Sequential optimization algorithms and their application. Part I

Mikhalevich, V. S.

doi:10.1007/bf01071444

Cited by 44 publications

(6 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A. D. Shatashvili was the first to use this idea in [33][34][35][36][37]. Moreover, V. S. Mikhalevich [15][16][17], R. L. Stratonovich [23,24], A. N. Shiryaev, R. Sh. Liptser, B. I. Grigelionis [12-14, 41-44, 6] and some other authors used densities of probability measures to determine optimal estimates in extrapolation and filtering problems for two Markov processes, where one component of a random process is observed and optimal estimate is determined for another component.…”

mentioning

confidence: 97%

See 1 more Smart Citation

A method for efficient computation of optimal estimates in the extrapolation of solutions of nonlinear evolutionary differential equations in a Hilbert space. I

Fomin-Shatashvili¹,

Shatashvili

2008

Cybern Syst Anal

View full text Add to dashboard Cite

519.21The paper considers the optimal extrapolation of a random process with values from a separable Hilbert space H. Formulas for random processes with bounded second moments are derived to efficiently calculate optimal (in the sense of minimum standard deviation) estimates in problems of random process extrapolation (prediction).Finding optimal estimates in problems of extrapolation and filtration of random processes is an important problem in the theory of random processes. It is of great significance in solving many topical applied problems of science and technology. Optimal extrapolation of a random process means predicting the future value of a random process in the best way from its past values.The optimal filtering problem is formulated as follows: let a random process be observed. It contains or combines a useful signal (first process) and random noise (second process). It is required to separate (filter) the noise from the signal and find the best (in a sense) approximation for the signal.The solution of the problem of linear extrapolation and filtration given earlier by A. N. Kolmogorov [10,11] and N. Wiener [1, 2] is optimal only for Gaussian processes, and is optimal for general processes only in the class of linear estimates and may appear not to be the best. Therefore, it is of theoretical and practical importance to develop methods for efficient solution of such problems. N. Wiener [1, 2], L. A. Zadeh [9], V. S. Mikhalevich [15-17], R. L. Stratonovich [23, 24], A. N. Shiryaev, R. Sh. Liptser, B. I. Grigelionis [12-14, 41-44, 6], et al. conducted studies in this field. However, the results they have obtained pertain mainly to different classes of Markov processes.The theory of linear extrapolation and filtering is completely developed, at least theoretically, by A. N. Kolmogorov, N. Wiener, A. M. Yaglom [1,2,10,11,[45][46][47][48] and many other authors. These results are presented in detail in the monograph by Yu. A. Rozanov [18]. As indicated above, linear extrapolation and filtering results in best estimates in the case of Gaussian processes. A. D. Shatashvili [33][34][35][36][37] showed for some class of random processes that if processes slightly differ from Gaussian ones, the order of deviation of the best estimates from linear ones is equal to the order of deviation of the process under study from the Gaussian one.It is natural that the necessity of considering all finite-dimensional distributions of processes under study substantially complicates the generalization of problems of linear extrapolation and filtering of non-Gaussian random processes. Therefore, problems of optimal estimation can efficiently be solved only if the information on all finite-dimensional distributions of the random process is obtained in closed form.From an abstract standpoint, the optimal solution to problems of extrapolation and filtration of random processes is rather simple and can be expressed using integration in a function space with respect to a measure depending on a functional 434

show abstract

mentioning

confidence: 97%

“…Thus, we have constructed an algorithm for efficient computation of the optimal prediction of the random process x t ( ) under study. Indeed, using expansion (16) and the fact that the expression l t T ( ) is a T T * -measurable quantity and the Gaussian term e T t ( ) does not depend on the s-algebra T T * , we obviously get the following important formula: …”

mentioning

confidence: 98%

A method for efficient computation of optimal estimates in the extrapolation of solutions of nonlinear evolutionary differential equations in a Hilbert space. I

Fomin-Shatashvili¹,

Shatashvili

2008

Cybern Syst Anal

View full text Add to dashboard Cite

show abstract

“…General scheme for sequential analysis of variants is developed by Myhalevich V.S. [11] based on ideas of theory of sequential solutions and dynamic programming. Methods of SAV are based on systematic solutions construction, or in other words: incremental specification of component values of the solution, and elimination of solutions which cannot be considered as a part of the optimal one.…”

Section: Algorithm Of Sequential Analysis Of Variants For Distribmentioning

confidence: 99%

The Algorithm for Sequential Analysis of Variants for Distribution of Virtual Machines in Data Center

