Backward Operator Research Articles

Distributed Charging Strategy of PEVs in SCS with Feeder Constraints Based on Generalized Nash Equilibria

In this article, a distributed charging strategy problem for plug-in electric vehicles (PEVs) with feeder constraints based on generalized Nash equilibria (GNE) in a novel smart charging station (SCS) is investigated. The purpose is to coordinate the charging strategies of all PEVs in SCS to minimize the energy cost of SCS. Therefore, we build a non-cooperative game framework and propose a new price-driven charging control game by considering the overload constraint of the assigned feeder, where each PEV minimizes the fees it pays to satisfy its optimal charging strategy. On this basis, the existence of GNE is given. Furthermore, we employ a distributed algorithm based on forward–backward operator splitting methods to find the GNE. The effectiveness of the employed algorithm is verified by the final simulation results.

Koszul duality, minimal model and L∞-structure for differential algebras with weight

A differential algebra with weight is an abstraction of both the derivation (weight zero) and the forward and backward difference operators (weight ±1). In 2010 Loday established the Koszul duality for the operad of differential algebras of weight zero. He did not treat the case of nonzero weight, noting that new techniques are needed since the operad is no longer quadratic. This paper continues Loday's work and establishes the Koszul duality in the case of nonzero weight. In the process, the minimal model and the Koszul dual homotopy cooperad of the operad governing differential algebras with weight are determined. As a consequence, a notion of homotopy differential algebras with weight is obtained and the deformation complex as well as its L∞-algebra structure for differential algebras with weight are deduced.

BELL DIFFERENCE SEQUENCE SPACES

In this article we define the new sequence spaces Bˆ ∞ (∆), Bˆ c (∆) and Bˆ 0 (∆) by introducing a new regular matrix Bˆ ∆ with the composition of the Bell matrix Bˆ and the backward difference operator ∆. Also we have determined the Schauder basis, certain inclusion relations, α−, β− and γ−duals along with matrix transformations.

Novel finite difference approach to discretize the symplectic dirac operator

Symplectic Dirac operator is an intertwining differential operator. Discretising symplectic Dirac operator gives a new direction to study the quantum space. The construction of discrete symplectic Dirac operator requires the theory of discrete symplectic Clifford analysis or the concept of discrete symplectic connections, which are not explained in literature. In this work, a discretization approach for symplectic Dirac operator is suggested by considering the forward and backward basis vectors on symplectic Clifford algebra. The suggested discrete symplectic Dirac operator is Ds=Ds++Ds- where the Ds+ and Ds- are the forward and backward discrete symplectic Dirac operators, respectively. The new discrete symplectic Dirac operator gives the factorization of discrete Laplacian on symplectic spaces. Further, we establish commutation relations involving forward and backward discrete symplectic Dirac operators in the representation theory.

A universal description of stochastic oscillators

Many systems in physics, chemistry, and biology exhibit oscillations with a pronounced random component. Such stochastic oscillations can emerge via different mechanisms, for example, linear dynamics of a stable focus with fluctuations, limit-cycle systems perturbed by noise, or excitable systems in which random inputs lead to a train of pulses. Despite their diverse origins, the phenomenology of random oscillations can be strikingly similar. Here, we introduce a nonlinear transformation of stochastic oscillators to a complex-valued function [Formula: see text](x) that greatly simplifies and unifies the mathematical description of the oscillator's spontaneous activity, its response to an external time-dependent perturbation, and the correlation statistics of different oscillators that are weakly coupled. The function [Formula: see text] (x) is the eigenfunction of the Kolmogorov backward operator with the least negative (but nonvanishing) eigenvalue λ1 = μ1 + iω1. The resulting power spectrum of the complex-valued function is exactly given by a Lorentz spectrum with peak frequency ω1 and half-width μ1; its susceptibility with respect to a weak external forcing is given by a simple one-pole filter, centered around ω1; and the cross-spectrum between two coupled oscillators can be easily expressed by a combination of the spontaneous power spectra of the uncoupled systems and their susceptibilities. Our approach makes qualitatively different stochastic oscillators comparable, provides simple characteristics for the coherence of the random oscillation, and gives a framework for the description of weakly coupled oscillators.

