Ergodic Problem Research Articles

Precoder Design in Statistical CSI Aided Non-Orthogonal Multiple Access

In this paper, the ergodic sum rate maximization problem is considered in the multiple-input single-output non-orthogonal multiple access (NOMA) system with statistical channel state information at the transmitter. First, we derive the analytic expression of the ergodic sum rate with given arbitrary precoders. Then, due to the non-convexity of the optimization problem, we use the gradient projection algorithm (GPA) to find the optimal precoders, approximately. In extremely low and extremely high signal-to-noise ratio (SNR) region, we deduce the closed-form optimal precoders and a simplified analytic expression of the ergodic sum rate, based on which another algorithm leveraging bisection algorithm (BA) is proposed in the arbitrary-SNR. Numerical results demonstrate that the GPA and BA can efficiently find the optimal precoding vectors, approximately, and the performance of NOMA is superior to the orthogonal multiple access and space division multiple access in intermediate- and high-SNR.

On Some Ergodic Impulse Control Problems with Constraint

This paper studies the impulse control of a general Markov process under the average (or ergodic) cost when the impulse instants are restricted to be the arrival times of an exogenous process, and ...

An Ergodic Control Problem for Many-Server Multiclass Queueing Systems with Cross-Trained Servers

A Markovian queueing network is considered with d independent customer classes and d server pools in Halfin–Whitt regime. Class i customers has priority for service in pool i for i = 1, …, d, and may access some other pool if the pool has an idle server and all the servers in pool i are busy. We formulate an ergodic control problem where the running cost is given by a non-negative convex function with polynomial growth. We show that the limiting controlled diffusion is modelled by an action space which depends on the state variable. We provide a complete analysis for the limiting ergodic control problem and establish asymptotic convergence of the value functions for the queueing model.

Some New Results on Sample Path Optimality in Ergodic Control of Diffusions

We present some new results on sample path optimality for the ergodic control problem of a class of non-degenerate diffusions controlled through the drift. The hypothesis most often used in the literature to ensure the existence of an a.s. sample path optimal stationary Markov control requires finite second moments of the first hitting times $\tau$ of bounded domains over all admissible controls. We show that this can be considerably weakened: ${\mathrm E}[\tau^2]$ may be replaced with ${\mathrm E}[\tau\ln^+(\tau)]$, thus reducing the required rate of convergence of averages from polynomial to logarithmic. A Foster-Lyapunov condition which guarantees this is also exhibited. Moreover, we study a large class of models that are neither uniformly stable, nor have a near-monotone running cost, and we exhibit sufficient conditions for the existence of a sample path optimal stationary Markov control.

Strong Law of Large Numbers for Solutions of Non-Autonomous Stochastic Differential Equations

Background. Asymptotic behavior at infinity of non-autonomous stochastic differential equation solutions is studied in the paper. Objective. The aim of the work is to find sufficient conditions for the strong law of large numbers for a random process which is a solution of non-autonomous stochastic differential equation.Methods. Basic results of the theory of stochastic differential equations related to stochastic integrals estimation.Results. Sufficient conditions for almost sure convergence to zero of normalized term related to diffusion of non-autonomous stochastic differential equation are obtained.Conclusions. Results of the paper can be used for further research on the asymptotic behavior of stochastic differential equation solutions, finding the stability condition of stochastic differential equation solution and ergodic type problems also.

Ergodic Control for Lévy-Driven Linear Stochastic Equations in Hilbert Spaces

In this paper, controlled linear stochastic evolution equations driven by square integrable Levy processes are studied in the Hilbert space setting. The control operator may be unbounded which makes the results obtained in the abstract setting applicable to parabolic SPDEs with boundary or point control. The first part contains some preliminary technical results, notably a version of Ito formula which is applicable to weak/mild solutions of controlled equations. In the second part, the ergodic control problem is solved: The feedback form of the optimal control and the formula for the optimal cost are found. As examples, various parabolic type controlled SPDEs are studied.

Ergodic Maximum Principle for Stochastic Systems

We present a version of the stochastic maximum principle (SMP) for ergodic control problems. In particular we give necessary (and sufficient) conditions for optimality for controlled dissipative systems in finite dimensions. The strategy we employ is mainly built on duality techniques. We are able to construct a dual process for all positive times via the analysis of a suitable class of perturbed linearized forward equations. We show that such a process is the unique bounded solution to a backward SDE on infinite horizon from which we can write a version of the SMP.

Lipschitz regularity results for nonlinear strictly elliptic equations and applications

Most of Lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable (subquadratic for instance). For equations with superquadratic growth power in gradient, one usually uses weak Bernstein-type arguments which require regularity and/or convex-type assumptions on the gradient nonlinearity. In this article, we obtain new Lipschitz regularity results for a large class of nonlinear strictly elliptic equations with possibly arbitrary growth power of the Hamiltonian with respect to the gradient variable using some ideas coming from Ishii–Lions' method. We use these bounds to solve an ergodic problem and to study the regularity and the large time behavior of the solution of the evolution equation.

