1,106 research outputs found
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
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The work of Michele Benzi was supported in part by the National Science Foundation grant DMS-0511336
BDDC and FETI-DP under Minimalist Assumptions
The FETI-DP, BDDC and P-FETI-DP preconditioners are derived in a particulary
simple abstract form. It is shown that their properties can be obtained from
only on a very small set of algebraic assumptions. The presentation is purely
algebraic and it does not use any particular definition of method components,
such as substructures and coarse degrees of freedom. It is then shown that
P-FETI-DP and BDDC are in fact the same. The FETI-DP and the BDDC
preconditioned operators are of the same algebraic form, and the standard
condition number bound carries over to arbitrary abstract operators of this
form. The equality of eigenvalues of BDDC and FETI-DP also holds in the
minimalist abstract setting. The abstract framework is explained on a standard
substructuring example.Comment: 11 pages, 1 figure, also available at
http://www-math.cudenver.edu/ccm/reports
GMRES-Accelerated ADMM for Quadratic Objectives
We consider the sequence acceleration problem for the alternating direction
method-of-multipliers (ADMM) applied to a class of equality-constrained
problems with strongly convex quadratic objectives, which frequently arise as
the Newton subproblem of interior-point methods. Within this context, the ADMM
update equations are linear, the iterates are confined within a Krylov
subspace, and the General Minimum RESidual (GMRES) algorithm is optimal in its
ability to accelerate convergence. The basic ADMM method solves a
-conditioned problem in iterations. We give
theoretical justification and numerical evidence that the GMRES-accelerated
variant consistently solves the same problem in iterations
for an order-of-magnitude reduction in iterations, despite a worst-case bound
of iterations. The method is shown to be competitive against
standard preconditioned Krylov subspace methods for saddle-point problems. The
method is embedded within SeDuMi, a popular open-source solver for conic
optimization written in MATLAB, and used to solve many large-scale semidefinite
programs with error that decreases like , instead of ,
where is the iteration index.Comment: 31 pages, 7 figures. Accepted for publication in SIAM Journal on
Optimization (SIOPT
Computation of Ground States of the Gross-Pitaevskii Functional via Riemannian Optimization
In this paper we combine concepts from Riemannian Optimization and the theory
of Sobolev gradients to derive a new conjugate gradient method for direct
minimization of the Gross-Pitaevskii energy functional with rotation. The
conservation of the number of particles constrains the minimizers to lie on a
manifold corresponding to the unit norm. The idea developed here is to
transform the original constrained optimization problem to an unconstrained
problem on this (spherical) Riemannian manifold, so that fast minimization
algorithms can be applied as alternatives to more standard constrained
formulations. First, we obtain Sobolev gradients using an equivalent definition
of an inner product which takes into account rotation. Then, the
Riemannian gradient (RG) steepest descent method is derived based on projected
gradients and retraction of an intermediate solution back to the constraint
manifold. Finally, we use the concept of the Riemannian vector transport to
propose a Riemannian conjugate gradient (RCG) method for this problem. It is
derived at the continuous level based on the "optimize-then-discretize"
paradigm instead of the usual "discretize-then-optimize" approach, as this
ensures robustness of the method when adaptive mesh refinement is performed in
computations. We evaluate various design choices inherent in the formulation of
the method and conclude with recommendations concerning selection of the best
options. Numerical tests demonstrate that the proposed RCG method outperforms
the simple gradient descent (RG) method in terms of rate of convergence. While
on simple problems a Newton-type method implemented in the {\tt Ipopt} library
exhibits a faster convergence than the (RCG) approach, the two methods perform
similarly on more complex problems requiring the use of mesh adaptation. At the
same time the (RCG) approach has far fewer tunable parameters.Comment: 28 pages, 13 figure
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