80 research outputs found
Approach to self-similarity in Smoluchowski's coagulation equations
We consider the approach to self-similarity (or dynamical scaling) in
Smoluchowski's equations of coagulation for the solvable kernels ,
and . In addition to the known self-similar solutions with
exponential tails, there are one-parameter families of solutions with algebraic
decay, whose form is related to heavy-tailed distributions well-known in
probability theory. For K=2 the size distribution is Mittag-Leffler, and for
and it is a power-law rescaling of a maximally skewed
-stable Levy distribution. We characterize completely the domains of
attraction of all self-similar solutions under weak convergence of measures.
Our results are analogous to the classical characterization of stable
distributions in probability theory. The proofs are simple, relying on the
Laplace transform and a fundamental rigidity lemma for scaling limits.Comment: Latex2e, 42 pages with 1 figur
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
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