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 Mat. Sb., 2019, Volume 210, Number 4, Pages 103–127 (Mi msb9068)

Linear collective collocation approximation for parametric and stochastic elliptic PDEs

Dinh Dũng

Information Technology Institute, Vietnam National University, Hanoi, Vietnam

Abstract: Consider the parametric elliptic problem
$$-\operatorname{div}(a(y)(x)\nabla u(y)(x))=f(x),\qquad x\in D,\quad y\in\mathbb I^\infty,\quad u|_{\partial D}=0,$$
where $D\subset\mathbb R^m$ is a bounded Lipschitz domain, $\mathbb I^\infty:=[-1,1]^\infty$, $f\in L_2(D)$, and the diffusion coefficients $a$ satisfy the uniform ellipticity assumption and are affinely dependent on $y$. The parameter $y$ can be interpreted as either a deterministic or a random variable. A central question to be studied is as follows. Assume that there is a sequence of approximations with a certain error convergence rate in the energy norm of the space $V:=H^1_0(D)$ for the nonparametric problem $-\operatorname{div}(a(y_0)(x)\nabla u(y_0)(x))=f(x)$ at every point $y_0\in\mathbb I^\infty$. Then under what assumptions does this sequence induce a sequence of approximations with the same error convergence rate for the parametric elliptic problem in the norm of the Bochner spaces $L_\infty(\mathbb I^\infty,V)$? We have solved this question using linear collective collocation methods, based on Lagrange polynomial interpolation on the parametric domain $\mathbb I^\infty$. Under very mild conditions, we show that these approximation methods give the same error convergence rate as for the nonparametric elliptic problem. In this sense the curse of dimensionality is broken by linear methods.
Bibliography: 22 titles.

Keywords: high-dimensional problems, parametric and stochastic elliptic PDEs, linear collective collocation approximation, affine dependence of the diffusion coefficients.

 Funding Agency Grant Number National Foundation for Science and Technology Development Vietnam 102.01-2017.05 This work was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant no. 102.01-2017.05.

DOI: https://doi.org/10.4213/sm9068

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English version:
Sbornik: Mathematics, 2019, 210:4, 565–588

Bibliographic databases:

UDC: 517.954+517.518
MSC: 41A10, 65N35, 65N30, 65N15, 65L10, 65D05, 65C30

Citation: Dinh Dũng, “Linear collective collocation approximation for parametric and stochastic elliptic PDEs”, Mat. Sb., 210:4 (2019), 103–127; Sb. Math., 210:4 (2019), 565–588

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb9068
• https://doi.org/10.4213/sm9068
• http://mi.mathnet.ru/eng/msb/v210/i4/p103

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This publication is cited in the following articles:
1. Zech J., Dung D., Schwab Ch., “Multilevel Approximation of Parametric and Stochastic Pdes”, Math. Models Meth. Appl. Sci., 29:9 (2019), 1753–1817
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