Adaptive wavelet methods for elliptic partial differential equations with random operators

Optimality of adaptive Galerkin methods for random parabolic partial differential equations

“…This is done in [4]; see also [8,15] for a variant of this method with quantitative improvements. All of these algorithms partition the argument w in (6.1) according to the modulus of its entries.…”

“…onto a finite span from a countable ensemble of tensorized orthogonal polynomials, in principle techniques for establishing optimality of Galerkin projection methods for the approximate solution of operator equations can be brought to bear. This programme has been implemented in [15] and the references therein for parametric operator equations.…”

“…Section 7 recapitulates from [15] general results on the optimality of adaptive Galerkin approximations of deterministic operator equations. Finally, Section 8 contains statements and proofs of the main result of the present paper, the optimality of the proposed adaptive Galerkin approximations in space, time and parameter domain by sparse, tensorized bases consisting of tensor products of Riesz bases Θ, Σ and of P .…”

“…The analysis differs substantially from the standard analysis of the power iteration for matrices since, in the present infinite-dimensional setting, no gap between the largest and second-largest eigenvalues is assumed. We refer to [15] for further details.…”

“…In order to derive s * -compressibility of B, we assume that the estimate (6.7) holds uniformly for all (n m,j ) j∈N , i.e. The following theorem is the first case of [15,Thm. 8.4].…”