Backward Operator Research Articles

The Spectrum of Second Order Quantum Difference Operator

In this study, we give another generalization of second order backward difference operator ∇2 by introducing its quantum analog ∇q2. The operator ∇q2 represents the third band infinite matrix. We construct its domains c0(∇q2) and c(∇q2) in the spaces c0 and c of null and convergent sequences, respectively, and establish that the domains c0(∇q2) and c(∇q2) are Banach spaces linearly isomorphic to c0 and c, respectively, and obtain their Schauder bases and α-, β- and γ-duals. We devote the last section to determine the spectrum, the point spectrum, the continuous spectrum and the residual spectrum of the operator ∇q2 over the Banach space c0 of null sequences.

Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noise

We provide a complete elaboration of the L^2-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behavior of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker–Planck framework and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to weakly differentiable diffusion coefficient matrices, introducing multiplicative noise for the corresponding stochastic differential equation. The rate of convergence is explicitly computed depending on the choice of these coefficients and the potential giving the outer force. In order to obtain a solution to the abstract Cauchy problem, we first prove essential self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert spaces, using prior elliptic regularity results and techniques from Bogachev, Krylov and Röckner. We apply operator perturbation theory to obtain essential m-dissipativity of the Kolmogorov operator, extending the m-dissipativity results from Conrad and Grothaus. We emphasize that the chosen Kolmogorov approach is natural, as the theory of generalized Dirichlet forms implies a stochastic representation of the Langevin semigroup as the transition kernel of a diffusion process which provides a martingale solution to the Langevin equation with multiplicative noise. Moreover, we show that even a weak solution is obtained this way.

Operators based on Pólya distribution and finite differences

The several generalizations of the well‐known Bernstein polynomials are available in the literature. The general form of Bernstein polynomial based on Pólya distribution and its Kantorovich variant available in the literature have no links. In order to introduce a modification of a known operator, it is important to have some connection with the original one. Here, we provide a link between the such operators and its Kantorovich type integral variant using the method of backward difference operator and find another generalization, different than the one proposed before by Razi. We also provide some direct results for the Kantorovich operators. Finally, we also introduce the higher order Pólya–Kantorovich operators.

Development of a Generalized Predictive Control System for Polynomial Reference Tracking

Some important applications require control systems capable of tracking high-degree polynomial references. However, polynomial reference tracking is difficult because the controller must have a fast response to track variable references. Generalized Predictive Control (GPC), used in industrial applications, has a fast response. However, up to now, GPC approaches in the literature are not designed for high-degree polynomial reference tracking. For that reason, this brief proposes a GPC system for the tracking of polynomial references of any degree. The high-order backward difference operator was used to develop an augmented prediction model designed to have an arbitrary number of embedded integrators. Thus, according to the internal model principle, the proposed predictive controller can track polynomial references of any degree. An optimization technique is used to define future control actions, while the control law is defined through the receding horizon principle. Simulation and experimental results prove that the proposed controller performs better than other approaches described in the literature.

Composition of Ces\xe0ro and backward difference operators

In this research, we combine the Cesàro and backward difference operators of different orders which results in introducing a matrix who has two different behaviors and includes several matrices. We also investigate the Köthe duals and inclusion relations of the associated sequence space of this new matrix. Moreover, we compute the norm of this matrix on some well-known sequence spaces.

A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

For an attracting periodic orbit (limit cycle) of a deterministic dynamical system, one defines the isochron for each point of the orbit as the cross-section with fixed return time under the flow. Equivalently, isochrons can be characterized as stable manifolds foliating neighborhoods of the limit cycle or as level sets of an isochron map. In recent years, there has been a lively discussion in the mathematical physics community on how to define isochrons for stochastic oscillations, i.e. limit cycles or heteroclinic cycles exposed to stochastic noise. The main discussion has concerned an approach finding stochastic isochrons as sections of equal expected return times versus the idea of considering eigenfunctions of the backward Kolmogorov operator. We discuss the problem in the framework of random dynamical systems and introduce a new rigorous definition of stochastic isochrons as random stable manifolds for random periodic solutions with noise-dependent period. This allows us to establish a random version of isochron maps whose level sets coincide with the random stable manifolds. Finally, we discuss links between the random dynamical systems interpretation and the equal expected return time approach via averaged quantities.

