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MATHEMATICAL METHODS OF DATA PROCESSING

Ill-posed problems, methods of consideration of a priori expert information

Robust methods

Methods of extraction of time series chaotic and deterministic components

Time series forecasting models

OPTIMISATION PROBLEMS AND SOLUTION METHODS

Optimisation problems in economics

Optimisation problems in finance

Optimisation problems in radiation therapy

Mathematical theory of nuclear reactors

 

 

 

 

 

 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 

 

 


 


 

 

 

 

 

 

 

 

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Mathematical methods of data processing

Ill-posed problems, methods of consideration of a priori expert information


A.V. Kryanev, G.V. Lukin. The metric analysis and data processing. M.: Fizmatlit, 2010.

The book main objective is to acquaint the reader with the most effective both approved classical and new stochastic and determined methods of estimation and forecasting, to learn to use these methods at the solution of specific targets of data processing.
In the book the basic concepts of parametrical and nonparametric statistics, including concepts of estimations, and also the requirements shown to properties of estimations from the point of view of their calculation at data processing on the computer are stated. Some new methods of robust estimation are presented to book, the account of the aprioristic information and forecasting, including algorithms of their numerical realization. Bases of a new direction of data processing - the metric analysis are presented, allowing to solve problems of interpolation and forecasting of one and various variables functions on the basis of an effective utilization of the information of the stochastic and determined characters about investigated functional dependence. It is supposed, that the reader has preliminary mastered a course of probability theory and the mathematical statistics.
This book is intended science officers, post-graduate students, students of older years of the various specialities using mathematical methods of data processing.

Arsenin V.Ya., Kryanev A.V. Generalized Maximum Likelihood Method and its Application for Ill-Posed Problems Solving. Ill-Posed Problems in Natural Sciences. A. Tikhanov(Ed.). VSP BV, Netherlands, pp.3-12, 1992.
 
 Arsenin V.Ya., Kryanev A.V. Generalized maximum likelihood method for finite dimensional ill-posed problems solving. USSR Computational Mathematics and Mathematical Physics, v.31, N5, pp.643-653, 1991. (In Russian)(581 kb)

This paper proposes a method of consideration of a priori information about the desired solution of finite-dimensional ill-posed problems, which is based on the maximum-likelihood method. The method allows considering available information as a stochastic character as well as deterministic one and to get the best approximated solution.

 Kryanev A.V. Statistical form of A.N. Tihonov's regularized least-squares method, - Dokl. Acad. Nauk SSSR, Tom 219, N4, Moscow, 1986. (In Russian) (217 Kb)
 
 Kryanev A.V. Iterational methods of ill-posed problems solving. USSR Computational Mathematics and Mathematical Physics, v. 14, N1, 1974. (In Russian) (516 kb)

This paper concentrates on one of the methods of ill-posed problems solving. The described method is convenient for a computer realization.

  Krjanev A.V. The solution of incorrectly posed problem by methods of successive approximations, - Dokl. Aad. Nauk SSSR, Tom 210, 1, .: 1973 (174 Kb)
 

Robust methods

I.Antoniou, P.Akritas, D.A.Burak, V.V.Ivanov, A.V.Kryanev, G.V.Lukin. Robust methods for stock market data analysis. Physica A, Vol 336, pp. 538 548. 2004. (402 Kb)

We consider the problem of extraction of trend and chaotic components from irregular stock market time series. The proposed methods also permit to extract a part of chaotic component, the so-called anomalous term, caused by the transient short-time surges with high amplitudes. This provides more accurate determination of the trend component. The methods are based on the M-evaluation with decision functions of Huber and Tukey type. The iterative numerical schemes for determination of trend and chaotic components are briefly presented, resulting in an acceptable solution within a finite number of iterations. The optimal level for extraction of the chaotic component is determined by a new numerical scheme based on the fractal dimension of the chaotic component of the analyzed series. Forecasting from the realized part of the analyzed series and a priori expert information is also discussed.

