A continuous analogue of the tensor-train decomposition

Gorodetsky, Alex; Karaman, Sertaç; Marzouk, Youssef

doi:10.1016/j.cma.2018.12.015

Cited by 67 publications

(87 citation statements)

References 44 publications

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“…Consequently, MC methods are often computationally intractable for high‐fidelity simulation models because they require a prohibitively large number of evaluations to obtain moderate accuracy in response statistics. In contrast, surrogate methods such as polynomial chaos, Gaussian processes, low‐rank decompositions, and sparse grid interpolation can be used to build an approximation of the input‐output response, often at a fraction of the cost of MC sampling. Once the surrogate has been constructed, various uncertainty quantification (UQ) tasks, such as sensitivity analysis, density estimation, etc., can then be performed on the approximation at negligible cost *…”

Section: Introductionmentioning

confidence: 99%

Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis

Jakeman

Eldred

Geraci

et al. 2019

Numerical Meth Engineering

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Summary In this paper, we present an adaptive algorithm to construct response surface approximations of high‐fidelity models using a hierarchy of lower fidelity models. Our algorithm is based on multi‐index stochastic collocation and automatically balances physical discretization error and response surface error to construct an approximation of model outputs. This surrogate can be used for uncertainty quantification (UQ) and sensitivity analysis (SA) at a fraction of the cost of a purely high‐fidelity approach. We demonstrate the effectiveness of our algorithm on a canonical test problem from the UQ literature and a complex multiphysics model that simulates the performance of an integrated nozzle for an unmanned aerospace vehicle. We find that, when the input‐output response is sufficiently smooth, our algorithm produces approximations that can be over two orders of magnitude more accurate than single fidelity approximations for a fixed computational budget.

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Section: Introductionmentioning

confidence: 99%

Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis

Jakeman

Eldred

Geraci

et al. 2019

Numerical Meth Engineering

Self Cite

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“…These sketching methods also naturally extend to out-of-core algorithms, and may even be used to increase scalability in distributed memory. Finally, it is natural to consider application of randomization to other decompositions such as Tucker [38], tensor train [28], or functional tensor decompositions [11,17].…”

Section: Discussionmentioning

confidence: 99%

A Practical Randomized CP Tensor Decomposition

Battaglino¹,

Ballard²,

Kolda³

2018

SIAM J. Matrix Anal. & Appl.

187

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The CANDECOMP/PARAFAC (CP) decomposition is a leading method for the analysis of multiway data. The standard alternating least squares algorithm for the CP decomposition (CP-ALS) involves a series of highly overdetermined linear least squares problems. We extend randomized least squares methods to tensors and show the workload of CP-ALS can be drastically reduced without a sacrifice in quality. We introduce techniques for efficiently preprocessing, sampling, and computing randomized least squares on a dense tensor of arbitrary order, as well as an efficient sampling-based technique for checking the stopping condition. We also show more generally that the Khatri-Rao product (used within the CP-ALS iteration) produces conditions favorable for direct sampling. In numerical results, we see improvements in speed, reductions in memory requirements, and robustness with respect to initialization.

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“…To achieve 1), one can for instance store the rows and columns of the Vandermonde matrix which are called with highest probability. To achieve 2) one can exploit further insight of the functions, for instance using a low-rank approximation [11] or functional analogues of tensor decomposition approximation [10].…”

Section: Discussionmentioning

confidence: 99%

Weighted Monte Carlo with Least Squares and Randomized Extended Kaczmarz for Option Pricing

et al. 2019

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We propose a methodology for computing single and multi-asset European option prices, and more generally expectations of scalar functions of (multivariate) random variables. This new approach combines the ability of Monte Carlo simulation to handle high-dimensional problems with the efficiency of function approximation. Specifically, we first generalize the recently developed method for multivariate integration in [arXiv:1806.05492] to integration with respect to probability measures. The method is based on the principle "approximate and integrate" in three steps i) sample the integrand at points in the integration domain, ii) approximate the integrand by solving a least-squares problem, iii) integrate the approximate function. In high-dimensional applications we face memory limitations due to large storage requirements in step ii). Combining weighted sampling and the randomized extended Kaczmarz algorithm we obtain a new efficient approach to solve large-scale least-squares problems. Our convergence and cost analysis along with numerical experiments show the effectiveness of the method in both low and high dimensions, and under the assumption of a limited number of available simulations.

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A continuous analogue of the tensor-train decomposition

Cited by 67 publications

References 44 publications

Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis

Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis

A Practical Randomized CP Tensor Decomposition

Weighted Monte Carlo with Least Squares and Randomized Extended Kaczmarz for Option Pricing

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