250,271 research outputs found
Robust preconditioners for a new stabilized discretization of the poroelastic equations
In this paper, we present block preconditioners for a stabilized
discretization of the poroelastic equations developed in [45]. The
discretization is proved to be well-posed with respect to the physical and
discretization parameters, and thus provides a framework to develop
preconditioners that are robust with respect to such parameters as well. We
construct both norm-equivalent (diagonal) and field-of-value-equivalent
(triangular) preconditioners for both the stabilized discretization and a
perturbation of the stabilized discretization that leads to a smaller overall
problem after static condensation. Numerical tests for both two- and
three-dimensional problems confirm the robustness of the block preconditioners
with respect to the physical and discretization parameters
Discretization-related issues in the KPZ equation: Consistency, Galilean-invariance violation, and fluctuation--dissipation relation
In order to perform numerical simulations of the KPZ equation, in any
dimensionality, a spatial discretization scheme must be prescribed. The known
fact that the KPZ equation can be obtained as a result of a Hopf--Cole
transformation applied to a diffusion equation (with \emph{multiplicative}
noise) is shown here to strongly restrict the arbitrariness in the choice of
spatial discretization schemes. On one hand, the discretization prescriptions
for the Laplacian and the nonlinear (KPZ) term cannot be independently chosen.
On the other hand, since the discretization is an operation performed on
\emph{space} and the Hopf--Cole transformation is \emph{local} both in space
and time, the former should be the same regardless of the field to which it is
applied. It is shown that whereas some discretization schemes pass both
consistency tests, known examples in the literature do not. The requirement of
consistency for the discretization of Lyapunov functionals is argued to be a
natural and safe starting point in choosing spatial discretization schemes. We
also analyze the relation between real-space and pseudo-spectral discrete
representations. In addition we discuss the relevance of the Galilean
invariance violation in these consistent discretization schemes, and the
alleged conflict of standard discretization with the fluctuation--dissipation
theorem, peculiar of 1D.Comment: RevTex, 23pgs, 2 figures, submitted to Phys. Rev.
Testing microscopic discretization
What can we say about the spectra of a collection of microscopic variables
when only their coarse-grained sums are experimentally accessible? In this
paper, using the tools and methodology from the study of quantum nonlocality,
we develop a mathematical theory of the macroscopic fluctuations generated by
ensembles of independent microscopic discrete systems. We provide algorithms to
decide which multivariate gaussian distributions can be approximated by sums of
finitely-valued random vectors. We study non-trivial cases where the
microscopic variables have an unbounded range, as well as asymptotic scenarios
with infinitely many macroscopic variables. From a foundational point of view,
our results imply that bipartite gaussian states of light cannot be understood
as beams of independent d-dimensional particle pairs. It is also shown that the
classical description of certain macroscopic optical experiments, as opposed to
the quantum one, requires variables with infinite cardinality spectra.Comment: Proof of strong NP-hardness. Connection with random walks. New
asymptotic results. Numerous typos correcte
Bourgain's discretization theorem
Bourgain's discretization theorem asserts that there exists a universal
constant with the following property. Let be Banach
spaces with . Fix and set .
Assume that is a -net in the unit ball of and that
admits a bi-Lipschitz embedding into with distortion at most
. Then the entire space admits a bi-Lipschitz embedding into with
distortion at most . This mostly expository article is devoted to a
detailed presentation of a proof of Bourgain's theorem.
We also obtain an improvement of Bourgain's theorem in the important case
when for some : in this case it suffices to take
for the same conclusion to hold true. The case
of this improved discretization result has the following consequence. For
arbitrarily large there exists a family of
-point subsets of such that if we write
then any embedding of , equipped with the
Earthmover metric (a.k.a. transportation cost metric or minimumum weight
matching metric) incurs distortion at least a constant multiple of
; the previously best known lower bound for this problem was
a constant multiple of .Comment: Proof of Lemma 5.1 corrected; its statement remains unchange
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