On a problem of linearized stability for fractional difference equations

This paper discusses the problem of linearized stability for nonlinear fractional difference equations. Computational methods based on appropriate linearization theorem are standardly applied in bifurcation analysis of dynamical systems. However, in the case of fractional discrete systems, a theoretical background justifying its use is still missing. Therefore, the main goal of this paper is to fill in the gap. We consider a general autonomous system of fractional difference equations involving the backward Caputo fractional difference operator and prove that any equilibrium of this system is asymptotically stable if the zero solution of the corresponding linearized system is asymptotically stable. Moreover, these asymptotic stability conditions for equilibria of the system are described via location of all the characteristic roots in a specific area of the complex plane. In the planar case, these conditions are given even explicitly in terms of trace and determinant of the appropriate Jacobi matrix. The results are applied to a fractional predator-prey model and the fractional Lorenz model. Related experiments are supported by a numerical code that is appended as well

Negative Difference Operator and Its Associated Sequence Space

Let be the backward difference operator of order −n. In this paper, we study some properties of the matrix domain associated with this operator and after computing α-, β-, γ-duals, we characterize some matrix classes on this space. Moreover, we investigate the problem of finding the norm of well-known operators such as Cesàro and Hilbert from into

On a new type of fractional difference operators on h-step isolated time scales

In this article, a new type of fractional sums and differences called the discrete weighted fractionaloperators are presented. The weighted backward and forward difference operators are defined on anisolated time scale with arbitrary step size and they obey the power law.

Discrete Subdiffusion Equations with Memory

In this paper, we study a discrete subdiffusion equation with memory. Based on the backward operator and the theory of fractional resolvent families, we find a discrete fractional resolvent sequence which allows to write the solution to this discrete subdiffusion equation as a variation of constant formula.

A survey on the spectra of the difference operators over the Banach space c

Recently, due to its numerous applications, the spectra of the bounded operators over Banach spaces have been studied extensively. This work aims to collect some of the investigations on the spectra of difference operators or matrices on the Banach space c in the literature and provide a foundation for related problems. To the best of our investigations, the problem has been solved over the sequence space c so far up to maximum order 2. In the present work, the fine spectra of the difference operator $$\Delta ^m, m\in \mathbb {N}$$ on c have been computed. The generalized difference operator $$\Delta ^m$$ on the Banach space c is defined by $$(\Delta ^mx)_k= \sum _{i=0}^m(-1)^i\left( {\begin{array}{c}m\\ i\end{array}}\right) x_{k-i},\;k=0,1,2,3,\dots $$ with $$ x_{k} = 0$$ for $$k<0$$ . Indeed, the operator $$\Delta ^m$$ is represented by an $$(m+1)$$ -th band matrix which generalizes several difference operators such as $$\Delta ,\Delta ^2,B(r,s)$$ and B(r, s, t) etc, under different limiting conditions. Initially, we provide some essential background results on the linearity and boundedness of the backward difference operator $$\Delta ^m$$ . Finally, the sets for the spectrum and fine spectra such as the point spectrum, the continuous spectrum and the residual spectrum of the defined operator on the space c have been computed. The geometrical interpretation for the spectral subdivisions of the above difference operator is also incorporated.

Backward Recursive Method for Structural Reliability Calculation Based on the Universal Generating Function

For structural performance functions with characteristics of multivariable, non-normality, small failure probability and nonlinearity, it is usually difficult to pursue satisfactory accuracy by means of traditional probabilistic methods such as first-order reliability method and second-order reliability method. According to the idea of adaptive subdivision, a backward recursive method based on the universal generating function is suggested for structural reliability calculation. In the sensitive region, the variable space is subdivided partially, the discrete interval length is reduced, and the random variables are discretized nonuniformly and adaptively by the backward recursion operation. Compared with the current method of forward recursion operator and the corresponding similar items merging operation, the backward recursion operator can not only avoid low accuracy owing to subdividing the state combinations in the sensitive region, but avert the high cost of calculation owing to merging state combinations in the nonsensitive region of the variable space as well. Three examples are presented to demonstrate the rationality of the backward recursive algorithm. The results indicate that the accuracy of the proposed method is significantly higher than that of traditional methods, and the computational efficiency of the proposed method meets the engineering requirement. The research work provides a new path for reliability calculation with consideration of complex performance functions.

