A note on the complexity and tractability of the heat equation
We wish to solve the heat equation ut = Delta u - qu in I-d x (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. (c) 2007 Elsevier Inc. All rights reserved.
tractability heat equation information-based complexity weighted reproducing kernel hilbert spaces