2,736 research outputs found
Composing Scalable Nonlinear Algebraic Solvers
Most efficient linear solvers use composable algorithmic components, with the
most common model being the combination of a Krylov accelerator and one or more
preconditioners. A similar set of concepts may be used for nonlinear algebraic
systems, where nonlinear composition of different nonlinear solvers may
significantly improve the time to solution. We describe the basic concepts of
nonlinear composition and preconditioning and present a number of solvers
applicable to nonlinear partial differential equations. We have developed a
software framework in order to easily explore the possible combinations of
solvers. We show that the performance gains from using composed solvers can be
substantial compared with gains from standard Newton-Krylov methods.Comment: 29 pages, 14 figures, 13 table
A Multigrid Optimization Algorithm for the Numerical Solution of Quasilinear Variational Inequalities Involving the -Laplacian
In this paper we propose a multigrid optimization algorithm (MG/OPT) for the
numerical solution of a class of quasilinear variational inequalities of the
second kind. This approach is enabled by the fact that the solution of the
variational inequality is given by the minimizer of a nonsmooth energy
functional, involving the -Laplace operator. We propose a Huber
regularization of the functional and a finite element discretization for the
problem. Further, we analyze the regularity of the discretized energy
functional, and we are able to prove that its Jacobian is slantly
differentiable. This regularity property is useful to analyze the convergence
of the MG/OPT algorithm. In fact, we demostrate that the algorithm is globally
convergent by using a mean value theorem for semismooth functions. Finally, we
apply the MG/OPT algorithm to the numerical simulation of the viscoplastic flow
of Bingham, Casson and Herschel-Bulkley fluids in a pipe. Several experiments
are carried out to show the efficiency of the proposed algorithm when solving
this kind of fluid mechanics problems
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
A Regularized Newton Method for Computing Ground States of Bose-Einstein condensates
In this paper, we propose a regularized Newton method for computing ground
states of Bose-Einstein condensates (BECs), which can be formulated as an
energy minimization problem with a spherical constraint. The energy functional
and constraint are discretized by either the finite difference, or sine or
Fourier pseudospectral discretization schemes and thus the original infinite
dimensional nonconvex minimization problem is approximated by a finite
dimensional constrained nonconvex minimization problem. Then an initial
solution is first constructed by using a feasible gradient type method, which
is an explicit scheme and maintains the spherical constraint automatically. To
accelerate the convergence of the gradient type method, we approximate the
energy functional by its second-order Taylor expansion with a regularized term
at each Newton iteration and adopt a cascadic multigrid technique for selecting
initial data. It leads to a standard trust-region subproblem and we solve it
again by the feasible gradient type method. The convergence of the regularized
Newton method is established by adjusting the regularization parameter as the
standard trust-region strategy. Extensive numerical experiments on challenging
examples, including a BEC in three dimensions with an optical lattice potential
and rotating BECs in two dimensions with rapid rotation and strongly repulsive
interaction, show that our method is efficient, accurate and robust.Comment: 25 pages, 6 figure
Line search multilevel optimization as computational methods for dense optical flow
We evaluate the performance of different optimization techniques developed in the context of optical flowcomputation with different variational models. In particular, based on truncated Newton methods (TN) that have been an effective approach for large-scale unconstrained optimization, we develop the use of efficient multilevel schemes for computing the optical flow. More precisely, we evaluate the performance of a standard unidirectional multilevel algorithm - called multiresolution optimization (MR/OPT), to a bidrectional multilevel algorithm - called full multigrid optimization (FMG/OPT). The FMG/OPT algorithm treats the coarse grid correction as an optimization search direction and eventually scales it using a line search. Experimental results on different image sequences using four models of optical flow computation show that the FMG/OPT algorithm outperforms both the TN and MR/OPT algorithms in terms of the computational work and the quality of the optical flow estimation
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