15,386 research outputs found
Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices
Let \orig{A} be any matrix and let be a slight random perturbation of
\orig{A}. We prove that it is unlikely that has large condition number.
Using this result, we prove it is unlikely that has large growth factor
under Gaussian elimination without pivoting. By combining these results, we
bound the smoothed precision needed by Gaussian elimination without pivoting.
Our results improve the average-case analysis of Gaussian elimination without
pivoting performed by Yeung and Chan (SIAM J. Matrix Anal. Appl., 1997).Comment: corrected some minor mistake
Smoothed Analysis for the Conjugate Gradient Algorithm
The purpose of this paper is to establish bounds on the rate of convergence
of the conjugate gradient algorithm when the underlying matrix is a random
positive definite perturbation of a deterministic positive definite matrix. We
estimate all finite moments of a natural halting time when the random
perturbation is drawn from the Laguerre unitary ensemble in a critical scaling
regime explored in Deift et al. (2016). These estimates are used to analyze the
expected iteration count in the framework of smoothed analysis, introduced by
Spielman and Teng (2001). The rigorous results are compared with numerical
calculations in several cases of interest
Smoothed analysis of symmetric random matrices with continuous distributions
We study invertibility of matrices of the form where is an
arbitrary symmetric deterministic matrix, and is a symmetric random matrix
whose independent entries have continuous distributions with bounded densities.
We show that with high probability. The bound is
completely independent of . No moment assumptions are placed on ; in
particular the entries of can be arbitrarily heavy-tailed.Comment: Several very small revisions mad
An algebraic multigrid method for mixed discretizations of the Navier-Stokes equations
Algebraic multigrid (AMG) preconditioners are considered for discretized
systems of partial differential equations (PDEs) where unknowns associated with
different physical quantities are not necessarily co-located at mesh points.
Specifically, we investigate a mixed finite element discretization of
the incompressible Navier-Stokes equations where the number of velocity nodes
is much greater than the number of pressure nodes. Consequently, some velocity
degrees-of-freedom (dofs) are defined at spatial locations where there are no
corresponding pressure dofs. Thus, AMG approaches leveraging this co-located
structure are not applicable. This paper instead proposes an automatic AMG
coarsening that mimics certain pressure/velocity dof relationships of the
discretization. The main idea is to first automatically define coarse
pressures in a somewhat standard AMG fashion and then to carefully (but
automatically) choose coarse velocity unknowns so that the spatial location
relationship between pressure and velocity dofs resembles that on the finest
grid. To define coefficients within the inter-grid transfers, an energy
minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific
coarsening schemes and grid transfer sparsity patterns, and so it is applicable
to the proposed coarsening. Numerical results highlighting solver performance
are given on Stokes and incompressible Navier-Stokes problems.Comment: Submitted to a journa
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