Non-intrusive low-rank separated approximation of high-dimensional stochastic models

By Alireza Doostan, Abdoulahad Validi, Gianluca Iaccarino. 2012

103 citations recorded by the source corpus.

This work proposes a sampling-based (non-intrusive) approach within the context of low-rank separated representations to tackle the issue of curse-of-dimensionality associated with the solution of models, e.g., PDEs/ODEs, with high-dimensional random inputs. Under some conditions discussed in details, the number of random realizations of the solution, required for a successful approximation, grows linearly with respect to the number of random inputs. The construction of the separated representation is achieved via a regularized alternating least-squares regression, together with an error indicator to estimate model parameters. The computational complexity of such a construction is quadratic in the number of random inputs. The performance of the method is investigated through its application to three numerical examples including two ODE problems with high-dimensional random inputs.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Statistics and Probability

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “…e.g., based on Hartree-Fock or DFT models [23,24,31,44,45,47,87,141–146,150,156–159]; • the solution of stochastic and parametric PDEs [68,69,76,77,161,165,171,189,202]; • approximation of Green’s functions of high-dimensional PDEs [30,113]; • the solution of the Boltzmann equation [129,…”
Low-rank methods for high-dimensional approximation and model order reduction (2015, unknown): “l subspace of L2 µ ν(Ξν) with basis Ψ (ν) = {ψ(ν) k }k ∈I . 5.2.1 Least-squares The standard discrete least-squares method for the approximation of gin a subset Sr of low-rank tensors in XI (see e.g. [7, 33, 19]) consists in solving min h∈Sr kg−hk2 2,K, with kg−hk22,K = 1 K XK k=1 (g(yk) −h(yk))2, w”
Polynomial-Chaos-based Kriging (2015, methodology): “…regression algorithms have been proposed in the literature, namely the least absolute shrinkage operator (LASSO) Tibshirani (1996), the Least Angle Regression (LAR) Efron et al. (2004), low-rank approximations Doostan et al. (2013); Peng et al. (2014) and compressive sensing Sargsyan et al. (2014).”
Numerical construction of Green's functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations (2012, background): “This is particularly useful in the context of solving stochastic ordinary/partial differential equations (SODE’s/SPDE’s): in such problems, the presence of random inputs can lead lead to rapid growth in the formal dimension [150, 80, 46].”