A hierarchical finite element Monte Carlo method for stochastic two-scale elliptic equations

We consider two-scale elliptic equations whose coefficients are random. In particular, we study two cases: in the first case, the coefficients are obtained from an ergodic dynamical system acting on a probability space, and in the second the case, the coefficients are periodic in the microscale but are random. We suppose that the coefficients also depend on the macroscopic slow variables. While the effective coefficient of the ergodic homogenization problem is deterministic, to approximate it, it is necessary to solve cell equations in a large but finite size “truncated” cube and compute an approximated effective coefficient from the solution of this equation. This approximated effective coefficient is, however, realization dependent; and the deterministic effective coefficient of the homogenization problem can be approximated by taking its expectation. In the periodic random setting, the effective coefficient for each realization is obtained from the solutions of cell equations which are posed in the unit cube, but to compute its average by the Monte Carlo method, we need to consider many uncorrelated realizations to accurately approximate the average. Straightforward employment of finite element approximation and the Monte Carlo method to compute this expectation with the same level of finite element resolution and the same number of Monte Carlo samples at every macroscopic point is prohibitively expensive. We develop a hierarchical finite element Monte Carlo algorithm to approximate the effective coefficients at a dense hierarchical network of macroscopic points. The method requires an optimal level of complexity that is essentially equal to that for computing the effective coefficient at one macroscopic point, and achieves essentially the same accuracy. The levels of accuracy for solving cell problems and for the Monte Carlo sampling are chosen according to the level in the hierarchy that the macroscopic points belong to. Solutions and the effective coefficients at the points where the cell problems are solved with higher accuracy and the effective coefficients are approximated with a larger number of Monte Carlo samples are employed as correctors for the effective coefficient at those points at which the cell problems are solved with lower accuracy and fewer Monte Carlo samples. The method combines the hierarchical finite element method for solving cell problems at a dense network of macroscopic points with the optimal complexity developed in Brown et al. (2013), with a hierarchical Monte Carlo sampling algorithm that uses different number of samples at different macroscopic points depending on the level in the hierarchy that the macroscopic points belong to. Proof of concept numerical examples confirm the theoretical results.

Asymptotic behaviour for operators of Grushin type: Invariant measure and singular perturbations

This paper concerns singular perturbation problems where the dynamics of the fast variable evolve in the whole space according to an operator whose infinitesimal generator is formed by a Grushin type second order part and a Ornstein-Uhlenbeck first order part. We prove that the dynamics of the fast variables admits an invariant measure and that the associated ergodic problem has a viscosity solution which is also regular and with logarithmic growth at infinity. These properties play a crucial role in the main theorem which establishes that the value functions of the starting perturbation problems converge to the solution of an effective problem whose operator and initial datum are given in terms of the associated invariant measure.

Blow-up solutions for some nonlinear elliptic equations involving a Finsler-Laplacian

In this paper we prove existence results and asymptotic behavior for strong solutions $u\in W^{2,2}_{\textrm{loc}}(\Omega)$ of the nonlinear elliptic problem \begin{equation} \tag{P} \label{abstr} \left\{ \begin{array}{ll} -\Delta_{H}u+H(\nabla u)^{q}+\lambda u=f&\text{in }\Omega, u\rightarrow +\infty &\text{on }\partial\Omega, \end{array} \right. \end{equation} where $H$ is a suitable norm of $\mathbb R^{n}$, $\Omega$ is a bounded domain, $\Delta_{H}$ is the Finsler Laplacian, $1 0$ and $f$ is a suitable function in $L^{\infty}_{\textrm{loc}}$. Furthermore, we are interested in the behavior of the solutions when $\lambda\rightarrow 0^{+}$, studying the so-called ergodic problem associated to \eqref{abstr}. A key role in order to study the ergodic problem will be played by local gradient estimates for \eqref{abstr}.

Resource Allocation for Weighted Sum-Rate Maximization in Multi-User Full-Duplex Device-to-Device Communications: Approaches for Perfect and Statistical CSIs

In this paper, we investigate the resource allocation problem for multi-user full-duplex device-to-device (D2D) underlay communication, considering both perfect channel state information (CSI) and statistical CSI scenarios. In perfect CSI scenario, the weighted sum-rate maximization problem under cellular users’ minimum rate constraints is formulated as a mixed integer programming problem. To solve the challenging problem, we decouple it into two subproblems as power allocation and channel assignment. Then we proposed a power allocation algorithm based on difference of two convex functions programming and a channel assignment algorithm based on Kuhn–Munkres algorithm, respectively. In statistical CSI scenario, we formulate the resource allocation problem as an outage probability constrained weighted ergodic sum-rate maximization problem. To solve the problem, the closed-form expressions of outage probability and weighted ergodic sum-rate are derived first. Then we decouple resource allocation problem into power allocation and channel assignment. An optimization solution that consists of a 2-D global searching and Kuhn–Munkres algorithm is then developed. Simulation results demonstrate that the proposed algorithms can improve the weighted sum-rate of full-duplex D2D communications significantly both in perfect CSI and statistical CSI scenarios and confirm the accuracy of our derived closed-form expressions.

