Spectra of Elliptic Operators on Quantum Graphs with Small Edges

We consider a general second order self-adjoint elliptic operator on an arbitrary metric graph, to which a small graph is glued. This small graph is obtained via rescaling a given fixed graph γ by a small positive parameter ε. The coefficients in the differential expression are varying, and they, as well as the matrices in the boundary conditions, can also depend on ε and we assume that this dependence is analytic. We introduce a special operator on a certain extension of the graph γ and assume that this operator has no embedded eigenvalues at the threshold of its essential spectrum. It is known that under such assumption the perturbed operator converges to a certain limiting operator. Our main results establish the convergence of the spectrum of the perturbed operator to that of the limiting operator. The convergence of the spectral projectors is proved as well. We show that the eigenvalues of the perturbed operator converging to limiting discrete eigenvalues are analytic in ε and the same is true for the associated perturbed eigenfunctions. We provide an effective recurrent algorithm for determining all coefficients in the Taylor series for the perturbed eigenvalues and eigenfunctions.

Developing a multi-step recurrent algorithm to maximize the criteria of correntropy

This paper considers the task of constructing a linear model of the object studied using a robust criterion. The functionality applied, in this case, is correntropy. That makes it possible to obtain estimates that have robust properties. The evaluation algorithm is a multi-step procedure that employs a limited number of information measurements, that is, it has limited memory. The feature of the algorithm is that the matrices and observation vectors involved in estimate construction are formed in the following way: they include information about the newly arrived measurements and exclude information about the oldest ones. Depending on the way these matrices and vectors are built (new information is added first, and then outdated is excluded, or the outdated is first excluded, and then a new one is added), two estimate forms are possible. The second Lyapunov method is used to study the convergence of the algorithm. The conditions of convergence for a multi-step algorithm have been defined. The analysis of the established regime has revealed that the algorithm ensures that unbiased estimates are obtained. It should be noted that all the estimates reported in this work depend on the choice of the width of the nucleus, the information weighting factor, and the algorithm memory, the task of determining which remains open. Therefore, these parameters' estimates should be applied for the practical use of such multi-step algorithms. The estimates obtained in this paper allow the researcher to pre-evaluate the possibilities of identification using a multi-step algorithm, as well as the effectiveness of its application when solving practical tasks

BFRVSR: A Bidirectional Frame Recurrent Method for Video Super-Resolution

Video super-resolution is a challenging task. One possible solution, called the sliding window method, tries to divide the generation of high-resolution video sequences into independent subtasks. Another popular method, named the recurrent algorithm, utilizes the generated high-resolution images of previous frames to generate the high-resolution image. However, both methods have some unavoidable disadvantages. The former method usually leads to bad temporal consistency and has higher computational cost, while the latter method cannot always make full use of information contained by optical flow or any other calculated features. Thus, more investigations need to be done to explore the balance between these two methods. In this work, a bidirectional frame recurrent video super-resolution method is proposed. To be specific, reverse training is proposed that also utilizes a generated high-resolution frame to help estimate the high-resolution version of the former frame. The bidirectional recurrent method guarantees temporal consistency and also makes full use of the adjacent information due to the bidirectional training operation, while the computational cost is acceptable. Experimental results demonstrate that the bidirectional super-resolution framework gives remarkable performance and it solves time-related problems.

RECURRENT REPRESENTATION FOR NON-STATIONARY PARAMETER ESTIMATE OF LEAST SQUARES METHOD WITH LEAST DEVIATIONS FROM "ATTRACTION" POINTS FOR BILINEAR DYNAMIC SYSTEMS

The estimation problem of non-stationary parameter matrices is considered for bilinear discrete dynamic system in the case when for these unknown parameter matrices their ‘attraction’ points are known at any moment. Explicit and recurrent forms of representation are obtained for these parameter estimates of the least squares method with variable forgetting factor and least deviation norm from ‘attraction’ points under non-classical assumptions. The recurrent algorithm is also proposed for corresponding weighted residual sum of squares.

Adaptation of the mathematical apparatus of the Markov chain theory for the probabilistic analysis of recurrent estimation of image inter-frame geometric deformations

The paper is devoted to the analysis of the possibilities of using Markov chains for analyzing the accuracy of stochastic gradient relay estimation of image geometric deformations. One of the ways to reduce computational costs is to discretize the domain of studied parameters. This approach allows to choose the dimension of transition probabilities matrix a priori. However, such a matrix has a rather complicated structure. It does not significantly reduce the number of computations. A modification of the transition probabilities matrix is proposed, it’s dimension does not depend on the dimension of estimated parameters vector. In this case, the obtained relations determine a recurrent algorithm for calculating the matrix at the estimation iterations. For the one-step transitions matrix, the calculated expressions for the probabilities of image deformation parameters estimates drift are given.

RECURSIVE ALGORITHM FOR ESTIMATING THE PARAMETERS OF MULTIDIMENSIONAL DISCRETE LINEAR DYNAMIC SYSTEMS OF DIFFERENT ORDERS WITH ERRORS ON THE INPUT

The paper presents a recurrent algorithm for estimating the parameters of multidimensional discrete linear dynamical systems of different orders with input errors, described by white noise. It is proved that the obtained estimates using stochastic gradient algorithm for minimization of quadratic forms are highly consistent

Recurrent algorithm for non-stationary parameter estimation by least squares method with least deviations from ‘attraction’ points for bilinear discrete dynamic systems

The estimation problem of slowly time-varying parameter matrices is considered for bilinear discrete dynamic system in the presence of disturbances. The least squares estimate with variable forgetting factor is investigated for this object in non-classical situation when this estimate may be not unique and additionally ‘attraction’ points for unknown parameter matrices are given at any moment. The set of all above-mentioned estimates of these unknown matrices is defined through the Moore-Penrose pseudo-inverse operator. The least squares estimate with variable forgetting factor and least deviation norm from given ‘attraction’ point at any moment is proposed as unique estimate on this set of all estimates. The explicit form of representation is obtained for this unique estimate of the parameter matrices by the least squares method with variable forgetting factor and least deviation norm from given ‘attraction’ points under non-classical assumptions. The recurrent algorithm for this estimate is also derived which does not require the usage of the matrix pseudo-inverse operator.