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首页> 外文期刊>Journal of complexity >Complexity and tractability for a class of elliptic partial integro-differential equations
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Complexity and tractability for a class of elliptic partial integro-differential equations

机译：Complexity and tractability for a class of elliptic partial integro-differential equations

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摘要

We study the complexity and tractability of computing s-approximations to a simple class of elliptic partial integro-differential equations (PIDEs). Given f is an element of F-d and q is an element of Q(2d), we find u is an element of H-1 (I-d) such that -Delta u + u + T(q)u = f on the d-dimensional unit cube, subject to homogeneous Neumann boundary conditions. Here, T-q is the Fredholm integral operator with kernel q, with F-d and Q(2d) being Hilbert subspaces of the dual spaces [H-1 (I-d)]* and [H-1 (I-2d)] of H-1 (I-d) and H-1 (I-2d). We consider the worst case setting over f in the F-d unit ball and q in the Q(2d) ball of radius M-1 is an element of (0, 1), measuring error in the H-1 (I-d) norm. We show that information complexity and tractability for this problem are essentially the same as that of the [H-1 (I-d)]* approximation problem or (equivalently) the elliptic partial differential equation with operator -Delta + I under homogeneous Neumann boundary conditions. In addition, we show how a Picard iteration can overcome various implementation issues, with a penalty of at most O(ln epsilon(-1)). (C) 2021 Elsevier Inc. All rights reserved.

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