Hypocoercivity for non-linear infinite-dimensional degenerate stochastic differential equations

The aim of this article is to construct solutions to second order in time stochastic partial differential equations and to show hypocoercivity of the corresponding transition semigroups. More generally, we analyze non-linear infinite-dimensional degenerate stochastic differential equations in terms of their infinitesimal generators. In the first part of this article we use resolvent methods developed by Beznea, Boboc and Röckner to construct diffusion processes with infinite lifetime and explicit invariant measures. The processes provide weak solutions to infinite-dimensional Langevin dynamics. The second part deals with a general abstract Hilbert space hypocoercivity method, developed by Dolbeaut, Mouhot and Schmeiser. In order to treat stochastic (partial) differential equations, Grothaus and Stilgenbauer translated these concepts to the Kolmogorov backwards setting taking domain issues into account. To apply these concepts in the context of infinite-dimensional Langevin dynamics we use an essential m-dissipativity result for infinite-dimensional Ornstein–Uhlenbeck operators, perturbed by the gradient of a potential. We allow unbounded diffusion operators as coefficients and apply corresponding regularity estimates. Furthermore, essential m-dissipativity of a non-sectorial Kolmogorov backward operator associated to the dynamic is needed. Poincaré inequalities for measures with densities w.r.t. infinite-dimensional non-degenerate Gaussian measures are studied. Deriving a stochastic representation of the semigroup generated by the Kolmogorov backward operator as the transition semigroup of a diffusion process enables us to show an L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-exponential ergodicity result for the weak solution. Finally, we apply our results to explicit infinite-dimensional degenerate diffusion equations.

Novel Algorithms with Inertial Techniques for Solving Constrained Convex Minimization Problems and Applications to Image Inpainting

In this paper, we propose two novel inertial forward–backward splitting methods for solving the constrained convex minimization of the sum of two convex functions, φ1+φ2, in Hilbert spaces and analyze their convergence behavior under some conditions. For the first method (iFBS), we use the forward–backward operator. The step size of this method depends on a constant of the Lipschitz continuity of ∇φ1, hence a weak convergence result of the proposed method is established under some conditions based on a fixed point method. With the second method (iFBS-L), we modify the step size of the first method, which is independent of the Lipschitz constant of ∇φ1 by using a line search technique introduced by Cruz and Nghia. As an application of these methods, we compare the efficiency of the proposed methods with the inertial three-operator splitting (iTOS) method by using them to solve the constrained image inpainting problem with nuclear norm regularization. Moreover, we apply our methods to solve image restoration problems by using the least absolute shrinkage and selection operator (LASSO) model, and the results are compared with those of the forward–backward splitting method with line search (FBS-L) and the fast iterative shrinkage-thresholding method (FISTA).

Stochastic variable metric proximal gradient with variance reduction for non-convex composite optimization

This paper introduces a novel algorithm, the Perturbed Proximal Preconditioned SPIDER algorithm (3P-SPIDER), designed to solve finite sum non-convex composite optimization. It is a stochastic Variable Metric Forward–Backward algorithm, which allows approximate preconditioned forward operator and uses a variable metric proximity operator as the backward operator; it also proposes a mini-batch strategy with variance reduction to address the finite sum setting. We show that 3P-SPIDER extends some Stochastic preconditioned Gradient Descent-based algorithms and some Incremental Expectation Maximization algorithms to composite optimization and to the case the forward operator can not be computed in closed form. We also provide an explicit control of convergence in expectation of 3P-SPIDER, and study its complexity in order to satisfy the approximate epsilon-stationary condition. Our results are the first to combine the non-convex composite optimization setting, a variance reduction technique to tackle the finite sum setting by using a minibatch strategy and, to allow deterministic or random approximations of the preconditioned forward operator. Finally, through an application to inference in a logistic regression model with random effects, we numerically compare 3P-SPIDER to other stochastic forward–backward algorithms and discuss the role of some design parameters of 3P-SPIDER.

