1,487 research outputs found
Local Martingale and Pathwise Solutions for an Abstract Fluids Model
We establish the existence and uniqueness of both local martingale and local
pathwise solutions of an abstract nonlinear stochastic evolution system. The
primary application of this abstract framework is to infer the local existence
of strong, pathwise solutions to the 3D primitive equations of the oceans and
atmosphere forced by a nonlinear multiplicative white noise. Instead of
developing our results specifically for the 3D primitive equations we choose to
develop them in a slightly abstract framework which covers many related forms
of these equations (atmosphere, oceans, coupled atmosphere-ocean, on the
sphere, on the {\beta}-plane approximation etc and the incompressible
Navier-Stokes equations). In applications, all of the details are given for the
{\beta}-plane approximation of the oceans equations
Convergence of a greedy algorithm for high-dimensional convex nonlinear problems
In this article, we present a greedy algorithm based on a tensor product
decomposition, whose aim is to compute the global minimum of a strongly convex
energy functional. We prove the convergence of our method provided that the
gradient of the energy is Lipschitz on bounded sets. The main interest of this
method is that it can be used for high-dimensional nonlinear convex problems.
We illustrate this method on a prototypical example for uncertainty propagation
on the obstacle problem.Comment: 36 pages, 9 figures, accepted in Mathematical Models and Methods for
Applied Science
Limit theorems for iterated random topical operators
Let A(n) be a sequence of i.i.d. topical (i.e. isotone and additively
homogeneous) operators. Let be defined by and
. This can modelize a wide range of systems including,
task graphs, train networks, Job-Shop, timed digital circuits or parallel
processing systems. When A(n) has the memory loss property, we use the spectral
gap method to prove limit theorems for . Roughly speaking, we show
that behaves like a sum of i.i.d. real variables. Precisely, we show
that with suitable additional conditions, it satisfies a central limit theorem
with rate, a local limit theorem, a renewal theorem and a large deviations
principle, and we give an algebraic condition to ensure the positivity of the
variance in the CLT. When A(n) are defined by matrices in the \mp semi-ring, we
give more effective statements and show that the additional conditions and the
positivity of the variance in the CLT are generic
Optimal distributed control of a stochastic Cahn-Hilliard equation
We study an optimal distributed control problem associated to a stochastic
Cahn-Hilliard equation with a classical double-well potential and Wiener
multiplicative noise, where the control is represented by a source-term in the
definition of the chemical potential. By means of probabilistic and analytical
compactness arguments, existence of an optimal control is proved. Then the
linearized system and the corresponding backward adjoint system are analysed
through monotonicity and compactness arguments, and first-order necessary
conditions for optimality are proved.Comment: Key words and phrases: stochastic Cahn-Hilliard equation, phase
separation, optimal control, linearized state system, adjoint state system,
first-order optimality condition
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