Constraint interface preconditioning for topology optimization problems

The discretization of constrained nonlinear optimization problems arising in the field of topology optimization yields algebraic systems which are challenging to solve in practice, due to pathological ill-conditioning, strong nonlinearity and size. In this work we propose a methodology which brings together existing fast algorithms, namely, interior-point for the optimization problem and a novel substructuring domain decomposition method for the ensuing large-scale linear systems. The main contribution is the choice of interface preconditioner which allows for the acceleration of the domain decomposition method, leading to performance independent of problem size.Comment: To be published in SIAM J. Sci. Com

Some Preconditioning Techniques for Saddle Point Problems

Saddle point problems arise frequently in many applications in science and engineering, including constrained optimization, mixed finite element formulations of partial differential equations, circuit analysis, and so forth. Indeed the formulation of most problems with constraints gives rise to saddle point systems. This paper provides a concise overview of iterative approaches for the solution of such systems which are of particular importance in the context of large scale computation. In particular we describe some of the most useful preconditioning techniques for Krylov subspace solvers applied to saddle point problems, including block and constrained preconditioners.\ud \ud The work of Michele Benzi was supported in part by the National Science Foundation grant DMS-0511336

The antitriangular factorisation of saddle point matrices

Mastronardi and Van Dooren recently introduced the block antitriangular ("Batman") decomposition for symmetric indefinite matrices. Here we show the simplification of this factorisation for saddle point matrices and demonstrate how it represents the common nullspace method. We show the relation of this factorisation to constraint preconditioning and how it transforms but preserves the block diagonal structure of block diagonal preconditioning

Incomplete factorization constraint preconditioners for saddle-point matrices

We consider the application of the conjugate gradient method to the solution of large symmetric, indefinite linear systems. Special emphasis is put on the use of constraint preconditioners and a new factorization that can reduce the number of flops required by the preconditioning step. Results concerning the eigenvalues of the preconditioned matrix and its minimum polynomial are given. Numerical experiments validate these conclusions

Randomized Riemannian Preconditioning for Orthogonality Constrained Problems

Optimization problems with (generalized) orthogonality constraints are prevalent across science and engineering. For example, in computational science they arise in the symmetric (generalized) eigenvalue problem, in nonlinear eigenvalue problems, and in electronic structures computations, to name a few problems. In statistics and machine learning, they arise, for example, in canonical correlation analysis and in linear discriminant analysis. In this article, we consider using randomized preconditioning in the context of optimization problems with generalized orthogonality constraints. Our proposed algorithms are based on Riemannian optimization on the generalized Stiefel manifold equipped with a non-standard preconditioned geometry, which necessitates development of the geometric components necessary for developing algorithms based on this approach. Furthermore, we perform asymptotic convergence analysis of the preconditioned algorithms which help to characterize the quality of a given preconditioner using second-order information. Finally, for the problems of canonical correlation analysis and linear discriminant analysis, we develop randomized preconditioners along with corresponding bounds on the relevant condition number