Volume 6, Issue 2
Stochastic Collocation via $l_1$-Minimisation on Low Discrepancy Point Sets with Application to Uncertainty Quantification

East Asian J. Appl. Math., 6 (2016), pp. 171-191.

Published online: 2018-02

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• Abstract

Various numerical methods have been developed in order to solve complex systems with uncertainties, and the stochastic collocation method using $ℓ_1$- minimisation on low discrepancy point sets is investigated here. Halton and Sobol’ sequences are considered, and low discrepancy point sets and random points are compared. The tests discussed involve a given target function in polynomial form, high-dimensional functions and a random ODE model. Our numerical results show that the low discrepancy point sets perform as well or better than random sampling for stochastic collocation via $ℓ_1$-minimisation.

• Keywords

Stochastic collocation, Quasi-Monte Carlo sequence, low discrepancy point sets, Legendre polynomials, $ℓ_1$-minimisation.

41A10, 60H35, 65N35

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@Article{EAJAM-6-171, author = {}, title = {Stochastic Collocation via $l_1$-Minimisation on Low Discrepancy Point Sets with Application to Uncertainty Quantification}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {6}, number = {2}, pages = {171--191}, abstract = {

Various numerical methods have been developed in order to solve complex systems with uncertainties, and the stochastic collocation method using $ℓ_1$- minimisation on low discrepancy point sets is investigated here. Halton and Sobol’ sequences are considered, and low discrepancy point sets and random points are compared. The tests discussed involve a given target function in polynomial form, high-dimensional functions and a random ODE model. Our numerical results show that the low discrepancy point sets perform as well or better than random sampling for stochastic collocation via $ℓ_1$-minimisation.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.090615.060216a}, url = {http://global-sci.org/intro/article_detail/eajam/10787.html} }
TY - JOUR T1 - Stochastic Collocation via $l_1$-Minimisation on Low Discrepancy Point Sets with Application to Uncertainty Quantification JO - East Asian Journal on Applied Mathematics VL - 2 SP - 171 EP - 191 PY - 2018 DA - 2018/02 SN - 6 DO - http://doi.org/10.4208/eajam.090615.060216a UR - https://global-sci.org/intro/article_detail/eajam/10787.html KW - Stochastic collocation, Quasi-Monte Carlo sequence, low discrepancy point sets, Legendre polynomials, $ℓ_1$-minimisation. AB -

Various numerical methods have been developed in order to solve complex systems with uncertainties, and the stochastic collocation method using $ℓ_1$- minimisation on low discrepancy point sets is investigated here. Halton and Sobol’ sequences are considered, and low discrepancy point sets and random points are compared. The tests discussed involve a given target function in polynomial form, high-dimensional functions and a random ODE model. Our numerical results show that the low discrepancy point sets perform as well or better than random sampling for stochastic collocation via $ℓ_1$-minimisation.

Yongle Liu & Ling Guo. (2020). Stochastic Collocation via $l_1$-Minimisation on Low Discrepancy Point Sets with Application to Uncertainty Quantification. East Asian Journal on Applied Mathematics. 6 (2). 171-191. doi:10.4208/eajam.090615.060216a
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