Contractivity of Runge-Kutta methods for convex gradient systems

J.M. Sanz-Serna, Konstantinos C Zygalakis

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the application of Runge-Kutta (RK) methods to gradient systems (d/dt)x=−∇V(x), where, as in many optimization problems, V is convex and ∇V (globally) Lipschitz-continuous with Lipschitz constant L. Solutions of this system behave contractively, i.e. the Euclidean distance between two solutions x(t) and x˜(t) is a nonincreasing function of t. It is then of interest to investigate whether a similar contraction takes place, at least for suitably small step sizes h, for the discrete solution. Dahlquist and Jeltsch results' imply that (1) there are explicit RK schemes that behave contractively whenever Lh is below a scheme-dependent constant and (2) Euler's rule is optimal in this regard. We prove however, by explicit construction of a convex potential using ideas from robust control theory, that there exists RK schemes that fail to behave contractively for any choice of the time-step h.
Original languageEnglish
Pages (from-to)2079–2092
Number of pages13
JournalSiam journal on numerical analysis
Volume58
Issue number4
DOIs
Publication statusPublished - 13 Jul 2020

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