A note on the complexity and tractability of the heat equation

We wish to solve the heat equation u t = Δ u - qu in I d × ( 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 standa...

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Bibliographic Details
Published inJournal of Complexity Vol. 23; no. 4; pp. 553 - 559
Main Author Werschulz, Arthur G.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.08.2007
Subjects
Heat equation
Information-based complexity
Tractability
Weighted reproducing kernel Hilbert spaces
Tractability
Weighted reproducing kernel Hilbert spaces
Heat equation
Information-based complexity
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Summary:We wish to solve the heat equation u t = Δ u - qu in I d × ( 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.
ISSN:0885-064X
1090-2708
DOI:10.1016/j.jco.2007.01.006