Optimality of adaptive Galerkin methods for random parabolic partial differential equations

Galerkin discretizations of a class of parametric and random parabolic partial differential equations (PDEs) are considered. The parabolic PDEs are assumed to depend on a vector y=(y1,y2,…) of possibly countably many parameters yj which are assumed to take values in [−1,1]. Well-posedness of weak formulations of these parametric equations in suitable Bochner spaces is established. Adaptive Galerkin discretizations of the equation based on a tensor product of a generalized polynomial chaos in the parameter domain Γ=[−1,1]N, and of suitable wavelet bases in the time interval I=[0,T] and the spatial domain D⊂Rd are proposed and their optimality is established.