the problem.An error estimator for the error between the exact solution of a parabolic PDE and the PGD approximation in a suitable norm is derived in [34] based on the constitutive relation error and the Prager-Synge theorem. The discretization error is thus also assessed, and the computational costs of the error estimator depend on the number of elements in the underlying spatial mesh. Relying on the same techniques an error estimator for the error between the exact solution of a parabolic PDE and a...

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Section: Numerical Experimentsmentioning

confidence: 99%

Randomized residual‐based error estimators for the proper generalized decomposition approximation of parametrized problems

Smetana

Zahm

2020

Numerical Meth Engineering

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“…To obtain the sharpest bounds, both these improvements would require controlling the spectral norm ofΣ X,Y X, − Σ µ rather than its Frobenius norm (which is bounded in Theorem 3.2 above). A detailed argument favoring the spectral norm in the context of active subspaces is also given in [38]. Controlling the spectral norm of the error however appears to be considerably more difficult.…”

Section: Remark 7 (Possible Improvements)mentioning

confidence: 99%

“…This completes the proof of Theorem 3.1. Next, for N X,min, > 0 to be set later, recall the "good" event E 2 in (38) whereby every x ∈ X has at least N X,min, neighbors in Y X, . Conditioned on the event E 2 , ... ΣX,Y X, takes the simpler form of (42), using which we prove the following result in Appendix H.…”

mentioning

confidence: 99%

“…Appendix G. Proof of Lemma 5.3. Throughout, X and ∈ (0, µ,X ] are fixed, and we further assume that the event E 2 holds (see (38)). For now, suppose in addition that the neighborhood structure N X, :…”

mentioning

confidence: 99%

“…where the third identity uses (90) and the last line above follows from (38). Because B µ, does not depend on N X, , it is easy to remove the conditioning on N X, in (100):…”

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confidence: 99%

See 2 more Smart Citations

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

Eftekhari¹,

Wakin²,

Li³

et al. 2019

Foundations of Data Science

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Consider an open set D ⊆ R n , equipped with a probability measure µ. An important characteristic of a smooth function f : D → R is its secondmoment matrix Σµ := ∇f (x)∇f (x) * µ(dx) ∈ R n×n , where ∇f (x) ∈ R n is the gradient of f (•) at x ∈ D and * stands for transpose. For instance, the span of the leading r eigenvectors of Σµ forms an active subspace of f (•), which contains the directions along which f (•) changes the most and is of particular interest in ridge approximation. In this work, we propose a simple algorithm for estimating Σµ from random point evaluations of f (•) without imposing any structural assumptions on Σµ. Theoretical guarantees for this algorithm are established with the aid of the same technical tools that have proved valuable in the context of covariance matrix estimation from partial measurements.

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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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Multifidelity Dimension Reduction via Active Subspaces

Cited by 35 publications

References 50 publications

Randomized residual‐based error estimators for the proper generalized decomposition approximation of parametrized problems

Randomized residual‐based error estimators for the proper generalized decomposition approximation of parametrized problems

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

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

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