Algorithms and Software for Large-Scale VIs: DifferentialÂ variational inequalities contain two terms: a differential equationÂ and a generalized algebraic equation represented by complementarityÂ constraints or a box-constrained variational inequality, for example,Â 0 â¤ u â¤ 1, that formalizes the concept of switches. Â SolvingÂ the differential variational inequality can yield a more accurateÂ solution in less time than a smoothed counterpart. Â A comprehensiveÂ theory for DVIs has been developed,1 and algorithms developed by theÂ optimization community over the last two decades for solving complementary problems based on a formulating them as nonsmoothÂ systems of equations, have been successfully applied to solveÂ applications.
Efficient methods for solving the resulting VIs onÂ massively parallel machines are now available in the development version of PETSc (petsc-dev),Â leveraging experience in TAO and PATH. Initial capabilities include semi-smooth and reduced-space active set VI solvers,Â asÂ initially motivated by heterogeneous materials problems, 2 and improvedÂ methods and implementations (including preconditioners, nonlinearÂ solvers, and adaptive mesh refinement) are actively being pursued.
- 1. J.-S. Pang and D. E. Stewart, Differential Variational Inequalities, Mathematical Programming, vol 113, number 2, 345-424. â
- 2. L. Wang, J. Lee, M. Anitescu, A. El Azab, L. C. McInnes, T. Munson, and B. Smith, A Differential Variational Inequality Approach for the Simulation of Heterogeneous Materials, Proceedings of SciDAC2011Â Conference, Denver, CO, July 10-14, 2011, also available as Preprint ANL/MCS-P1895-0511.â