Multilevel and Domain Decomposition Methods in Optimization

The goal of this project is to utilize efficient methods of numerical linear algebra (namely multigrid and domain decomposition) in large-scale numerical optimization. So far we targeted problems of PDE constrained optimization, however, our aim is to extend the approach to general purpose large-scale optimization.

In [1] we propose a new nonlinear domain decomposition technique for scalar elliptic PDEs.

In [2] we solve convex optimization problems with bound constraints and a possible linear equality constraint using nonlinear multigrid techniques. We generalize the multigrid algorithms developed for linear complementarity problems. Our aim was to use solely first-order information, thus avoiding expensive Hessian computation.

In [3-4] we solve the specific problem of topology optimization, a (possibly very-) large-scale convex optimization problem with nonlinear constraints. The problem is solved by interior-point method, whereas the (Newton) linear systems are solved by Krylov-subspace iterative methods preconditioned by domain decomposition techniques [3] or by multigrid [4].