8,500 research outputs found
A Self-learning Algebraic Multigrid Method for Extremal Singular Triplets and Eigenpairs
A self-learning algebraic multigrid method for dominant and minimal singular
triplets and eigenpairs is described. The method consists of two multilevel
phases. In the first, multiplicative phase (setup phase), tentative singular
triplets are calculated along with a multigrid hierarchy of interpolation
operators that approximately fit the tentative singular vectors in a collective
and self-learning manner, using multiplicative update formulas. In the second,
additive phase (solve phase), the tentative singular triplets are improved up
to the desired accuracy by using an additive correction scheme with fixed
interpolation operators, combined with a Ritz update. A suitable generalization
of the singular value decomposition is formulated that applies to the coarse
levels of the multilevel cycles. The proposed algorithm combines and extends
two existing multigrid approaches for symmetric positive definite eigenvalue
problems to the case of dominant and minimal singular triplets. Numerical tests
on model problems from different areas show that the algorithm converges to
high accuracy in a modest number of iterations, and is flexible enough to deal
with a variety of problems due to its self-learning properties.Comment: 29 page
Minimal convex extensions and finite difference discretization of the quadratic Monge-Kantorovich problem
We present an adaptation of the MA-LBR scheme to the Monge-Amp{\`e}re
equation with second boundary value condition, provided the target is a convex
set. This yields a fast adaptive method to numerically solve the Optimal
Transport problem between two absolutely continuous measures, the second of
which has convex support. The proposed numerical method actually captures a
specific Brenier solution which is minimal in some sense. We prove the
convergence of the method as the grid stepsize vanishes and we show with
numerical experiments that it is able to reproduce subtle properties of the
Optimal Transport problem
Asynchronous Networks and Event Driven Dynamics
Real-world networks in technology, engineering and biology often exhibit
dynamics that cannot be adequately reproduced using network models given by
smooth dynamical systems and a fixed network topology. Asynchronous networks
give a theoretical and conceptual framework for the study of network dynamics
where nodes can evolve independently of one another, be constrained, stop, and
later restart, and where the interaction between different components of the
network may depend on time, state, and stochastic effects. This framework is
sufficiently general to encompass a wide range of applications ranging from
engineering to neuroscience. Typically, dynamics is piecewise smooth and there
are relationships with Filippov systems. In the first part of the paper, we
give examples of asynchronous networks, and describe the basic formalism and
structure. In the second part, we make the notion of a functional asynchronous
network rigorous, discuss the phenomenon of dynamical locks, and present a
foundational result on the spatiotemporal factorization of the dynamics for a
large class of functional asynchronous networks
Monte Carlo approximations of the Neumann problem
We introduce Monte Carlo methods to compute the solution of elliptic
equations with pure Neumann boundary conditions. We first prove that the
solution obtained by the stochastic representation has a zero mean value with
respect to the invariant measure of the stochastic process associated to the
equation. Pointwise approximations are computed by means of standard and new
simulation schemes especially devised for local time approximation on the
boundary of the domain. Global approximations are computed thanks to a
stochastic spectral formulation taking into account the property of zero mean
value of the solution. This stochastic formulation is asymptotically perfect in
terms of conditioning. Numerical examples are given on the Laplace operator on
a square domain with both pure Neumann and mixed Dirichlet-Neumann boundary
conditions. A more general convection-diffusion equation is also numerically
studied
Correlation Decay up to Uniqueness in Spin Systems
We give a complete characterization of the two-state anti-ferromagnetic spin
systems which are of strong spatial mixing on general graphs. We show that a
two-state anti-ferromagnetic spin system is of strong spatial mixing on all
graphs of maximum degree at most \Delta if and only if the system has a unique
Gibbs measure on infinite regular trees of degree up to \Delta, where \Delta
can be either bounded or unbounded. As a consequence, there exists an FPTAS for
the partition function of a two-state anti-ferromagnetic spin system on graphs
of maximum degree at most \Delta when the uniqueness condition is satisfied on
infinite regular trees of degree up to \Delta. In particular, an FPTAS exists
for arbitrary graphs if the uniqueness is satisfied on all infinite regular
trees. This covers as special cases all previous algorithmic results for
two-state anti-ferromagnetic systems on general-structure graphs.
Combining with the FPRAS for two-state ferromagnetic spin systems of
Jerrum-Sinclair and Goldberg-Jerrum-Paterson, and the very recent hardness
results of Sly-Sun and independently of Galanis-Stefankovic-Vigoda, this gives
a complete classification, except at the phase transition boundary, of the
approximability of all two-state spin systems, on either degree-bounded
families of graphs or family of all graphs.Comment: 27 pages, submitted for publicatio
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