A.V. Kryanev, G.V. Lukin The metric analysis and data processing (40 Kb)
 

Borog V.V., Kryanev A.V., Udumyan D.K. Synthetic-method determination of latent anomalies in variations of galactic space beams. Geomagnetism and aeronomy. Vol. 51, Issue 4, pp. 475 482. 2011. (111 Kb)

A method has been proposed for detecting hidden periodicities in nonstationary finite time series with strong noise. The successive application of the spectral singular expansion and wavelet transform makes it possible to calculate the energy of short signals with an arbitrary form against a suppressing noise back-ground, the amplitude of which is several times as large as that of a useful signal. This method has been used to identify variations in the galactic cosmic ray (GCR) flux related to IMF disturbances due to the earthward propagation of solar magnetic clouds in the heliosphere.

Kryanev A.V. Application of modern methods of mathematical statistics for regression analysis on computer. MEPhI, 1988. (in Russian) (3,236 kb) Additions and changes, 2001. (in Russian) (127 Kb)

This manual is based on the material of the course for five-year students, 'Mathematical methods of data processing'. The manual states methods and algorithms of regression dependence recovery, including solutions of planning optimal experiments problem. The main purpose of the manual is to familiarize the reader with the most effective and approved classic and new statistical methods of recovery, and to learn using these methods for specific problems solving.

Kryanev A.V. Application of Modern Methods of Parametric and Non-Parametric Statistics in Case of Experiment Data Processing on Computer. MEPhI, 1987. (in Russian) (3,137 Kb) Additions and changes, 2001. (in Russian) (108 Kb)

This manual is based on the material of the first part of the course 'Mathematical methods of data processing'. The manual introduces the reader to the most effective and approved modern methods of mathematical statistic, it shows how to apply these methods to the specific problems solving. The manual states basic notions of parametric and non-parametric statistics, including notions of estimation and properties, producing to estimation from the calculation point of view at the data processing on computer.

Arsenin V.Ya., Kryanev A.V., Tsupko-Sitnikov M.V. Application of Robust Methods for Ill-Posed Problems Solving. USSR Computational Mathematics and Mathematical Physics, v.29, N5, pp.653-661, 1989. (in Russian) (439 Kb)

We propose the robust method of ill-posed problems solving for integral first kind equations and liner algebraic equation systems. The described method is robust to data errors, which exceed the bound of set mistake. We suggest the effective algorithm of robust smoothing functional minimization and describe the application of this method for charged beam diagnostics problem solving.

 

Methods of extraction of time series chaotic and deterministic components


A.V. Kryanev, G.V. Lukin The metric analysis and data processing (40 Kb)
 

Ivanov V.V., Kryanev A.V., Lukin G.V. NON-STATIONARY SINGULAR-SPECTRUM ANALYSIS OF DETERMINED AND CHAOTIC TIME PROCESS (95 Kb)
 

A.V. Kryanev, G.V.Lukin, D.K.Udumyan. Metric Analysis and Applications. Numerical Methods and Programming, Vol.10, pp. 408 414. 2009.
http://srcc.msu.su/num-meth/english/ (354 Kb)

A new approach to the interpolation of functions of one or several variables and to the time series forecasting (extrapolation) as well as some schemes and algorithms based on it are proposed. This approach is called the metric analysis. The metric analysis shows a high accuracy in interpolating the functions of several variables, even in the case of a comparatively small number of points with known function values. It is found that the metric analysis allows one to extrapolate and forecast the values of functions of a wider class than the known singular spectrum analysis.

Kryanev A.V., Chernyi A.I. Robust linear smoothing splines and their applications. MEPhI Preprint, 006-97, 1997. (In Russian)
 
Kryanev A.V. Robust methods for extraction of trend, anomalous and cycle components from chaotic time-series. Paper work of ISIPC, 1999.
 


