27,662 research outputs found
Numerical Methods for Solving Convection-Diffusion Problems
Convection-diffusion equations provide the basis for describing heat and mass
transfer phenomena as well as processes of continuum mechanics. To handle flows
in porous media, the fundamental issue is to model correctly the convective
transport of individual phases. Moreover, for compressible media, the pressure
equation itself is just a time-dependent convection-diffusion equation.
For different problems, a convection-diffusion equation may be be written in
various forms. The most popular formulation of convective transport employs the
divergent (conservative) form. In some cases, the nondivergent (characteristic)
form seems to be preferable. The so-called skew-symmetric form of convective
transport operators that is the half-sum of the operators in the divergent and
nondivergent forms is of great interest in some applications.
Here we discuss the basic classes of discretization in space: finite
difference schemes on rectangular grids, approximations on general polyhedra
(the finite volume method), and finite element procedures. The key properties
of discrete operators are studied for convective and diffusive transport. We
emphasize the problems of constructing approximations for convection and
diffusion operators that satisfy the maximum principle at the discrete level
--- they are called monotone approximations.
Two- and three-level schemes are investigated for transient problems.
Unconditionally stable explicit-implicit schemes are developed for
convection-diffusion problems. Stability conditions are obtained both in
finite-dimensional Hilbert spaces and in Banach spaces depending on the form in
which the convection-diffusion equation is written
Convergent finite difference methods for one-dimensional fully nonlinear second order partial differential equations
This paper develops a new framework for designing and analyzing convergent
finite difference methods for approximating both classical and viscosity
solutions of second order fully nonlinear partial differential equations (PDEs)
in 1-D. The goal of the paper is to extend the successful framework of
monotone, consistent, and stable finite difference methods for first order
fully nonlinear Hamilton-Jacobi equations to second order fully nonlinear PDEs
such as Monge-Amp\`ere and Bellman type equations. New concepts of consistency,
generalized monotonicity, and stability are introduced; among them, the
generalized monotonicity and consistency, which are easier to verify in
practice, are natural extensions of the corresponding notions of finite
difference methods for first order fully nonlinear Hamilton-Jacobi equations.
The main component of the proposed framework is the concept of "numerical
operator", and the main idea used to design consistent, monotone and stable
finite difference methods is the concept of "numerical moment". These two new
concepts play the same roles as the "numerical Hamiltonian" and the "numerical
viscosity" play in the finite difference framework for first order fully
nonlinear Hamilton-Jacobi equations. In the paper, two classes of consistent
and monotone finite difference methods are proposed for second order fully
nonlinear PDEs. The first class contains Lax-Friedrichs-like methods which also
are proved to be stable and the second class contains Godunov-like methods.
Numerical results are also presented to gauge the performance of the proposed
finite difference methods and to validate the theoretical results of the paper.Comment: 23 pages, 8 figues, 11 table
Monotone Hurwitz numbers in genus zero
Hurwitz numbers count branched covers of the Riemann sphere with specified
ramification data, or equivalently, transitive permutation factorizations in
the symmetric group with specified cycle types. Monotone Hurwitz numbers count
a restricted subset of the branched covers counted by the Hurwitz numbers, and
have arisen in recent work on the the asymptotic expansion of the
Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study
of monotone Hurwitz numbers. We prove two results that are reminiscent of those
for classical Hurwitz numbers. The first is the monotone join-cut equation, a
partial differential equation with initial conditions that characterizes the
generating function for monotone Hurwitz numbers in arbitrary genus. The second
is our main result, in which we give an explicit formula for monotone Hurwitz
numbers in genus zero.Comment: 22 pages, submitted to the Canadian Journal of Mathematic
Stochastic Variance Reduction Methods for Saddle-Point Problems
We consider convex-concave saddle-point problems where the objective
functions may be split in many components, and extend recent stochastic
variance reduction methods (such as SVRG or SAGA) to provide the first
large-scale linearly convergent algorithms for this class of problems which is
common in machine learning. While the algorithmic extension is straightforward,
it comes with challenges and opportunities: (a) the convex minimization
analysis does not apply and we use the notion of monotone operators to prove
convergence, showing in particular that the same algorithm applies to a larger
class of problems, such as variational inequalities, (b) there are two notions
of splits, in terms of functions, or in terms of partial derivatives, (c) the
split does need to be done with convex-concave terms, (d) non-uniform sampling
is key to an efficient algorithm, both in theory and practice, and (e) these
incremental algorithms can be easily accelerated using a simple extension of
the "catalyst" framework, leading to an algorithm which is always superior to
accelerated batch algorithms.Comment: Neural Information Processing Systems (NIPS), 2016, Barcelona, Spai
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