Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs

The numerical approximation of parametric partial differential equations $D(u,y)$ =0 is a computational challenge when the dimension $d$ of the parameter vector $y$ is large, due to the so-called $curse$ $of$ $dimensionality$. It was recently shown in [5, 6] that, for a certain class of elliptic PDEs with diffusion coefficients depending on the parameters in an affine manner, there exist polynomial approximations to the solution map $y$ → $u(y)$ with an algebraic convergence rate that is independent of the parametric dimension $d$. The analysis in [5, 6] used, however, the affine parameter dependence of the operator. The present paper proposes a strategy for establishing similar results for some classes parametric PDEs that do not necessarily fall in this category. Our approach is based on building an analytic extension $z$→ $u(z)$ of the solution map on certain tensor product of ellipses in the complex domain, and using this extension to estimate the Legendre coefficients of $u$. The varying radii of the ellipses in each coordinate $z_j$ reflect the anisotropy of the solution map with respect to the corresponding parametric variables $y_j$. This allows us to derive algebraic convergence rates for tensorized Legendre expansions in the case $d$ = ∞. We also show that such rates are preserved when using certain interpolation procedures, which is an instance of a non-intrusive method. As examples of parametric PDE’s that are covered by this approach, we consider (i) elliptic diffusion equations with coefficients that depend on the parameter vector $y$ in a not necessarily affine manner, (ii) parabolic diffusion equations with similar dependence of the coefficient on $y$, (iii) nonlinear, monotone parametric elliptic PDE’s, and (iv) elliptic equations set on a domain that is parametrized by the vector $y$ We give general strategies that allows us to derive the analytic extension in a unified abstract way for all these examples, in particular based on the holomorphic version of the implicit function theorem in Banach spaces, generalizing recent results in [13, 15]. We expect that this approach can be applied to a large variety of parametric PDEs, showing that the curse of dimensionality can be overcome under mild assumptions