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].
References:
- M. Kocvara, D. Loghin and J.
Turner. A nonlinear domain decomposition technique for scalar
elliptic PDEs. In: J. Erhel, M. Gander, L. Halpern, G. Pichot, T.
Sassi, O. Widlund (eds): Domain Decomposition Methods in Science and
Engineering XXI, pp. 869--878, Springer, 2014.
- M. Kocvara and S. Mohammed. A first-order multigrid method for
bound-constrained convex optimization.Optimization Methods and
Software 31(3):622-644, 2016. DOI:
10.1080/10556788.2016.1146267. arXiv:1602.03771
- M. Kocvara, D. Loghin and J. Turner. Constraint interface
preconditioning for topology optimization problems. SIAM Journal
on Scientific Computation 38(1):A128-A145, 2016. arXiv:1510.04568
- M. Kocvara and S. Mohammed. Primal-dual interior-point multigrid
method for topology optimization. SIAM Journal on Scientific
Computation. 38(5):B685-B709, 2016.
Michal
Kocvara
February 3, 2017