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The paper considers the problem of parameter identification of discrete-time linear stochastic systems in the state space with additive and multiplicative noise. It is assumed that the state and measurements equations of a discrete-time linear stochastic system depend on an unknown parameter to be identified.

A new approach to the construction of gradient parameter identification methods in the class of discrete-time linear stochastic systems with additive and multiplicative noise is presented, based on the application of modified weighted Gram – Schmidt orthogonalization (MWGS) and the discrete-time information-type filtering algorithms.

The main theoretical results of this research include: 1) a new identification criterion in terms of an extended information filter; 2) a new algorithm for calculating derivatives with respect to an uncertainty parameter in a discrete-time linear stochastic system based on an extended information LD filter using the direct procedure of modified weighted Gram – Schmidt orthogonalization; and 3) a new method for calculating the gradient of identification criteria using a “differentiated” extended information LD filter.

The advantages of this approach are that it uses MWGS orthogonalization which is numerically stable against machine roundoff errors, and it forms the basis of all the developed methods and algorithms. The information LD-filter maintains the symmetry and positive definiteness of the information matrices. The algorithms have an array structure that is convenient for computer implementation.

All the developed algorithms were implemented in MATLAB. A series of numerical experiments were carried out. The results obtained demonstrated the operability of the proposed approach, using the example of solving the problem of parameter identification for a mathematical model of a complex mechanical system.

The results can be used to develop methods for identifying parameters in mathematical models that are represented in state space by discrete-time linear stochastic systems with additive and multiplicative noise.

This paper establishes quantitative conditions for the stability of the Quantum Phase Estimation (QPE) algorithm under the assumption of a uniform distribution of eigenvalues of the unitary operator. Using perturbation theory for linear operators, we demonstrate that the accuracy of phase estimation is fundamentally limited by a logarithmic dependence on the perturbation magnitude: the number of reliably recoverable binary digits of the phase satisfies the condition $n=o(-\log_2^{}(\epsilon))$. Furthermore, we show that distinct phases remain resolvable only if the perturbation does not exceed the minimal distance $\frac{1}{m}$ between adjacent phases, which leads to the condition $m=o\left(\epsilon^{-1}\right)$. These results reveal fundamental limitations on the resolving power of QPE in the presence of imperfect input data and are of direct practical relevance for the design of robust quantum algorithms that employ QPE as a~subroutine.

The paper addresses the state estimation problem for pairwise Markov models with Gaussian noises. The class of pairwise Markov models generalizes the classical hidden Markov models. The key difference lies in the assumption that the Markov property holds not for the hidden process alone, but for the pair consisting of the state and the observation. This allows modeling more complex dependencies and, in particular, eliminates the requirement of Markovianity for the hidden process. For linear Gaussian pairwise models, Kalman filtering methods remain applicable, leading to the concept of the pairwise Kalman filter.

This work proposes a new modification of the pairwise Kalman filter based on the application of modified weighted Gram – Schmidt orthogonalization and the LD decomposition of covariance matrices. The main results are as follows: a novel LD modification of the pairwise Kalman filter (Theorem 1); a new LD-PKF algorithm for state estimation of pairwise Markov models, based on a direct procedure of modified weighted Gram–Schmidt orthogonalization and LD decomposition of covariance matrices (algorithm 2); results of comparative analysis on the numerical properties of pairwise discrete filtering algorithms.

The obtained theoretical results complement the theory of pairwise filtering in the class of linear discrete pairwise Markov models with Gaussian noises.

The developed algorithm is implemented in MATLAB. A series of numerical experiments are conducted, and the results demonstrate its effectiveness and numerical advantages over other existing modifications of the pairwise Kalman filter.

The presented results can be further used to develop new methods for parameter identification of pairwise Markov models.

Method which allows to obtain explicit form of the solution as a part of power series of the argument step is developed. Formalization of characteristics of the algorithm analogous to operations of a computer is performed. The operation of transfer from one rank to another leads to a probability scheme of the algorithm that averages unknown intermediate steps in higher ranks of the series. The stochastic characteristics of the method are studied and illustrated. Examples of solving nonlinear equations and systems of nonlinear differential equations are presented.

Difference of software development for parallel computing technology from sequential programming is dicussed. Arguements for introduction of new phases into technology of software engineering are given. These phases are: decomposition of algorithms, assignment of jobs to performers, conducting and mapping of logical to physical performers. Issues of performance evaluation of algorithms are briefly discussed. Decomposition of algorithms and programs into parts that can be executed in parallel is dicussed.

We developed an algorithm for fast simulation of VLSI CMOS (Very Large Scale Integration with Complementary Metal-Oxide-Semiconductors) with an accuracy control. The algorithm provides an ability of parallel numerical experiments in multiprocessor computational environment. There is computation speed up by means of block-matrix and structural (DCCC) decompositions application. A feature of the approach is both in a choice of moments and ways of parameters synchronization and application of multi-rate integration methods. Due to this fact we have ability to estimate and control error of given characteristics.

We propose a generalized probabilistic topic model of text corpora which can incorporate heuristics of Bayesian regularization, sampling, frequent parameters update, and robustness in any combinations. Wellknown models PLSA, LDA, CVB0, SWB, and many others can be considered as special cases of the proposed broad family of models. We propose the robust PLSA model and show that it is more sparse and performs better that regularized models like LDA.

In this paper we consider one of the core subsystems of the retail payment system named “Developer”. The Queuing System for modeling this subsystem was developed and information about it is provided. The task for the assignment problem was set up and solved (the modification of the Hungarian algorithm was used). Information about Agent Based Model for subsystem “Developer” and the results of the simulation experiments are given.

Methods of the solution of a problem of focal approximation — approach on point-by-point given smooth closed empirical curve by multifocal lemniscates are investigated. Criteria and convergence of the developed approached methods with use of the description, both in real, and in complex variables are analyzed. Topological equivalence of the used criteria is proved.

The stationary flame spread rate has been calculated using the relationship based on the thermodynamic variational principle. It has been shown that proposed numerical algorithm provides the stable convergence under any initial approximation, which could be noticeably far from the searched solution.