Algorithms and Software for Large-Scale VIs: Differential variational inequalities contain two terms: a differential equation and a generalized algebraic equation represented by complementarity constraints or a box-constrained variational inequality, for example, 0 ≤ u ≤ 1, that formalizes the concept of switches.  Solving the differential variational inequality can yield a more accurate solution in less time than a smoothed counterpart.  A comprehensive theory for DVIs has been developed,1 and algorithms developed by the optimization community over the last two decades for solving complementary problems based on a formulating them as nonsmooth systems of equations, have been successfully applied to solve applications.

Efficient methods for solving the resulting VIs on massively parallel machines are now available in the development version of PETSc (petsc-dev), leveraging experience in TAO and PATH. Initial capabilities include semi-smooth and reduced-space active set VI solvers, as initially motivated by heterogeneous materials problems, 2 and improved methods and implementations (including preconditioners, nonlinear solvers, and adaptive mesh refinement) are actively being pursued.