Past

The use of weighted matchings is becoming increasingly standard in the

solution of sparse linear systems. While non-symmetric permutations based on these

matchings have been the state-of-the-art for

several years (especially for direct solvers), approaches for symmetric

matrices have only recently gained attention.

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In this talk we discuss results of our work on using weighted matchings in

the preconditioning of symmetric indefinite linear systems, following ideas

introduced by Duff and Gilbert. In order to maintain symmetry,

the weighted matching is symmetrized and the cycle structure of the

resulting matching is used to build reorderings that form small diagonal

blocks from the matched entries.

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For the preconditioning we investigated two approaches. One is an

incomplete $LDL^{T}$ preconditioning, that chooses 1x1 or 2x2 diagonal pivots

based on a simple tridiagonal pivoting criterion. The second approach

targets distributed computing, and is based on factorized sparse approximate

inverses, whose existence, in turn, is based on the existence of an $LDL^{T}$

factorization. Results for a number of comprehensive test sets are given,

including comparisons with sparse direct solvers and other preconditioning

approaches.

Coherent structures, or defects, are interfaces between wave trains with

possibly different wavenumbers: they are time-periodic in an appropriate

coordinate frame and connect two, possibly different, spatially-periodic

travelling waves. We propose a classification of defects into four

different classes which have all been observed experimentally. The

characteristic distinguishing these classes is the sign of the group

velocities of the wave trains to either side of the defect, measured

relative to the speed of the defect. Using a spatial-dynamics description

in which defects correspond to homoclinic and heteroclinic orbits, we then

relate robustness properties of defects to their spectral stability

properties. If time permits, we will also discuss how defects interact with

each other.

We describe a nice example of duality between coagulation and fragmentation associated with certain Dirichlet distributions. The fragmentation and coalescence chains we derive arise naturally in the context of the genealogy of Yule processes.

We propose two models of the evolution of a pair of competing populations. Both are lattice based. The first is a compromise between fully spatial models, which do not appear amenable to analytic results, and interacting particle system models, which don't, at present, incorporate all the competitive strategies that a population might adopt. The second is a simplification of the first in which competition is only supposed to act within lattice sites and the total population size within each lattice point is a constant. In a special case, this second model is dual to a branching-annihilating random walk. For each model, using a comparison with N-dependent oriented percolation, we show that for certain parameter values both populations will coexist for all time with positive probability.

As a corollary we deduce survival for all time of branching annihilating random walk for sufficiently large branching rates.

We also present conjectures relating to the role of space in the survival probabilities for the two populations.

/notices/abstracts/stochastic-analysis/ht04/crisan.shtml

Ergodic Markov processes possess invariant measures. In the case if transition probabilities or SDE coefficients depend on a parameter, it is important to know whether these measures depend regularly on this parameter. Results of this kind will be discussed. Another close topic is whether approximations to Markov diffusions possess ergodic properties similar to those of the limiting processes. Some partial answer to this question will be presented.

An interior-point method for solving mathematical programs with

equilibrium constraints (MPECs) is proposed. At each iteration of the

algorithm, a single primal-dual step is computed from each subproblem of

a sequence. Each subproblem is defined as a relaxation of the MPEC with

a nonempty strictly feasible region. In contrast to previous

approaches, the proposed relaxation scheme preserves the nonempty strict

feasibility of each subproblem even in the limit. Local and superlinear

convergence of the algorithm is proved even with a less restrictive

strict complementarity condition than the standard one. Moreover,

mechanisms for inducing global convergence in practice are proposed.

Numerical results on the MacMPEC test problem set demonstrate the

fast-local convergence properties of the algorithm.