Convergence of regularized time-stepping methods for differential variational inequalities

Xiaojun Chen, Zhengyu Wang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

30 Citations (Scopus)

Abstract

This paper provides convergence analysis of regularized time-stepping methods for the differential variational inequality (DVI), which consists of a system of ordinary differential equations and a parametric variational inequality (PVI) as the constraint. The PVI often has multiple solutions at each step of a time-stepping method, and it is hard to choose an appropriate solution for guaranteeing the convergence. In [L. Han, A. Tiwari, M. K. Camlibel and J.-S. Pang, SIAM J. Numer. Anal., 47 (2009) pp. 3768-3796], the authors proposed to use "least-norm solutions" of parametric linear complementarity problems at each step of the time-stepping method for the monotone linear complementarity system and showed the novelty and advantages of the use of the least-norm solutions. However, in numerical implementation, when the PVI is not monotone and its solution set is not convex, finding a least-norm solution is difficult. This paper extends the Tikhonov regularization approximation to the P0-function DVI, which ensures that the PVI has a unique solution at each step of the regularized time-stepping method. We show the convergence of the regularized time-stepping method to a weak solution of the DVI and present numerical examples to illustrate the convergence theorems.
Original languageEnglish
Pages (from-to)1647-1671
Number of pages25
JournalSIAM Journal on Optimization
Volume23
Issue number3
DOIs
Publication statusPublished - 29 Oct 2013

Keywords

  • Differential variational inequalities
  • Epiconvergence
  • P -function 0
  • Tikhonov regularization

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science

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