Existence and uniqueness for variational data assimilation in continuous time

<p style='text-indent:20px;'>A variant of the optimal control problem is considered which is nonstandard in that the performance index contains "stochastic" integrals, that is, integrals against very irregular functions. The motivation for considering such performance indices comes from dynamical estimation problems where observed time series need to be "fitted" with trajectories of dynamical models. The observations may be contaminated with white noise, which gives rise to the nonstandard performance indices. Problems of this kind appear in engineering, physics, and the geosciences where this is referred to as data assimilation. The fact that typical models in the geosciences do not satisfy linear growth nor monotonicity conditions represents an additional difficulty. Pathwise existence of minimisers is obtained, along with a maximum principle as well as preliminary results in dynamic programming. The results also extend previous work on the maximum aposteriori estimator of trajectories of diffusion processes.</p>

Global Lorentz gradient estimates for quasilinear equations with measure data for the strongly singular case: 1 < p ≤ 3n−2 2n−1

In this paper, we study the global regularity estimates in Lorentz spaces for gradients of solutions to quasilinear elliptic equations with measure data of the form [Formula: see text] where [Formula: see text] is a finite signed Radon measure in [Formula: see text], [Formula: see text] is a bounded domain such that its complement [Formula: see text] is uniformly [Formula: see text]-thick and [Formula: see text] is a Carathéodory vector-valued function satisfying growth and monotonicity conditions for the strongly singular case [Formula: see text]. Our result extends the earlier results [19,22] to the strongly singular case [Formula: see text] and a recent result [18] by considering rough conditions on the domain [Formula: see text] and the nonlinearity [Formula: see text].

Existence, uniqueness and monotonic behavior of the solution of classical flow distribution problem for hydraulic networks with pressure-dependent closure relations

Existence, uniqueness and monotonic behavior of the solution of classical flow distribution problem for hydraulic networks with pressure-dependent closure relations was proved. The closure relation can have very general form, restricted only by continuity and monotonicity conditions necessary for providing existence, uniqueness and continuity of flow distribution problem for each branch. It is shown that network as a whole “inherits” monotonicity and continuity of its branches behavior, and this provides existence and uniqueness of solution.

CONTRACTION-MAPPING ALGORITHM FOR THE EQUILIBRIUM PROBLEM OVER THE FIXED POINT SET OF A NONEXPANSIVE SEMIGROUP

In this paper, we consider the proximal mapping of a bifunction. Under the Lipschitz-type and the strong monotonicity conditions, we prove that the proximal mapping is contractive. Based on this result, we construct an iterative process for solving the equilibrium problem over the fixed point sets of a nonexpansive semigroup and prove a weak convergence theorem for this algorithm. Also, some preliminary numerical experiments and comparisons are presented.

Relaxation of monotone coupling conditions: Poisson approximation and beyond

It is well known that assumptions of monotonicity in size-bias couplings may be used to prove simple, yet powerful, Poisson approximation results. Here we show how these assumptions may be relaxed, establishing explicit Poisson approximation bounds (depending on the first two moments only) for random variables which satisfy an approximate version of these monotonicity conditions. These are shown to be effective for models where an underlying random variable of interest is contaminated with noise. We also state explicit Poisson approximation bounds for sums of associated or negatively associated random variables. Applications are given to epidemic models, extremes, and random sampling. Finally, we also show how similar techniques may be used to relax the assumptions needed in a Poincaré inequality and in a normal approximation result.