We consider the stochastic Allen-Cahn equation driven by mollified space-time white noise. We show that, as the mollifier is removed, the solutions converge weakly to 0, independently of the initial condition. If the intensity of the noise simultaneously converges to 0 at a sufficiently fast rate, then the solutions converge to those of the deterministic equation. At the critical rate, the limi...

The subgradient method is used frequently to optimize dual functions in Lagrangian relaxation for separable integer programming problems. In the method, all subproblems must be solved optimally to obtain a subgradient direction. In this paper, the surrogate subgradient method is developed, where a proper direction can be obtained without solving optimally all the subproblems. In fact, only an a...

in an open bounded domain Ω ⊂ R2, where u(t, x) > 0, ω(t, x) ∈ R2 and E denote the population density, the average velocity of dispersing individuals, and the individual net reproduction rate, respectively. In this model, the individuals are assumed to disperse randomly in space (δ ∆u) with a bias −∇ · (u ω) in the direction of increasing reproduction rate, the term ε ∆ω acting as a mollifier t...

We introduce a generalised subgradient for law-invariant closed convex risk measures on L and establish its relationship with optimal risk allocations and equilibria. Our main result gives sufficient conditions ensuring a non-empty generalised subgradient.

A study of the convergence properties of spectral projected subgradient method is presented and the convergence is shown. The convergence is based on spectral projected gradient approach. Some updates of the spectral projected subgradient are described.