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

Abstract

We study the complexity and tractability of computing ε-approximations to a simple class of elliptic partial integro-differential equations (PIDEs). Given f∈Fd and q∈Q2d, we find u∈H1(Id) such that −Δu+u+Tqu=f on the d-dimensional unit cube, subject to homogeneous Neumann boundary conditions. Here, Tq is the Fredholm integral operator with kernel q, with Fd and Q2d being Hilbert subspaces of the dual spaces [H1(Id)]⁎ and [H1(I2d)]⁎ of H1(Id) and H1(I2d). We consider the worst case setting over f in the Fd unit ball and q in the Q2d ball of radius M1∈(0,1), measuring error in the H1(Id) norm. We show that information complexity and tractability for this problem are essentially the same as that of the [H1(Id)]⁎ approximation problem or (equivalently) the elliptic partial differential equation with operator −Δ+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⁡ε−1).