Multigrid algorithms based on cache aware smoothers produce a high performance, portable, cache aware multigrid solver for problems with one or more degrees of freedom on unstructured grids. This paper analyzes the work cost of a necessary preprocessing step in cache aware multigrid. This mesh renumbering step is not present in standard multigrid codes. The cost is found to be small relative to...

Seidel and Smith have constructed an invariant of links as the Floer cohomology for two Lagrangians inside a complex affine variety Y. This variety is the intersection of a semisimple orbit with a transverse slice at a nilpotent in the Lie algebra sl2m. We exhibit bijections between a set of generators for the Seidel-Smith cochain complex, the generators in Bigelow’s picture of the Jones polyno...

A fourth-order compact difference scheme with unequal mesh sizes in different coordinate directions is employed to discretize a two-dimensional Poisson equation in a rectangular domain. Multigrid methods using a partial semicoarsening strategy and line Gauss–Seidel relaxation are designed to solve the resulting sparse linear systems. Numerical experiments are conducted to test the accuracy of t...

In this note, we introduce a relative (or Lagrangian) version of the Seidel homomorphism that assigns to each homotopy class of paths in Ham(M), starting at the identity and ending on the subgroup that preserves a given Lagrangian submanifold L, an element in the Floer homology of L. We show that these elements are related to the absolute Seidel elements by the Albers map. We also study, for la...

For the linear least squares problem with coefficient matrix columns being highly correlated, we develop a greedy randomized Gauss-Seidel method oblique direction. Then corresponding convergence result is deduced. Numerical examples demonstrate that our proposed superior to and

Short and accurate analytic formulas for the Seidel coefficients of gradient-index lenses with arbitrary radial refractive-index distributions have been obtained. Starting from analytic ray-tracing formulas, we have developed a technique for decomposing the two components of the transverse aberration of an arbitrary skew ray in surface and inhomogeneous transfer contributions and a technique fo...

We consider Newton-Krylov methods for solving discretized compressible Euler equations. A good preconditioner in the Krylov subspace method is crucial for the efficiency of the solver. In this paper we consider a point-block Gauss-Seidel method as preconditioner. We describe and compare renumbering strategies that aim at improving the quality of this preconditioner. A variant of reordering meth...