Linear collective collocation approximation for parametric and stochastic elliptic PDEs

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.

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