Powerset operators induced by fuzzy relations as a basis for fuzzification of various mathematical structures

We propose a unified approach to the theory of forward and backward powerset operators induced by fuzzy relations with special attention to the algebraic and topological properties of these operators. On the basis of this theory, we reconsider previous studies done by various authors in the fields of fuzzy rough sets, fuzzy concept analysis and fuzzy mathematical morphology as well as obtain some new results in these areas. In this way we contribute to the development of the theory of fuzzy rough sets, fuzzy concept analysis and fuzzy mathematical morphology on the basis of fuzzy powerset operators induced by fuzzy relations.

On the characterization of matrix domains using Cesàro and backward difference operators

In this manuscript, the new sequence space Z? for Z ? {c, c0, ??} and ? = Cm? have been introduced using the Ces?ro matrix and backward difference operator. Some of the topological properties of these spaces have been studied and the existence of the Schauder basis for new spaces have been verified. Also, the ?, ? and ?-duals have been computed along with the characterization of the matrix transformation between new spaces.

Lensless complex amplitude demodulation based on deep learning in holographic data storage

To increase the storage capacity in holographic data storage (HDS), the information to be stored is encoded into a complex amplitude. Fast and accurate retrieval of amplitude and phase from the reconstructed beam is necessary during data readout in HDS. In this study, we proposed a complex amplitude demodulation method based on deep learning from a single-shot diffraction intensity image and verified it by a non-interferometric lensless experiment demodulating four-level amplitude and four-level phase. By analyzing the correlation between the diffraction intensity features and the amplitude and phase encoding data pages, the inverse problem was decomposed into two backward operators denoted by two convolutional neural networks (CNNs) to demodulate amplitude and phase respectively. The experimental system is simple, stable, and robust, and it only needs a single diffraction image to realize the direct demodulation of both amplitude and phase. To our investigation, this is the first time in HDS that multilevel complex amplitude demodulation is achieved experimentally from one diffraction intensity image without iterations.

A remark on the spectrum of fractional difference operator

This work computes the spectrum of the backward fractional difference operator over various Banach spaces and . Note that recently, the spectrum of the fractional difference operator , ( being either a constant or strictly decreasing sequence of positive real numbers satisfying some conditions) on the sequence space has been computed by Baliarsingh P. [On the spectrum of fractional difference operator. Linear Multilinear Algebra. 2021;1–13] and it has been noticed that the results obtained are unified and more general, but lack of accuracy for the case . In this work, we improve the results by providing the sharper estimations and some illustrative examples. Also, individual sets describing the point, the residual and the continuous spectrum of the operator on the Banach space c and are computed.

A generalized forward–backward splitting operator: degenerate analysis and applications

A generalized forward–backward splitting operator: degenerate analysis and applications

Stochastic Generalized Nash Equilibrium-Seeking in Merely Monotone Games

We solve the stochastic generalized Nash equilibrium (SGNE) problem in merely monotone games with expected value cost functions. Specifically, we present the first distributed SGNE-seeking algorithm for monotone games that require one proximal computation (e.g., one projection step) and one pseudogradient evaluation per iteration. Our main contribution is to extend the relaxed forward–backward operator splitting by the Malitsky (Mathematical Programming, 2019) to the stochastic case and in turn to show almost sure convergence to an SGNE when the expected value of the pseudogradient is approximated by the average over a number of random samples.

Symmetric Difference Operator in Quantum Calculus

The main focus of this paper is to develop certain types of fundamental theorems using q, q(α), and h difference operators. For several higher order difference equations, we get two forms of solutions: one is closed form and another is summation form. However, most authors concentrate only on the summation part. This motivates us to develop closed-form solutions, and we succeed. The key benefit of this research is finding the closed-form solutions for getting better results when compared to the summation form. The symmetric difference operator is the combination of forward and backward difference symmetric operators. Using this concept, we employ the closed and summation form for q, q(α), and h difference symmetric operators on polynomials, polynomial factorials, logarithmic functions, and products of two functions that act as a solution for symmetric difference equations. The higher order fundamental theorems of q and q(α) are difficult to find when the order becomes high. Hence, by inducing the h difference symmetric operator in q and q(α) symmetric operators, we find the solution easily and quickly. Suitable examples are given to validate our findings. In addition, we plot the figures to examine the value stability of q and q(α) difference equations.

