1,089 research outputs found
Two-points problem for an evolutional first order equation in Banach space
Two-point nonlocal problem for the first order differential evolution equation with an operator co-
efficient in a Banach space X is considered. An exponentially convergent algorithm is proposed and
justified in assumption that the operator coefficient is strongly positive and some existence and unique-
ness conditions are fulfilled. This algorithm leads to a system of linear equations that can be solved
by fixed-point iteration. The algorithm provides exponentially convergence in time that in combination
with fast algorithms on spatial variables can be efficient treating such problems. The efficiency of the
proposed algorithms is demonstrated by numerical examples
Greedy algorithms for high-dimensional non-symmetric linear problems
In this article, we present a family of numerical approaches to solve
high-dimensional linear non-symmetric problems. The principle of these methods
is to approximate a function which depends on a large number of variates by a
sum of tensor product functions, each term of which is iteratively computed via
a greedy algorithm. There exists a good theoretical framework for these methods
in the case of (linear and nonlinear) symmetric elliptic problems. However, the
convergence results are not valid any more as soon as the problems considered
are not symmetric. We present here a review of the main algorithms proposed in
the literature to circumvent this difficulty, together with some new
approaches. The theoretical convergence results and the practical
implementation of these algorithms are discussed. Their behaviors are
illustrated through some numerical examples.Comment: 57 pages, 9 figure
Dobrushin ergodicity coefficient for Markov operators on cones, and beyond
The analysis of classical consensus algorithms relies on contraction
properties of adjoints of Markov operators, with respect to Hilbert's
projective metric or to a related family of seminorms (Hopf's oscillation or
Hilbert's seminorm). We generalize these properties to abstract consensus
operators over normal cones, which include the unital completely positive maps
(Kraus operators) arising in quantum information theory. In particular, we show
that the contraction rate of such operators, with respect to the Hopf
oscillation seminorm, is given by an analogue of Dobrushin's ergodicity
coefficient. We derive from this result a characterization of the contraction
rate of a non-linear flow, with respect to Hopf's oscillation seminorm and to
Hilbert's projective metric
A unified approach to compute foliations, inertial manifolds, and tracking initial conditions
Several algorithms are presented for the accurate computation of the leaves
in the foliation of an ODE near a hyperbolic fixed point. They are variations
of a contraction mapping method in [25] to compute inertial manifolds, which
represents a particular leaf in the unstable foliation. Such a mapping is
combined with one for the leaf in the stable foliation to compute the tracking
initial condition for a given solution. The algorithms are demonstrated on the
Kuramoto-Sivashinsky equation
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