The numerical approximation of parametric partial differential equations is a
computational challenge, in particular when the number of involved parameter is large.
This paper considers a model class of second order, linear, parametric, elliptic PDEs on a
bounded domain D with diffusion coefficients depending on the parameters
in an affine manner. For such models, it was shown in [9, 10] that under very weak assumptions
on the diffusion coefficients, the entire family of solutions to such equations can be
simultaneously approximated in the Hilbert space
V = H01(D) by multivariate sparse polynomials in the parameter
vector y with a controlled number N of terms. The
convergence rate in terms of N does not depend on the number of
parameters in V, which may be arbitrarily large or countably infinite,
thereby breaking the curse of dimensionality. However, these approximation results do not
describe the concrete construction of these polynomial expansions, and should therefore
rather be viewed as benchmark for the convergence analysis of numerical methods. The
present paper presents an adaptive numerical algorithm for constructing a sequence of
sparse polynomials that is proved to converge toward the solution with the optimal
benchmark rate. Numerical experiments are presented in large parameter dimension, which
confirm the effectiveness of the adaptive approach.