Novel Discontinuous Galerkin Methods for Deterministic and Stochastic Optimization Problems with Inequality Constraints

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The project aims to develop new numerical methods for solving optimization problems that have applications in elasticity theory, fluid filtration in porous media, constrained heating, cancer therapy, shape optimization, and financial mathematics. The computational simulations from this project will provide insights on the understanding of complicated physical models with random perturbations. Another emphasis of this project will be the training of graduate students in numerical methods and their analysis while also training the students in theory. The students will further be trained in the efficient implementation of the computer codes so that they are better prepared for careers in industry.The project is on the design, implementation, and rigorous analysis of a new class of discontinuous Galerkin (DG) methods for variational inequalities and optimal control problems with inequality constraints that are fundamental for the modeling of nonlinear problems arising from applications in materials science, mechanical engineering, shape optimization, and financial science. Furthermore, the underlying problems may involve small parameters and random perturbations such that the complete numerical analyses are more subtle. The formulations of classical DG methods usually require large positive penalty parameters that depend on the shape regularity of the mesh and other unknown constants. The project will design novel DG methods for variational inequalities, optimal control problems with partial differential equations constraints, and related singularly perturbed and stochastically perturbed problems. Another goal of the project is to design robust, reliable, and efficient a posteriori error estimators for the corresponding deterministic and stochastic problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.