Time series forecasting models

A.V. Kryanev, G.V. Lukin The metric analysis and data processing (40 Kb)
 

Ivanov V.V., Kryanev A.V., Lukin G.V. NON-STATIONARY SINGULAR-SPECTRUM ANALYSIS OF DETERMINED AND CHAOTIC TIME PROCESS (95 Kb)
 

A.V. Kryanev, G.V.Lukin, D.K.Udumyan. Metric Analysis and Applications. Numerical Methods and Programming, Vol.10, pp. 408 414. 2009.
http://srcc.msu.su/num-meth/english/ (354 Kb)

A new approach to the interpolation of functions of one or several variables and to the time series forecasting (extrapolation) as well as some schemes and algorithms based on it are proposed. This approach is called the metric analysis. The metric analysis shows a high accuracy in interpolating the functions of several variables, even in the case of a comparatively small number of points with known function values. It is found that the metric analysis allows one to extrapolate and forecast the values of functions of a wider class than the known singular spectrum analysis.

I. Antoniou, Yu.S. Gorshkov, V.V. Ivanov, A.V. Kryanev. Forecasting financial derivative prices. Chaos, Solitons and Fractals 11, 2000.

In this paper we consider the problem of forecasting the prices of financial market derivatives. A model of changing the underlying asset prices in the form of general Ito stochastic process is developed. The derivative prices can be obtained from the solution of the reverse Cauchy problem for appropriate parabolic equations on the basis of the reverse Kolmogorov equation. We present here the numerical scheme for solving the reverse Cauchy problem for call option and put option prices based on the implicit finite element difference method.




Optimisation problems and solution methods

Optimisation problems in economics

I.Antoniou, V.V.Ivanov, A.V.Kryanev, V.V.Matokhin, M.V.Shapovalov. On the efficient resources distribution in economics based on entropy. Physica A Vol. 336, pp. 549 - 562. 2004.
http://www.sciencedirect.com/science/article/pii/S0378437103012329 (799 Kb)

We generalize the approach developed in our previous work on the efficient resources distribution in economic systems with a small number of elements based on entropy. In order to take into account the asymmetric resources distribution, a new set of two-parameter interpolating functions is introduced. We demonstrate that the maximal value of entropy is reached only in case of asymmetric distribution of resources. First results on application of the generalized approach to the analysis of incomes distribution for Sweden and Russia populations have shown that the new scheme permits to estimate effectively the state of the analyzed system and control the resources distribution process.

V. V. Haritonov, A.V. Kryanev, V.V. Matokhin. Adaptable potential of economic systems. International Journal of Nuclear Governance, Economy and Ecology. Vol. 2, pp. 131 145. 2008 (21 Kb)

This paper describes an analysis of the stability of economic systems and the preferable distributions of resources. The approach describes the adaptable potential of numerical sequences corresponding to the distribution of resources in economic systems, and can be used for the analysis of the stability of economic systems and in forecasting the occurrence of crisis situations.

N.I. Geraskin, V.V.Haritonov, S.G.Klimanov, A.V. Kryanev and V.I. Savander. Dynamic behavior of fuel cost component in total electricity cost of nuclear power plants with WWER-type power units. International Journal of Nuclear Energy Science and Technology. Vol. 3, pp. 314 323. 2007 (21 Kb)

This paper describes the technical and economic analyses of changes in the fuel component of prime electrical energy cost that are caused by implementing various versions of nuclear fuel design and nuclear fuel cycle schemes in WWER-440 power reactors.

I.Antoniou, V.V. Ivanov, Yu.L. Korolev, A.V.Kryanev, V.V.Matokhin, Z.Suchanecki. Analysis of resources distribution in economics based on entropy Physica A: Statistical Mechanics and its Applications, 304 (3-4) (2002) pp. 525-534

We propose a new approach to the problem of efficient resources distribution in different types of economic systems. We also propose to use entropy as an indicator of the efficiency of resources distribution. Our approach is based on methods of statistical physics in which the states of economic systems are described in terms of the density functions rho(g,alpha) of the variable g parameterised by alpha. The parameter alpha plays a role of the integral characteristic of the state of the economic system. Having the density function rho(g,alpha) we can use the corresponding entropy to evaluate the efficiency of the resources distribution. Our theoretical study have been tested on real data related to the portfolio investment.