A Correction to “A Relative Value Iteration Algorithm for Nondegenerate Controlled Diffusions''

In A Relative Value Iteration Algorithm for Nondegenerate Controlled Diffusions, [SIAM J. Control Optim., 50 (2012), pp. 1886--1902], convergence of the relative value iteration for the ergodic control problem for a nondegenerate diffusion controlled through its drift was established, under the assumption of geometric ergodicity, using two methods: (a) the theory of monotone dynamical systems and (b) the theory of reverse martingales. However, in the proof using (a) it is wrongly claimed that the semiflow is strong order preserving. In this note, we provide a simple generic proof and also comment on how to relax the uniform geometric ergodicity hypothesis.

Qualitative properties of generalized principal eigenvalues for superquadratic viscous Hamilton–Jacobi equations

This paper is concerned with the ergodic problem for superquadratic viscous Hamilton-Jacobi equations with exponent m \textgreater{} 2. We prove that the generalized principal eigenvalue of the equation converges to a constant as m $\rightarrow$ $\infty$, and that the limit coincides with the generalized principal eigenvalue of an ergodic problem with gradient constraint. We also investigate some qualitative properties of the generalized principal eigenvalue with respect to a perturbation of the potential function. It turns out that different situations take place according to m = 2, 2 \textless{} m \textless{} $\infty$, and the limiting case m = $\infty$.

On unbounded solutions of ergodic problems in ℝm for viscous Hamilton–Jacobi equations

ABSTRACTIn this article, we study ergodic problems in the whole space ℝm for viscous Hamilton–Jacobi equations in the case of locally Lipschitz continuous and coercive right-hand sides. We prove in particular the existence of a critical value λ* for which (i) the ergodic problem has solutions for all λ≤λ*, (ii) bounded from below solutions exist and are associated to λ*, (iii) such solutions are unique (up to an additive constant). We obtain these properties without additional assumptions in the superquadratic case, while, in the subquadratic one, we assume the right-hand side to behave like a power. These results are slight generalizations of analogous results by Ichihara but they are proved in the present paper by partial differential equation (pde) methods, contrarily to Ichihara who is using a combination of pde technics with probabilistic arguments.

Selection problems for a discount degenerate viscous Hamilton–Jacobi equation

We prove that the solution of the discounted approximation of a degenerate viscous Hamilton–Jacobi equation with convex Hamiltonians converges to that of the associated ergodic problem. We characterize the limit in terms of stochastic Mather measures by using the nonlinear adjoint method and deriving a commutation lemma. This convergence result was first proven by Davini, Fathi, Iturriaga, and Zavidovique for first order Hamilton–Jacobi equations.

The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus

We develop a variational approach to the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations. Under mild assumptions, we introduce viscosity Mather measures for such partial differential equations, which are natural extensions of the Mather measures. Using the viscosity Mather measures, we prove that the whole family of solutions vλ of the discount problem with the factor λ>0 converges to a solution of the ergodic problem as λ→0.

Ergodic diffusion control of multiclass multi-pool networks in the Halfin–Whitt regime

We consider Markovian multiclass multi-pool networks with heterogeneous server pools, each consisting of many statistically identical parallel servers, where the bipartite graph of customer classes and server pools forms a tree. Customers form their own queue and are served in the first-come first-served discipline, and can abandon while waiting in queue. Service rates are both class and pool dependent. The objective is to study the limiting diffusion control problems under the long run average (ergodic) cost criteria in the Halfin–Whitt regime. Two formulations of ergodic diffusion control problems are considered: (i) both queueing and idleness costs are minimized, and (ii) only the queueing cost is minimized while a constraint is imposed upon the idleness of all server pools. We develop a recursive leaf elimination algorithm that enables us to obtain an explicit representation of the drift for the controlled diffusions. Consequently, we show that for the limiting controlled diffusions, there always exists a stationary Markov control under which the diffusion process is geometrically ergodic. The framework developed in [Ann. Appl. Probab. 25 (2015) 3511–3570] is extended to address a broad class of ergodic diffusion control problems with constraints. We show that the unconstrained and constrained problems are well posed, and we characterize the optimal stationary Markov controls via HJB equations.

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.

Karl Wilhelm Butzer: (19 August 1934–4 May 2016)

Previous articleNext article No AccessKarl Wilhelm Butzer: (19 August 1934–4 May 2016)Richard G. KleinRichard G. KleinStanford University Search for more articles by this author Stanford UniversityPDFPDF PLUSFull Text Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by Journal of Anthropological Research Volume 72, Number 3Fall 2016 Article DOIhttps://doi.org/10.1086/688413 Views: 70Total views on this site Citations: 1Citations are reported from Crossref HistoryPublished online July 19, 2016 © 2016 by The University of New Mexico. All rights reserved.PDF download Crossref reports the following articles citing this article:Paul L. Butzer, Rudolf L. Stens Exterior Dirichlet and Neumann Problems and the Linked Ergodic Inverse Problems in the Entire Space, Journal of Fourier Analysis and Applications 28, no.22 (Mar 2022).https://doi.org/10.1007/s00041-022-09929-3