The Complexity of Multivariate Elliptic Problems with Analytic Data

Abstract

Let F be a class of functions de ned on a d-dimensional domain. Our task is to compute H m-norm "-approximations to solutions of 2mth-order elliptic boundary-value problems Lu = f for a xed L and for f 2 F. W e assume that the only information we can compute about f 2 F is the value of a nite number of continuous linear functionals of f, e a c h e v aluation having cost c(d). Previous work has assumed that F was the unit ball of a Sobolev space H r of xed smoothness r, and it was found that the complexity o f computing an "-approximation was comp(" d) = (c(d)(1=") d=(r+m)). Since the exponent of 1=" depends on d, w e see that the problem is intractable in 1=" for any s u c h F of xed smoothness r. In this paper, we ask whether we can break intractability b y letting F be the unit ball of a space of in nite smoothness. To be speci c, we l e t F b e t h e u n i t b a l l o f a Hardy space of analytic functions de ned over a complex d-dimensional ball of radius greater than one. We t h e n s h o w that the problem is tractable in 1=". More precisely, w e p r o ve that comp(" d) = (c(d)(ln 1=") d), where the-constant depends on d. Since for any p > 0, there is a function K() s u c h that comp(" d) c(d)K(d)(1=") p for su ciently small ", w e see that the problem is tractable, with (minimal) exponent 0. F urthermore, we s h o w h o w to construct a nite element p-method (in the sense of Babu ska) that can compute an "-approximation with cost (c(d)(ln 1=") d). Hence this nite element method is a nearly optimal complexity algorithm for d-dimensional elliptic problems with analytic data.

The Complexity of Definite Elliptic Problems with Noisy Data

“…Most previous work (see, e.g., [10][11][12], as well as the references cited therein) on the complexity of elliptic PDEs has assumed that we have complete information about the coefficients of L, and exact (but partial) information about the right-hand side f. As a typical result, consider the 2mth order elliptic boundary value problem Lu ϭ f (with homogeneous Dirichlet boundary conditions), defined on a d-dimensional domain ⍀. The right-hand sides f belong to the unit ball BW r,p (⍀) of the Sobolev space W r,p (⍀), so that they have r derivatives in the L p sense.…”