Kharitonov V.V., Kryanev A.V., Matokhin V.V. Technology of Making of Investment Attractive Packs of Innovation Projects. MEPhI Scientific digest-2002. (in Russian)
 
Simonov B.P., Matokhin V.V., Sobolev A.U., Anisimov K.V., Grigoriev A.U. Technology of R&D Results Commercialisation. 'Scientific session in MEPhI-2001', Digest, v.6, 2001 (in Russian)
 
Kharitonov V.V., Kryanev A.V., Korolev Yu.L., Matokhin V.V. Informational Internet Technology of Innovation Projects Initial Examination for Optimal Portfolio (Pack) Making. Report on conference, Obninsk, 2001. (in Russian)
 
Kharitonov V.V., Kryanev A.V., Matokhin V.V. Technology of R&D Packs Commercialisation and Its Financing Optimisation. Economics and Forecasting in Nuclear Energy, Digest N3, 2001. (in Russian)
 
Korolev Yu.L., Kryanev A.V., Malginova E.G., Matokhin V.V. Dynamic Changes Analysis of Consumption Structure, MEPhI, 2000. (in Russian)
 
Matokhin V.V., Simonov B.P., Yunev D.A., Shchurov A.M., Ananiev A.A. Estimation of Innovation Project Commercialisation. Innovation magazine, Saint-Petersburg: 'Transfer', N9-10, 1999. (in Russian)
 
Kryanev A.V., Matokhin V.V., Klimanov S.G. Statistical functions of resources distribution in economy, MEPhI Preprint, 1998. (In Russian) (450 Kb)

We propose the development of statistical functions of resource distribution from control function used for Lorenz diagram interpolation. It is shown that smoothing change of resource distribution non-linear power leads to distinctions in kind of statistical functions. It proves the special significance of distribution corresponding to the circle on Lorenz plane. The material of the preprint is using as a study-book in Economic-Analytical Institute of the MEPhI.

Dudyrev K., Kryanev A.V., Nekrasov V.I. Monitoring Optimisation in Surface Contamination Reconstruction. MEPhI Preprint, 1994. (in Russian)
 

Optimisation problems in finance

Kryanev A.V., Lukin G.V. MATHEMATICAL SIMULATION OF INVESTMENT PORTFOLIOS OPTIMIZATION PROBLEMS (99 Kb)
 

I. Antoniou, V.V. Ivanov, A.V. Kryanev On a Binomial Model of Option Pricing (99 Kb)

The conventional models of option pricing are based on the artificial proposition of the market neutrality to the risk, which is equivalent to the assumption that the expected return µ equals the risk-free return r. However, in real markets µ may significantly differ from r. This may lead to the noticeable difference between model and real values of option prices. We consider here a modiffcation of the Cox-Ross-Rubenstein binomial model that takes into account the dependence of option prices on µ.

I. Antoniou, Yu. Gorshkov, V.V. Ivanov, A.V. Kryanev. Forecasting financial derivative prices. Chaos, Solitons & Fractals, Vol. 11, pp. 223 229. 2000 (14 Kb)

In this paper we consider the problem of forecasting the prices of financial market derivatives. A model of changing the underlying asset prices in the form of general Ito stochastic process is developed. The derivative prices can be obtained from the solution of the reverse Cauchy problem for appropriate parabolic equations on the basis of the reverse Kolmogorov equation. We present here the numerical scheme for solving the reverse Cauchy problem for call option and put option prices based on the implicit finite element difference method.

I.Antoniou, V.V. Ivanov, A.V.Kryanev. On a binomial model of option pricing. Journal of Computational Methods in Sciences and Engineering vol. 2, no. 1s-2s, 2002, pp. 105-109

The conventional models of option pricing are based on the artificial propositionof the market neutrality to the risk, which is equivalent to the assumption that the expected return mu equals the risk-free return r. However, in real markets mu may significantly differ from r. This may lead to the noticeable difference between model and real values of option prices. We consider here a modification of the Cox-Ross-Rubenstein binomial model that takes into account the dependence of option prices on mu.
 
Kryanev A.V. Basis of Financial Analysis and Portfolio Investments in Market Economy. MEPhI, 2001. (in Russian)

This manual proposes a summary of basis of mathematical theory for portfolio investment. Here you can find the basic elements of financial analysis, cash flow analysis, description of Markowitz scheme for portfolio investment and its application to the Russian reality. Also tasks for self-control are presented.
The manual means for students of Economic-Analytical Institute and extension course.

