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Stochastic Computational Methods and Experiment Design
The study presents a brief overview of key results of research conducted at the Statistical Modeling Department of St. Petersburg State University. These results include mathematical substantiation of the comp...
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Article
Separation of Roots of Systems of Nonlinear Equations. Stochastic Approach
In this work, we consider the actual problem of separating the roots of nonlinear systems of equations in the case of many variables. The known method of reducing the problem of solving the system to an equiva...
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Article
On the Choice of Regression Basis Functions and Machine Learning
As is known, regression-analysis tools are widely used in machine-learning problems to establish the relationship between the observed variables and to store information in a compact manner. Most often, a regr...
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Article
Backward Iterations for Solving Integral Equations with Polynomial Nonlinearity
The theory of adjoint operators is widely used in solving applied multidimensional problems with the Monte Carlo method. Efficient algorithms are constructed using the duality principle for many problems descr...
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Article
The Monte Carlo Method for Solving Large Systems of Linear Ordinary Differential Equations
The Monte Carlo method to solve the Cauchy problem for large systems of linear differential equations is proposed in this paper. Firstly, a quick overview of previously obtained results from applying the appro...
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Monte Carlo Method for Solving ODE Systems
The Monte Carlo method is applied to solve Cauchy problems for a system of linear and nonlinear ordinary differential equations. The Monte Carlo method is relevant for the solution of large systems of equation...
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Article
On randomization of Halton quasi-random sequences
The problem of estimating the error of quasi-Monte Carlo methods by means of randomization is considered. The well known Koksma–Hlawka inequality enables one to estimate asymptotics for the error, but it is no...
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Article
Towards the Analysis of the simulated annealing method in the multiextremal case
There are many applied problems in which it is necessary to calculate global extrema whose number is large or even infinite. These problems include, for example, some experimental design problems, and the prob...
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Article
On Monte Carlo methods in distributed memory systems
The way for solving a system of linear algebraic equations (SLAEs) with computers with distributed memory is presented. It is assumed that there are M computing nodes, each of which has a limited fast memory, and...
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Article
On the partial synchronization of iterative methods
The rapidly growing field of parallel computing systems promotes the study of parallel algorithms, with the Monte Carlo method and asynchronous iterations being among the most valuable types. These algorithms ...
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Article
Empirically estimating error of integration by quasi-Monte Carlo method
The possibility of applying probability-theoretical methods to a deterministic procedure for estimating the error of evaluating multiple integrals by the quasi-Monte Carlo method is considered. The existing me...
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Article
An extension of the Krylov method for calculating the coefficients of the minimal polynomial
The concept of a k-minimal polynomial of an operator is introduced, and a method for approximate calculation of the coefficients of this polynomial is proposed. The method uses the calculated values of certain fu...
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Article
On estimates of the spectrum of a linear operator
The paper is concerned with new approaches to the analysis of spectra of linear operators. New algorithms are proposed to calculate the coefficients of the minimal polynomial of a matrix; they are based on the...
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Article
Stochastic and quasistochastic computations
The work contains a brief survey of research related to stochastic and quasistochastic methods of computation carried out in recent years by the author and his colleagues. The areas of research include, in par...
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Article
Parametrically splitting algorithms
The notion of parametrically splitting algorithms is introduced, which are characterized by their capability of solving a given problem on a large number of processors with few data transfers. These algorithms...
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Article
The “walk in hemispheres” process and its applications to solving boundary value problems
Representations of solutions of boundary value problems for simple domains in the Monte Carlo algorithms are widely distributed [2]. In particular, widespread use is made of such a representation for the ball....
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Article
On construction of parallel algorithms in problems of computational mathematics
It is shown by using the example of solution methods for systems of linear algebraic equations that algorithms possessing the important properties of being parallel and asynchronous can be constructed by reduc...
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Article
On stochastic and quasistochastic methods for solving equations
A modification of the quasi-Monte Carlo method previously suggested by the authors is discussed. This modification is successfully applied to solve integral equations of the second kind thanks to a number of a...
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On Regression Experiment Design in the Presence of Systematic Error
Different approaches to experimental design in the presence of systematic error are considered. Randomisation of designs allows us to study the problems from a unified viewpoint. Some new results concerning ra...
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