Mixed Type Duality for Set-valued Optimization Problems via Higher-order Radial Epiderivatives

In this article, we introduce a notion of higher-order radial epiderivative for set-valued maps and study its properties. A generalized concept of higher-order strict minimizers in set-valued optimization is proposed as well. By virtue of the radial epiderivative, we establish a mixed dual problem, and then weak, strong, and converse duality theorems are obtained in dealing with generalized strict minimizers.     

An inverse problem of finding the time‐dependent diffusion coefficient from an integral condition

We consider the inverse problem of determining the time‐dependent diffusivity in one‐dimensional heat equation with periodic boundary conditions and nonlocal over‐specified data. The problem is highly nonlinear and it serves as a mathematical model for the technological process of external guttering applied in cleaning admixtures from silicon chips. First, the well‐posedness conditions for the existence, uniqueness, and continuous dependence upon the data of the classical solution of the problem are established. Then, the problem is discretized using the finite‐difference method and recasts as a nonlinear least‐squares minimization problem with a simple positivity lower bound on the unknown diffusivity. Numerically, this is effectively solved using the lsqnonlin routine from the MATLAB toolbox. In order to investigate the accuracy, stability, and robustness of the numerical method, results for a few test examples are presented and discussed. Copyright © 2015 John Wiley & Sons, Ltd.     

Finite element Galerkin approximations to a class of nonlinear and nonlocal parabolic problems

In this article, a finite element Galerkin method is applied to a general class of nonlinear and nonlocal parabolic problems. Based on an exponential weight function, new a priori bounds which are valid for uniform in time are derived. As a result, existence of an attractor is proved for the problem with nonhomogeneous right hand side which is independent of time. In particular, when the forcing function is zero or decays exponentially, it is shown that solution has exponential decay property which improves even earlier results in one dimensional problems. For the semidiscrete method, global existence of a unique discrete solution is derived and it is shown that the discrete problem has an attractor. Moreover, optimal error estimates are derived in both and ‐norms with later estimate is a new result in this context. For completely discrete scheme, backward Euler method with its linearized version is discussed and existence of a unique discrete solution is established. Further, optimal estimates in ‐norm are proved for fully discrete schemes. Finally, several numerical experiments are conducted to confirm our theoretical findings. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1232–1264, 2016     

Zonal PANS: evaluation of different treatments of the RANS–LES interface

The partially Reynolds-averaged Navier–Stokes (PANS) model can be used to simulate turbulent flows either as RANS, large eddy simulation (LES) or DNS. Its main parameter is f_k whose physical meaning is the ratio of the modelled to the total turbulent kinetic energy. In RANS f_k = 1, in DNS f_k = 0 and in LES f_k takes values between 0 and 1. Three different ways of prescribing f_k are evaluated for decaying grid turbulence and fully developed channel flow: f_k = 0.4, f_k = k^3/2 _tot/? and, from its definition, f_k = k/k_tot where k_tot is the sum of the modelled, k, and resolved, k_res, turbulent kinetic energy. It is found that the f_k = 0.4 gives the best results. In Girimaji and Wallin, a method was proposed to include the effect of the gradient of f_k. This approach is used at RANS– LES interface in the present study. Four different interface models are evaluated in fully developed channel flow and embedded LES of channel flow: in both cases, PANS is used as a zonal model with f_k = 1 in the unsteady RANS (URANS) region and f_k = 0.4 in the LES region. In fully developed channel flow, the RANS– LES interface is parallel to the wall (horizontal) and in embedded LES, it is parallel to the inlet (vertical). The importance of the location of the horizontal interface in fully developed channel flow is also investigated. It is found that the location – and the choice of the treatment at the interface – may be critical at low Reynolds number or if the interface is placed too close to the wall. The reason is that the modelled turbulent shear stress at the interface is large and hence the relative strength of the resolved turbulence is small. In RANS, the turbulent viscosity – and consequently also the modelled Reynolds shear stress – is only weakly dependent on Reynolds number. It is found in the present work that it also applies in the URANS region.     

Second order asymptotic behaviour of subordinated sequences with longtailed subordinator

Suppose that {a(n)} is a discrete probability distribution on the set N₀={0,1,2,…} and {p(n)} is some non-negative sequence defined on the same set. The equation defines a new sequence {b(n)}. Here {a*k(n)} denotes the k-fold convolution of the distribution {a(n)}. In the paper the asymptotic behaviour of the sequence {b(n)} is investigated. It is known that for the large classes of the sequences {a(n)} and {p(n)}, b(n)∼σp([σn]), where 1/σ is the mean of the distribution {a(n)}. The main object of the present work is to estimate the difference b(n)−σp([σn]) for some classes of the sequences {a(n)} and {p(n)}.     

Strong Feller properties for distorted Brownian motion with reflecting boundary condition and an application to continuous N-particle systems with singular interactions

Assuming that Ω⊂Rn, n?2, is an open, relatively compact set with boundary ∂Ω of Lebesgue measure zero we prove strong Feller properties for a class of distorted Brownian motions in with reflecting boundary condition. Dirichlet form techniques give the existence of a weak solution to the corresponding stochastic differential equation for quasi all starting points in the sense of the associated martingale problem. Combining this result with the strong Feller properties we can construct a weak solution for specified starting points. If Ω has C²-boundary the construction works for all starting points, where the drift term is not singular, even on the boundary. But also for a certain class of sets with less smooth boundary our approach works for all points in Ω, where the drift term is not singular, and at least some points from ∂Ω. Our techniques allow very singular drift terms. This enables us to construct continuous N-particle gradient stochastic dynamics in cuboids Λ⊂Rd, d∈N, with reflecting boundary condition and singular interactions for dN?2. We can start the stochastic dynamics in all initial configurations having at most one particle in ∂Λ, provided ∂Λ is locally smooth there.     

Periodic solutions of nonlinear functional difference equations at nonresonance case

Sufficient conditions for the existence of at least one T-periodic solution of nonlinear functional difference equation

Δx(n)+a(n)x(n)=f(n,u(n)),     

A new exclusion test for finding the global minimum

Exclusion algorithms have been used recently to find all solutions of a system of nonlinear equations or to find the global minimum of a function over a compact domain. These algorithms are based on a minimization condition that can be applied to each cell in the domain. In this paper, we consider Lipschitz functions of order α and give a new minimization condition for the exclusion algorithm. Furthermore, convergence and complexity results are presented for such algorithm.     

A sufficient condition for Hamiltonian cycles in bipartite tournaments

In this paper, we present a new sufficient condition on degrees for a bipartite tournament to be Hamiltonian, that is, if an n × n bipartite tournament T satisfies the condition W(n - 3), then T is Hamiltonian, except for four exceptional graphs. This result is shown to be best possible in a sense.     

LINEAR AND NONLINEAR SEMICLASSICAL THEORY OF WHISPERING-GALLERY-MODE MICROSPHERE LASER

A complete whispering-gallery-mode microsphere laser theory is developed, which combines usual laser semiclassical theory with Lorenz-Mie theory. The linear and nonlinear problems of the microsphere lasing are dealt with in the universal mode picture. The threshold condition, average amplitude of the steady state and frequency effect are obtained. A comparison between the theory and experiments is also made.