Gorchkov Yu.S., Klimanov S.G., Kryanev A.V. Investment Portfolio Optimisation with Expert Efficiency Forecasting, Digest of Scientific session MEPhI-1998, 1998. (in Russian)
 
Kryanev A.V., Chernyi A.I. Mathematical Models of Investment Portfolio Optimisation Problems, MEPhI Preprint, 1997. (in Russian) (508 Kb)

This preprint proposes schemes of formulation and solving the portfolio investment optimisation problem, which are different from Markowitz scheme and advise a new risk definition. In particular a new portfolio optimisation problem formulation considers for large-scale portfolio.

Kryanev A.V., Chernyi A.I. Numerical Solutions of Optimisation Problems for Mathematical Models in Investment Theory, Mathematical modelling magazine, v.8, N8, 1996. (in Russian) (277 Kb)
 
Kryanev A.V., Chernyi A.I. Mathematical Models of Efficiency Investments Interaction, MEPhI Preprint, 1996. (in Russian) (373 Kb)

This preprint proposes mathematical model, which help to link effectiveness of different classes, including reconstruction of class effectiveness covariance matrix. Mathematical models take into account the presence of effectiveness link through regions and industries that helps to detect among parameters of issuers those who connected with regions and industries along with own parameters.


Optimisation problems in radiation therapy

Klimanov V.A., Kryanev A.V., Klimanov S.G., Beliakov A.I., Golovkin Yu.V. Impact of Errors in Input Data on Value of Probability of Control over Tumour under Optimal Irradiation Plan. Scientific works of MEPhI-2002. (in Russian) (129 Kb)

This paper considers basic profile beams optimisation problem with taking into account constraints on dose distribution in risk organs and tumour.

Klimanov V.A., Kryanev A.V. Profiles Beams Optimisation Problem with Application of Physical Criterion Functions. Thesis of Intern. conf. Inverse and ill-posed problems, MSU, p.37. (in Russian) (69 Kb)

This paper considers basic profile beams optimisation problem with taking into account constraints on dose distribution in risk organs and tumour.

Rubinsky D.A., Klimanov V.A., Klimanov S.G., Krianev A.V. Profiles beams optimization problem for remote radiation therapy as multi-criterion problem. Proceeding of IX Mediterranean Conference on Medical and Biological Engineering and Computing, Croatia, 2001. (151 Kb)

Profiles beams optimisation problem for tumours in remote radiation therapy has been researched in lot of papers during last years. This paper deals with formulation of profiles intensity optimisation problem as a multi-criterion one and application of finite-size pencil beam algorithm and large-scaled elements method for its numeric solution.

Klimanov V.A., Klimanov S.G., Kryanev A.V. Formulation and Numerical Solution of the Radiation Intensity Profile Optimisation Problem as Multi-Criteria Problem Using Physical and Biological Objective Function. Medical Physics, 11, 2001.
 
Kryanev A.V., Klimanov V.A., Klimanov S.G., Rubinsky D.A., Zrajun A.G. Profiles Beams Optimization Problems for Remote Radiation Therapy as Multicriterion Problem. Chicago-2000 World Congress. (409 Kb)

Profiles beams optimisation problem for tumours in remoteradiation therapy has been researched in lot of papers during last years. This paper deals with formulation of profiles intensity optimisation problem as a multi-criterion one and application of finite-size pencil beam algorithm and large-scaled elements method for its numeric solution.

Klimanov V.A., Kryanev A.V. Formulation of Radiation Therapy Planning Optimisation Problem. Medical Physics, N7, 2000. (in Russian) (160 Kb)

Radiation therapy optimisation is one of the modern research directions in physical medicine. This paper researches lot of papers for last ten years, which deal with formulation and solving of radiation therapy optimisation problems as radiation intensity profiling. It is emphasising that now three schemes of radiation intensity profile optimisation problem formulation is used: with and without using physical or biological criterion functions. Advantages and disadvantages of each of foregoing possible scheme are analysed here. Some formulation recommendations are given.

