198,077 research outputs found
Lectures on Randomized Numerical Linear Algebra
This chapter is based on lectures on Randomized Numerical Linear Algebra from
the 2016 Park City Mathematics Institute summer school on The Mathematics of
Data.Comment: To appear in the edited volume of lectures from the 2016 PCMI summer
schoo
Fast randomized iteration: diffusion Monte Carlo through the lens of numerical linear algebra
We review the basic outline of the highly successful diffusion Monte Carlo
technique commonly used in contexts ranging from electronic structure
calculations to rare event simulation and data assimilation, and propose a new
class of randomized iterative algorithms based on similar principles to address
a variety of common tasks in numerical linear algebra. From the point of view
of numerical linear algebra, the main novelty of the Fast Randomized Iteration
schemes described in this article is that they work in either linear or
constant cost per iteration (and in total, under appropriate conditions) and
are rather versatile: we will show how they apply to solution of linear
systems, eigenvalue problems, and matrix exponentiation, in dimensions far
beyond the present limits of numerical linear algebra. While traditional
iterative methods in numerical linear algebra were created in part to deal with
instances where a matrix (of size ) is too big to store, the
algorithms that we propose are effective even in instances where the solution
vector itself (of size ) may be too big to store or manipulate.
In fact, our work is motivated by recent DMC based quantum Monte Carlo schemes
that have been applied to matrices as large as . We
provide basic convergence results, discuss the dependence of these results on
the dimension of the system, and demonstrate dramatic cost savings on a range
of test problems.Comment: 44 pages, 7 figure
Two-dimensional Banach spaces with Polynomial numerical index zero
We study two-dimensional Banach spaces with polynomial numerical indices
equal to zero.Comment: 12 pages, to appear in Linear Algebra App
Gaussian elimination as an iterative algorithm
Gaussian elimination (GE) for solving an linear system of equations is the archetypical direct method of numerical linear algebra, as opposed to iterative. In this note we want to point out that GE has an iterative side too
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