Robust newsvendor problems: effect of discrete demands

Distribution-free newsvendor models often assume continuous demand distributions to facilitate analysis and computation. However, in practice, discrete demand is a natural phenomenon. So far, there exists no analytical and computational results in the literature under this setting. Thus, the goal of this paper is to investigate the newsvendor problems with partial information when the demand is discrete and solve them using the so-called discrete moment problems. Numerical results are presented to illustrate the value of discrete information.

     

On solutions of a discretized heat equation in discrete Clifford analysis

The main purpose of this paper was to study solutions of the heat equation in the setting of discrete Clifford analysis. More precisely we consider this equation with discrete space and continuous time. Thereby we focus on representations of solutions by means of dual Taylor series expansions. Furthermore we develop a discrete convolution theory, apply it to the inhomogeneous heat equation and construct solutions for the related Cauchy problem by means of heat polynomials.     

A Spectral Property of Discrete Schrödinger Operators with Non-Negative Potentials

In the context of an infinite weighted graph of bounded degree, we give a sufficient condition for the discrete Schrödinger operator with a non-negative potential to have a strictly positive bottom of the spectrum. The main result is a discrete analogue of a theorem of Shen in the setting of complete Riemannian manifolds.     

The Multichannel Deconvolution Problem: A Discrete Analysis

We study the multichannel deconvolution problem (MDP) in a discrete setting by developing the theory for converting the method used in the continuous setting in [36]. We give a method for solving the MDP when the convolvers are characteristic functions, derive the explicit form of the linear system, and obtain an upper bound on the condition number of the system in a particular case. We compare the Schiske reconstruction [28] to our solution in the discrete setting, and give an explicit formula for the corresponding error. We then give the algorithm for solving the general MDP and discuss in detail the local reconstruction aspects of the problem. Finally, we describe a method for improving the reconstruction by regularization and give some explicit estimates on error bounds in the presence of noise.     

Discrete time queues and matrix-analytic methods

This is an expository paper dealing with discrete time analysis of queues using matrix-analytic methods (MAM). Discrete time analysis queues has always been popular with engineers who are very keen on obtaining numerical values out of their analyses for the sake of experimentation and design. As telecommunication systems are based more on digital technology these days than analog the need to use discrete time analysis for queues has become more important. Besides, we find that several queues which are difficult to analyse by the continuous time approach are sometimes easier to analyse using discrete time method. Of course, there are some queueing problems which are easier to analyse using continuous time approach instead of discrete time. We discuss, in this paper, both the advantages and disadvantages of discrete time analysis. The paper focusses on setting up several queueing systems as discrete time quasi-birth-and-death processes and then shows how to use matrix-geometric method (MGM), a class of MAM, to analyse the problems. We point out that there are other methods for analysing such queues but MGM provides a much simpler approach for setting up the problems in order to obtain semi-explicit results for computational tractability. We also point out some of the shortcomings of MGM. The paper mainly focusses on the Geo/Geo/1, PH/PH/1, GI/G/1 and GI/G/1/K systems and some of the related problems, including vacation models with time-limited visits.     

The number of excellent discrete Morse functions on graphs

In Nicolaescu (2008) [7] the number of non-homologically equivalent excellent Morse functions defined on S² was obtained in the differentiable setting. We carried out an analogous study in the discrete setting for some kinds of graphs, including S¹, in Ayala et al. (2009) [1]. This paper completes this study, counting excellent discrete Morse functions defined on any infinite locally finite graph.     

Operator renewal theory for continuous time dynamical systems with finite and infinite measure

We develop operator renewal theory for flows and apply this to obtain results on mixing and rates of mixing for a large class of finite and infinite measure semiflows. Examples of systems covered by our results include suspensions over parabolic rational maps of the complex plane, and nonuniformly expanding semiflows with indifferent periodic orbits. In the finite measure case, the emphasis is on obtaining sharp rates of decorrelations, extending results of Gouëzel and Sarig from the discrete time setting to continuous time. In the infinite measure case, the primary question is to prove results on mixing itself, extending our results in the discrete time setting. In some cases, we obtain also higher order asymptotics and rates of mixing.     

Matching and stabilization of discrete mechanical systems

Controlled Lagrangian and matching techniques are developed for the stabilization of equilibria of discrete mechanical systems with symmetry as well as broken symmetry. Interesting new phenomena arise in the controlled Lagrangian approach in the discrete context that are not present in the continuous theory. Specifically, a nonconservative force that is necessary for matching in the discrete setting is introduced. The paper also discusses digital and model predictive controllers. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonuniform Exponential Dichotomy for Discrete Dynamical Systems in Banach Spaces

The aim of this paper is to present three concepts of nonuniform exponential dichotomies for linear discrete systems and their correspondents in the uniform case. With the aid of particular discrete systems defined on infinite-dimensional Banach spaces, we emphasize the connections between the presented concepts. The main result of the paper is given by a result of boundedness of the dichotomic sequence of projections in the nonuniform setting.     