Rolik

Bodaniuk²,

Kolesnik

et al. 2017

Annals of Computer Science and Information Systems

View full text Add to dashboard Cite

Abstract-this work proposes an algorithm of sequential analysis of variants (SAV) to solve the distributional problem of allocation of virtual machines to physical servers in a data center. The set of tests and rules of the SAV algorithm is defined. The experimental results for problems of different dimensions are given. The comparison of the proposed algorithm with heuristic and genetic algorithms is accomplished. The time of finding solution required by the SAV algorithm depending on the dimension of the problem is evaluated. The recommendations for using the SAV algorithm are given. For tasks requiring high precision distribution it is better to use the SAV algorithm as it finds the optimal solution, whereas heuristic and evolutionary algorithms can quickly get an effective solution. The speed of the heuristic and evolutionary algorithms is not significantly dependent on the problem's size, but the quality of their solutions is worse than equivalent solution received with the SAV algorithm.

show abstract

“…Many publications [2][3][4][5][6][7][8][9][10][11][12] consider theoretical issues of accuracy and efficiency of numerical methods, and also focus on their comparative analysis (by some criteria) and construction of efficient (including optimal) solution methods for various classes of problems. We describe the main directions in the development of computational mathematics and present some of our own results that reflect a certain design conception of speed-optimal and accuracy-optimal (or nearly optimal) algorithms for various classes of problems, as well as a certain approach to optimization of computer computations.…”

mentioning

confidence: 99%

“…Many publications [2][3][4][5][6][7][8][9][10][11][12] consider theoretical issues of accuracy and efficiency of numerical methods, and also focus on their comparative analysis (by some criteria) and construction of efficient (including optimal) solution methods for various classes of problems. These studies have focused on discretization of continuous problems, effectively testable conditions of applicability of various methods, rate of convergence bounds, prior and posterior error estimates, choice of initial approximations, robustness of numerical methods to errors in input data and to rounding errors, etc.…”

mentioning

confidence: 99%

Optimization of computations

Mikhalevich¹,

Сергиенко²,

Zadiraka³

et al. 1994

Cybern Syst Anal

Self Cite

View full text Add to dashboard Cite

519.685.7 This article examines some topics of optimization of computations, which have been discussed at 25 seminar-schools and symposia organized by the V. M. Glushkov Institute of Cybernetics of the Ukrainian Academy of Sciences since 1969. We describe the main directions in the development of computational mathematics and present some of our own results that reflect a certain design conception of speed-optimal and accuracy-optimal (or nearly optimal) algorithms for various classes of problems, as well as a certain approach to optimization of computer computations.The vigorous development of computer science in the 1950s-1960s has left its imprint on computational mathematics, which studies the methods of generating solutions of various mathematical problems in numerical form (whether approximate or exact). The traditional divisions of computational mathematics mainly include [ 1] function evaluation, computational methods of linear algebra, numerical solution of algebraic and transcendental equations, numerical differentiation and integration, numerical solution of differential and integral equations, and numerical minimization methods for functions and functionals. New directions are developing in parallel with this traditional body of knowledge, including theory and numerical methods of solution of ill-posed problems, linear and nonlinear programming, theory of data processing systems, fitting of functions and functionals, optimization methods, etc. Each of these disciplines is characterized by its own numerical methods, which are based on different ideas. Alongside the iterative methods, which have rapidly grown and are efficiently used today for many classes of problems, we are witnessing particularly wide development of various approximation (discretization) methods that replace the original (exact) problem with an approximate (in some sense) problem for which an exact or an approximate solution is obtained. These include projection methods, finite-difference methods, projection-iteration methods, and others. Approximating equations are constructed by these methods in such a way that their solution normally reduces to solving a finite system of scalar equations. We know that the quality and the efficiency of specific numerical methods is determined by the set of their characteristics, some of which are applicable only to numerical methods of a certain class (such as initial approximation and rate of convergence that are applicable for iterative methods) while others apply to numerical methods of all classes. Such general characteristics include various errors of the numerical method, running time on a computer, and the computer memory requirements.Many mathematicians are engaged in the theoretical analysis of numerical methods with the object of estimating their various characteristics. These studies have focused on discretization of continuous problems, effectively testable conditions of applicability of various methods, rate of convergence bounds, prior and posterior error estimates, choice of init...

show abstract

Sequential optimization algorithms and their application. Part I

Cited by 44 publications

References 2 publications

A method for efficient computation of optimal estimates in the extrapolation of solutions of nonlinear evolutionary differential equations in a Hilbert space. I

A method for efficient computation of optimal estimates in the extrapolation of solutions of nonlinear evolutionary differential equations in a Hilbert space. I

The Algorithm for Sequential Analysis of Variants for Distribution of Virtual Machines in Data Center

Optimization of computations

Contact Info

Product

Resources

About