890 research outputs found
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
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
Accelerated graph-based spectral polynomial filters
Graph-based spectral denoising is a low-pass filtering using the
eigendecomposition of the graph Laplacian matrix of a noisy signal. Polynomial
filtering avoids costly computation of the eigendecomposition by projections
onto suitable Krylov subspaces. Polynomial filters can be based, e.g., on the
bilateral and guided filters. We propose constructing accelerated polynomial
filters by running flexible Krylov subspace based linear and eigenvalue solvers
such as the Block Locally Optimal Preconditioned Conjugate Gradient (LOBPCG)
method.Comment: 6 pages, 6 figures. Accepted to the 2015 IEEE International Workshop
on Machine Learning for Signal Processin
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