39,877 research outputs found
Super-Exponential Solution in Markovian Supermarket Models: Framework and Challenge
Marcel F. Neuts opened a key door in numerical computation of stochastic
models by means of phase-type (PH) distributions and Markovian arrival
processes (MAPs). To celebrate his 75th birthday, this paper reports a more
general framework of Markovian supermarket models, including a system of
differential equations for the fraction measure and a system of nonlinear
equations for the fixed point. To understand this framework heuristically, this
paper gives a detailed analysis for three important supermarket examples: M/G/1
type, GI/M/1 type and multiple choices, explains how to derive the system of
differential equations by means of density-dependent jump Markov processes, and
shows that the fixed point may be simply super-exponential through solving the
system of nonlinear equations. Note that supermarket models are a class of
complicated queueing systems and their analysis can not apply popular queueing
theory, it is necessary in the study of supermarket models to summarize such a
more general framework which enables us to focus on important research issues.
On this line, this paper develops matrix-analytical methods of Markovian
supermarket models. We hope this will be able to open a new avenue in
performance evaluation of supermarket models by means of matrix-analytical
methods.Comment: Randomized load balancing, supermarket model, matrix-analytic method,
super-exponential solution, density-dependent jump Markov process, Batch
Markovian Arrival Process (BMAP), phase-type (PH) distribution, fixed poin
Shenfun -- automating the spectral Galerkin method
With the shenfun Python module (github.com/spectralDNS/shenfun) an effort is
made towards automating the implementation of the spectral Galerkin method for
simple tensor product domains, consisting of (currently) one non-periodic and
any number of periodic directions. The user interface to shenfun is
intentionally made very similar to FEniCS (fenicsproject.org). Partial
Differential Equations are represented through weak variational forms and
solved using efficient direct solvers where available. MPI decomposition is
achieved through the {mpi4py-fft} module (bitbucket.org/mpi4py/mpi4py-fft), and
all developed solver may, with no additional effort, be run on supercomputers
using thousands of processors. Complete solvers are shown for the linear
Poisson and biharmonic problems, as well as the nonlinear and time-dependent
Ginzburg-Landau equation.Comment: Presented at MekIT'17, the 9th National Conference on Computational
Mechanic
Fast finite difference solvers for singular solutions of the elliptic Monge-Amp\`ere equation
The elliptic Monge-Ampere equation is a fully nonlinear Partial Differential
Equation which originated in geometric surface theory, and has been applied in
dynamic meteorology, elasticity, geometric optics, image processing and image
registration. Solutions can be singular, in which case standard numerical
approaches fail. In this article we build a finite difference solver for the
Monge-Ampere equation, which converges even for singular solutions. Regularity
results are used to select a priori between a stable, provably convergent
monotone discretization and an accurate finite difference discretization in
different regions of the computational domain. This allows singular solutions
to be computed using a stable method, and regular solutions to be computed more
accurately. The resulting nonlinear equations are then solved by Newton's
method. Computational results in two and three dimensions validate the claims
of accuracy and solution speed. A computational example is presented which
demonstrates the necessity of the use of the monotone scheme near
singularities.Comment: 23 pages, 4 figures, 4 tables; added arxiv links to references, added
coment
High-frequency averaging in semi-classical Hartree-type equations
We investigate the asymptotic behavior of solutions to semi-classical
Schroedinger equations with nonlinearities of Hartree type. For a weakly
nonlinear scaling, we show the validity of an asymptotic superposition
principle for slowly modulated highly oscillatory pulses. The result is based
on a high-frequency averaging effect due to the nonlocal nature of the Hartree
potential, which inhibits the creation of new resonant waves. In the proof we
make use of the framework of Wiener algebras.Comment: 13 pages; Version 2: Added Remark 2.
Continuum Mechanics and Thermodynamics in the Hamilton and the Godunov-type Formulations
Continuum mechanics with dislocations, with the Cattaneo type heat
conduction, with mass transfer, and with electromagnetic fields is put into the
Hamiltonian form and into the form of the Godunov type system of the first
order, symmetric hyperbolic partial differential equations (SHTC equations).
The compatibility with thermodynamics of the time reversible part of the
governing equations is mathematically expressed in the former formulation as
degeneracy of the Hamiltonian structure and in the latter formulation as the
existence of a companion conservation law. In both formulations the time
irreversible part represents gradient dynamics. The Godunov type formulation
brings the mathematical rigor (the well-posedness of the Cauchy initial value
problem) and the possibility to discretize while keeping the physical content
of the governing equations (the Godunov finite volume discretization)
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