Sparse Reconstruction for Near-Field MIMO Radar Imaging Using Fast Multipole Method

Radar imaging using multiple input multiple output systems are becoming popular recently. These applications typically contain a sparse scene and the imaging system is challenged by the requirement of high quality real-time image reconstruction from under-sampled measurements via compressive sensing. In this paper, we deal with obtaining sparse solution to near- field radar imaging problems by developing efficient sparse reconstruction, which avoid storing and using large-scale sensing matrices. We demonstrate that the “fast multipole method” can be employed within sparse reconstruction algorithms to efficiently compute the sensing operator and its adjoint (backward) operator, hence improving the computation speed and memory usage, especially for large-scale 3-D imaging problems. For several near-field imaging scenarios including point scatterers and 2-D/3-D extended targets, the performances of sparse reconstruction algorithms are numerically tested in comparison with a classical solver. Furthermore, effectiveness of the fast multipole method and efficient reconstruction are illustrated in terms of memory requirement and processing time.

On Λ-Fibonacci difference sequence spaces of fractional order

In this article, we introduce -Fibonacci difference operator of fractional order which is obtained by the composition of -Fibonacci matrix and backward fractional difference operator defined by and introduce the sequence spaces and We give some topological properties, and obtain the Schauder basis of the new spaces.

A Distributed Forward–Backward Algorithm for Stochastic Generalized Nash Equilibrium Seeking

We consider the stochastic generalized Nash equilibrium problem (SGNEP) with expected-value cost functions. Inspired by Yi and Pavel (2019), we propose a distributed generalized Nash equilibrium seeking algorithm based on the preconditioned forward–backward operator splitting for SGNEPs, where, at each iteration, the expected value of the pseudogradient is approximated via a number of random samples. Our main contribution is to show almost sure convergence of the proposed algorithm if the pseudogradient mapping is restricted (monotone and) cocoercive.

PDE Evolutions for M-Smoothers in One, Two, and Three Dimensions

Local M-smoothers are interesting and important signal and image processing techniques with many connections to other methods. In our paper, we derive a family of partial differential equations (PDEs) that result in one, two, and three dimensions as limiting processes from M-smoothers which are based on local order-p means within a ball the radius of which tends to zero. The order p may take any nonzero value $$>-1$$ , allowing also negative values. In contrast to results from the literature, we show in the space-continuous case that mode filtering does not arise for $$p \rightarrow 0$$ , but for $$p \rightarrow -1$$ . Extending our filter class to p-values smaller than $$-1$$ allows to include, e.g. the classical image sharpening flow of Gabor. The PDEs we derive in 1D, 2D, and 3D show large structural similarities. Since our PDE class is highly anisotropic and may contain backward parabolic operators, designing adequate numerical methods is difficult. We present an $$L^\infty $$ -stable explicit finite difference scheme that satisfies a discrete maximum–minimum principle, offers excellent rotation invariance, and employs a splitting into four fractional steps to allow larger time step sizes. Although it approximates parabolic PDEs, it consequently benefits from stabilisation concepts from the numerics of hyperbolic PDEs. Our 2D experiments show that the PDEs for $$p<1$$ are of specific interest: Their backward parabolic term creates favourable sharpening properties, while they appear to maintain the strong shape simplification properties of mean curvature motion.

Single-Timescale Distributed GNE Seeking for Aggregative Games Over Networks via Forward–Backward Operator Splitting

We consider aggregative games with affine coupling constraints, where agents have partial information on the aggregate value and can only communicate with neighbouring agents. We propose a single-layer distributed algorithm that reaches a variational generalized Nash equilibrium, under constant step sizes. The algorithm works on a single timescale, i.e., does not require multiple communication rounds between agents before updating their action. The convergence proof leverages an invariance property of the aggregate estimates and relies on a forward-backward splitting for two preconditioned operators and their restricted (strong) monotonicity properties on the consensus subspace.

Compressive sensing‐based Stolt migration imaging algorithm for impulse through‐the‐wall radar

In this Letter, the authors propose using compressive sensing (CS)-based Stolt migration imaging approach to reduce the amount of measurement data and provide an enhanced imaging quality. The authors utilise both forward and backward Stolt migration imaging operators to deal with the measurement and image the space data in two-dimensional matrix form for sparse recovery computation. This imaging process accounts for the electromagnetic wave propagation and does not require prior knowledge of the transmitted pulse, which is practically not available in practice. Results of the proposed numerical simulation demonstrate the validity and effectiveness of the proposed imaging algorithm.

A Discrete Version of Plane Wave Solutions of the Dirac Equation in the Joyce Form

We construct a discrete version of the plane wave solution to a discrete Dirac-Kähler equation in the Joyce form. A geometric discretisation scheme based on both forward and backward difference operators is used. The conditions under which a discrete plane wave solution satisfies a discrete Joyce equation are discussed.

Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator.

Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.