Klimanov V.A., Klimanov S.G., Kryanev A.V., Rubinsky D.A. Mathematical Modelling of Radiation Therapy Dose Planning Optimisation Problem Based on the Pencil Beam Algorithm and Large-Scaled Elements Methods. MEPhI Scientific digest-2000. (in Russian)
 
Klimanov V.A., Klimanov S.G., Kryanev A.V. Choice of Optimal Solution for Set Radiation Ports Placement with Use of Physical Criterion Functions. MEPhI Scientific digest-2000. (in Russian) (114 Kb)

We developed an algorithm of set radiation ports optimal placement choice that we propose in this paper. This algorithm use physical criterion functions. As a directing parameter in the ports choice optimisation problem the geometrical placement of radiation intensity ports is used in conditions of its constancy.

Klimanov V.A., Kryanev A.V., Rubinsky D.A. Numeric Solution for Radiation Therapy Dose Planning Optimization Problem Based on the Pencil Beam Algorithm and Large-Scaled Elements Methods. Physica Medica, v.15, p. 166, 1999.

 
Klimanov V.A., Klimanov S.G., Kryanev A.V., Rubinsky D.A., Zrajun A.G. Profiles Beams Optimisation Problems for Remote Radiation Therapy. (in Russian) (91 Kb)
 
Klimanov V.A., Kryanev A.V., Rubinsky D.A. Radiation therapy dose planning optimization based on the pencil beam algorithm and large-scaled elements method. Third "Medical Physics-97" International Conference meeting reports theses. Obninsk, p.11, 1997
 
Klimanov V.A., Klimanov S.G., Kryanev A.V., Rubinsky D.A., Zrajun A.G. Profiles Beams Optimisation Problems for Remote Radiation Therapy. - 2 (in Russian) (110 Kb)

Profiles beams optimisation problem for tumours in remote radiation therapy has been researched in lot of papers during last years. This paper deals with formulation of profiles intensity optimisation problem as a multi-criterion one and application of finite-size pencil beam algorithm and large-scaled elements method for its numeric solution.


Mathematical theory of nuclear reactors


Vdovina O.P., Kryanev A.V., Sevostianov A.E. Numeric Solutions of Inverse Problem Solving for Steady-State Transport Equation and Computer-Aided Tomography Problems Using Worth Function. Mathematical modelling, v.6, N9, 1994. (in Russian)
 
Kryanev A.V. Analytical and Numerical Methods of Mathematical Transport Theory. MEPhI Preprint, 1991. (in Russian) (6,682 Kb)

This manual is based on the material of the year course 'Questions of mathematical transport theory'. Here formulation and decision methods of basic mathematical transport theory are presented. As analytical as well as numerical methods of transport theory problems solving are considered. The purpose of the manual is to introduce reader to basic mathematical transport theory results, which help to solve many transport particles and emissions applications.

Shchikhov S.B., Kryanev A.V. Research of Stationary Temperature and Neutron Fields Stability of Reactor Core in Distributed Models. Atomic energy, v.68, issue 2, 1990. (in Russian)
 
Kryanev A.V. Qualitative and Numerical Methods for Correction Problems of Mathematical Reactors Theory Problems. Abstract form doctoral thesis, MEPhI, 1989. (in Russian).
 
Kryanev A.V. Existence, Uniqueness and Stability of Stationary Temperature-Neurone Distributions in Multiplying Mediums, Preprint, USSR Academy of Science, 1986. (in Russian)
 
Kryanev A.V., Shchikhov S.B. Questions of Mathematical Theory of Reactors. Non-linear Analysis. Energoatomisdat, 1983. (in Russian) (12,615 Kb)

This work deals with basic nuclear reactor non-linear distribution research qualitative methods. We got the conditions guaranteeing existence and uniqueness of stationary regimes with taking into account feedbacks including the temperature ones. Such properties of non-stationary regimes as duration, stability, and periodicity are researched here. We used cone theory methods, in particular, stability on cone. Methods, developed in the work, apply for direct nuclear reactors models research.


 

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