85 research outputs found
On barrier and modified barrier multigrid methods for 3d topology optimization
One of the challenges encountered in optimization of mechanical structures,
in particular in what is known as topology optimization, is the size of the
problems, which can easily involve millions of variables. A basic example is
the minimum compliance formulation of the variable thickness sheet (VTS)
problem, which is equivalent to a convex problem. We propose to solve the VTS
problem by the Penalty-Barrier Multiplier (PBM) method, introduced by R.\
Polyak and later studied by Ben-Tal and Zibulevsky and others. The most
computationally expensive part of the algorithm is the solution of linear
systems arising from the Newton method used to minimize a generalized augmented
Lagrangian. We use a special structure of the Hessian of this Lagrangian to
reduce the size of the linear system and to convert it to a form suitable for a
standard multigrid method. This converted system is solved approximately by a
multigrid preconditioned MINRES method. The proposed PBM algorithm is compared
with the optimality criteria (OC) method and an interior point (IP) method,
both using a similar iterative solver setup. We apply all three methods to
different loading scenarios. In our experiments, the PBM method clearly
outperforms the other methods in terms of computation time required to achieve
a certain degree of accuracy
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
A Subgradient Method for Free Material Design
A small improvement in the structure of the material could save the
manufactory a lot of money. The free material design can be formulated as an
optimization problem. However, due to its large scale, second-order methods
cannot solve the free material design problem in reasonable size. We formulate
the free material optimization (FMO) problem into a saddle-point form in which
the inverse of the stiffness matrix A(E) in the constraint is eliminated. The
size of A(E) is generally large, denoted as N by N. This is the first
formulation of FMO without A(E). We apply the primal-dual subgradient method
[17] to solve the restricted saddle-point formula. This is the first
gradient-type method for FMO. Each iteration of our algorithm takes a total of
foating-point operations and an auxiliary vector storage of size O(N),
compared with formulations having the inverse of A(E) which requires
arithmetic operations and an auxiliary vector storage of size . To
solve the problem, we developed a closed-form solution to a semidefinite least
squares problem and an efficient parameter update scheme for the gradient
method, which are included in the appendix. We also approximate a solution to
the bounded Lagrangian dual problem. The problem is decomposed into small
problems each only having an unknown of k by k (k = 3 or 6) matrix, and can be
solved in parallel. The iteration bound of our algorithm is optimal for general
subgradient scheme. Finally we present promising numerical results.Comment: SIAM Journal on Optimization (accepted
A first-order multigrid method for bound-constrained convex optimization
The aim of this paper is to design an efficient multigrid method for
constrained convex optimization problems arising from discretization of some
underlying infinite dimensional problems. Due to problem dependency of this
approach, we only consider bound constraints with (possibly) a single equality
constraint. As our aim is to target large-scale problems, we want to avoid
computation of second derivatives of the objective function, thus excluding
Newton like methods. We propose a smoothing operator that only uses first-order
information and study the computational efficiency of the resulting method
- …