Note: A note on the complexity and tractability of the heat equation

Authors:
Arthur G. Werschulz
Affiliations:
Department of Computer and Information Sciences, Fordham University, New York, NY 10023, USA and Department of Computer Science, Columbia University, New York, NY 10027, USA
Venue:
Journal of Complexity
Year:
2007

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Abstract

We wish to solve the heat equation u"t=@Du-qu in I^dx(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.