A distributed Bregman forward-backward algorithm for a class of Nash equilibrium problems

We present a distributed Nash equilibrium seeking method based on the Bregman forward-backward splitting, which allows us to have a mirror mapping instead of the standard projection as the backward operator. Our main technical contribution is to show convergence to a Nash equilibrium when the game has cocoercive pseudogradient mapping. Furthermore, when the feasible sets of the agents are simplices, a suitable choice of a Legendre function results in an exponentiated pseudogradient method, which, in our numerical experience, performs out the standard projected pseudogradient and dual averaging methods.

On Polynomials Orthogonal with Respect to an Inner Product Involving Higher-Order Differences: The Meixner Case

In this contribution we consider sequences of monic polynomials orthogonal with respect to the Sobolev-type inner product f,g=⟨uM,fg⟩+λTjf(α)Tjg(α), where uM is the Meixner linear operator, λ∈R+, j∈N, α≤0, and T is the forward difference operator Δ or the backward difference operator ∇. Moreover, we derive an explicit representation for these polynomials. The ladder operators associated with these polynomials are obtained, and the linear difference equation of the second order is also given. In addition, for these polynomials, we derive a (2j+3)-term recurrence relation. Finally, we find the Mehler–Heine type formula for the particular case α=0.

A definition of the asymptotic phase for quantum nonlinear oscillators from the Koopman operator viewpoint.

We propose a definition of the asymptotic phase for quantum nonlinear oscillators from the viewpoint of the Koopman operator theory. The asymptotic phase is a fundamental quantity for the analysis of classical limit-cycle oscillators, but it has not been defined explicitly for quantum nonlinear oscillators. In this study, we define the asymptotic phase for quantum oscillatory systems by using the eigenoperator of the backward Liouville operator associated with the fundamental oscillation frequency. By using the quantum van der Pol oscillator with a Kerr effect as an example, we illustrate that the proposed asymptotic phase appropriately yields isochronous phase values in both semiclassical and strong quantum regimes.

An Improved Incipient Fault Diagnosis Method of Bearing Damage Based on Hierarchical Multi-Scale Reverse Dispersion Entropy.

The amplitudes of incipient fault signals are similar to health state signals, which increases the difficulty of incipient fault diagnosis. Multi-scale reverse dispersion entropy (MRDE) only considers difference information with low frequency range, which omits relatively obvious fault features with a higher frequency band. It decreases recognition accuracy. To defeat the shortcoming with MRDE and extract the obvious fault features of incipient faults simultaneously, an improved entropy named hierarchical multi-scale reverse dispersion entropy (HMRDE) is proposed to treat incipient fault data. Firstly, the signal is decomposed hierarchically by using the filter smoothing operator and average backward difference operator to obtain hierarchical nodes. The smoothing operator calculates the mean sample value and the average backward difference operator calculates the average deviation of sample values. The more layers, the higher the utilization rate of filter smoothing operator and average backward difference operator. Hierarchical nodes are obtained by these operators, and they can reflect the difference features in different frequency domains. Then, this difference feature is reflected with MRDE values of some hierarchical nodes more obviously. Finally, a variety of classifiers are selected to test the separability of incipient fault signals treated with HMRDE. Furthermore, the recognition accuracy of these classifiers illustrates that HMRDE can effectively deal with the problem that incipient fault signals cannot be easily recognized due to a similar amplitude dynamic.

TgEDMD: Approximation of the Kolmogorov Operator in Tensor Train Format

Extracting information about dynamical systems from models learned off simulation data has become an increasingly important research topic in the natural and engineering sciences. Modeling the Koopman operator semigroup has played a central role in this context. As the approximation quality of any such model critically depends on the basis set, recent work has focused on deriving data-efficient representations of the Koopman operator in low-rank tensor formats, enabling the use of powerful model classes while avoiding over-fitting. On the other hand, detailed information about the system at hand can be extracted from models for the infinitesimal generator, also called Kolmogorov backward operator for stochastic differential equations. In this work, we present a data-driven method to efficiently approximate the generator using the tensor train (TT) format. The centerpiece of the method is a TT representation of the tensor of generator evaluations at all data sites. We analyze consistency and complexity of the method, present extensions to practically relevant settings, and demonstrate its applicability to benchmark numerical examples.