  Numerically solving high-dimensional random parametric PDEs poses a
challenging computational problem. It is well-known that numerical methods can
greatly benefit from adaptive refinement algorithms, in particular when
functional approximations in polynomials are computed as in stochastic Galerkin
finite element methods. This work investigates a residual based adaptive
algorithm, akin to classical adaptive FEM, used to approximate the solution of
the stationary diffusion equation with lognormal coefficients, i.e. with a
non-affine parameter dependence of the data. It is known that the refinement
procedure is reliable but the theoretical convergence of the scheme for this
class of unbounded coefficients remains a challenging open question. This paper
advances the theoretical state-of-the-art by providing a quasi-error reduction
result for the adaptive solution of the lognormal stationary diffusion problem.
The presented analysis generalizes previous results in that guaranteed
convergence for uniformly bounded coefficients follows directly as a corollary.
Moreover, it highlights the fundamental challenges with unbounded coefficients
that cannot be overcome with common techniques. A computational benchmark
example illustrates the main theoretical statement.
