A note on the complexity and tractability of the heat equation

Abstract

We wish to solve the heat equation $u_t=\mathrm{\Delta }u-\mathit{qu}$ in $I^d\times (0,T)$, where I is the unit interval and T is a maximum time value, subject to homogeneous Dirichlet boundary conditions and to initial conditions $u(·,0)=f$ over $I^d$. We show that this problem is intractable if f belongs to standard Sobolev spaces, even if we have complete information about q. However, if f and q belong to a reproducing kernel Hilbert space with finite-order weights, we can show that the problem is tractable, and can actually be strongly tractable.