Entropy of discrete Choquet capacities

We introduce a measure of entropy for any discrete Choquet capacity and we interpret it in the setting of aggregation by the Choquet integral.     

Adaptive efficient analysis for big data ergodic diffusion models

Statistical Inference for Stochastic Processes - We consider drift estimation problems for high dimension ergodic diffusion processes in nonparametric setting based on observations at discrete...     

Discrete stochastic approximations of the Mumford–Shah functional

We propose a new Γ-convergent discrete approximation of the Mumford–Shah functional. The discrete functionals act on functions defined on stationary stochastic lattices and take into account general finite differences through a non-convex potential. In this setting the geometry of the lattice strongly influences the anisotropy of the limit functional. Thus we can use statistically isotropic lattices and stochastic homogenization techniques to approximate the vectorial Mumford–Shah functional in any dimension.     

Variational and Geometric Structures of Discrete Dirac Mechanics

In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew’s triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange–Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange–d’Alembert–Pontryagin and Hamilton–d’Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.     

A new efficient energy-preserving finite volume element scheme for the improved Boussinesq equation

In this paper, a new linearized energy-preserving Crank-Nicolson finite volume element scheme is derived for the improved Boussinesq equation. The fully discrete scheme can be shown to conserve both mass and energy in the discrete setting. It is proved that the scheme is uniquely solvable and convergent with the rate of order two in a discrete L₂ norm. At last, a series of numerical experiments on typical improved Boussinesq and Sine–Gordon equations are provided to verify our theoretical results and to show the efficiency and accuracy of the proposed scheme.     

Low noise regimes for ℓ regularization : continuous and discrete settings

We focus on support recovery for signal deconvolution with sparsity assumption. We adopt the continuous setting defined by several recent works and we try to reconstruct a sum of Dirac masses from its low frequencies (possibly perturbed by some noise), by using a total variation prior for Radon measures (i.e. the generalization to measures of the ℓ¹ norm). We show that, under a non degenerate source condition, there exists a small noise regime in which the model recovers exactly the same number of spikes as the original signal, and the spikes converge to those of the original signal as the noise vanishes. This continuous setting, by allowing the spikes to “move”, provides robust support recovery for signals composed of well separated spikes. In a discrete setting, where the spikes are reconstructed on a grid, similar low noise regimes which guarantee the exact recovery of the support also exist (see [3]). Yet, this property only concerns a small class of signals. Considering the asymptotics of the discrete problems as the size of the grid tends to zero, we show that the support of the original signal cannot be stable on thin grids, and that the discrete models actually reconstruct pairs of spikes near each original spike. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

离散的三项分布风险模型的渐近解

本文研究了离散的三项分布风险模型,在调节系数存在的前提下,借助于离散更新方程的一个极限定理,对于充分大的初始盈余给出了最终破产概率、破产前一刻的盈余和破产时赤字的概率的渐近解．其结果可以在离散的多项分布风险模型中得到推广．     

Exact Solutions of Some Discrete Deconvolution Problems

In this work, we study the existence of solutions of the deconvolution problems in the discrete setting. More precisely, we prove the existence of solutions of the discrete multichannel deconvolution problems DMDP with convolvers being the characteristic functions of finite sets of positive integers. Also, we provide the reader with a simple method and a fast algorithm for finding the closed forms of the discrete deconvolvers with minimal supports that constitute exact solutions of the DMDP. Moreover, we show that unlike the singular value decomposition scheme, the multichannel deconvolution scheme based on the use of these discrete deconvolvers is not very sensitive to small 2-norm perturbation of the data. Finally, we show how to generalize our method for solving the 2-D version of the DMDP.     

Solvability of discrete Neumann boundary value problems

In this article we gain solvability to a nonlinear, second-order difference equation with discrete Neumann boundary conditions. Our methods involve new inequalities on the right-hand side of the difference equation and Schaefer's Theorem in the finite-dimensional space setting.     

Ergodic backward stochastic difference equations

We consider ergodic backward stochastic differential equations in a discrete time setting, where noise is generated by a finite state Markov chain. We show existence and uniqueness of solutions, along with a comparison theorem. To obtain this result, we use a Nummelin splitting argument to obtain ergodicity estimates for a discrete time Markov chain which hold uniformly under suitable perturbations of its transition matrix. We conclude with an application of this theory to a treatment of an ergodic control problem.     

Implicitly and densely discrete black-box optimization problems

This paper addresses derivative-free optimization problems where the variables lie implicitly in an unknown discrete closed set. The evaluation of the objective function follows a projection onto the discrete set, which is assumed dense (and not sparse as in integer programming). Such a mathematical setting is a rough representation of what is common in many real-life applications where, despite the continuous nature of the underlying models, a number of practical issues dictate rounding of values or projection to nearby feasible figures. We discuss a definition of minimization for these implicitly discrete problems and outline a direct-search algorithm framework for its solution. The main asymptotic properties of the algorithm are analyzed and numerically illustrated.