# The Monte Carlo Method for a Solution of ODE System

Research output

### Abstract

The Monte Carlo method is an effective method for a solution of large dimension problems.An elaboration of methods of solution of Cauchu problems for large systems of ODE basedon a simulation of Markov processes is very interesting, but this problem is not enoughinvestigated now.Here this problem is discussed and some peculiarities are indicated. As a rule differentialequations are replaced by equivalent integral Volterra equations, and then the well knownNeumann-Ulam system is used (see Ermakov S.M., 2009).The appearing peculiarities are discussed. First, the linearization that complicate theMarkov process construction is to be used. The case of a polynomial nonlinearity is anexclusive one, because there exist algorithms based on the approximation of nonlinearfunctions by polynomials and on branching processes simulation. In the general case theinterval [0,t] is divided by subintervals of length and the approximation is performed ineach subinterval. Two types of errors, systematic and random, appear and they are to beinvestigated.Second, in the linear case the valuetmay be arbitrary large, but in the nonlinear casethe choice of depends as on the small value of error of polynomial approximation, so onthe value of the Picard interval of the solution existence.As examples, some linear and nonlinear systems ODE by the Monte Carlo method aresolved. The simulation of branching Markov chains is used. The obtained solutions arecompared with the solutions found by the Runge-Kutta method.The error of the proposed methods is to be investigated in future, but the consideredexamples point of their perspective.The work is supported by RFBR,No17-01-00267-a.
Original language English 10th International Workshop on Simulation and Statistics Workshop booklet Salzburg Universitat Salzburg 90 Published - Sep 2019 10th International Workshop on Simulation and Statistics - SalzburgDuration: 2 Sep 2019 → 6 Sep 2019

### Conference

Conference 10th International Workshop on Simulation and Statistics Australia Salzburg 2/09/19 → 6/09/19

### Fingerprint

Monte Carlo method
Solution Existence
Polynomial
Systematic Error
Random Error
Process Simulation
Volterra Integral Equations
Branching process
Polynomial Approximation
Approximation
Runge-Kutta Methods
Markov Process
Linearization
Branching
Markov chain
Simulation
Nonlinear Systems
Linear Systems
Nonlinearity
Interval

### Scopus subject areas

• Mathematics(all)

### Cite this

Tovstik, T. M., & Ermakov, S. M. (2019). The Monte Carlo Method for a Solution of ODE System. In 10th International Workshop on Simulation and Statistics: Workshop booklet (pp. 90). Salzburg: Universitat Salzburg.
Tovstik, Tatiana Mikhailovna ; Ermakov, Sergei Mikhailovich. / The Monte Carlo Method for a Solution of ODE System. 10th International Workshop on Simulation and Statistics: Workshop booklet. Salzburg : Universitat Salzburg, 2019. pp. 90
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title = "The Monte Carlo Method for a Solution of ODE System",
abstract = "The Monte Carlo method is an effective method for a solution of large dimension problems.An elaboration of methods of solution of Cauchu problems for large systems of ODE basedon a simulation of Markov processes is very interesting, but this problem is not enoughinvestigated now.Here this problem is discussed and some peculiarities are indicated. As a rule differentialequations are replaced by equivalent integral Volterra equations, and then the well knownNeumann-Ulam system is used (see Ermakov S.M., 2009).The appearing peculiarities are discussed. First, the linearization that complicate theMarkov process construction is to be used. The case of a polynomial nonlinearity is anexclusive one, because there exist algorithms based on the approximation of nonlinearfunctions by polynomials and on branching processes simulation. In the general case theinterval [0,t] is divided by subintervals of length and the approximation is performed ineach subinterval. Two types of errors, systematic and random, appear and they are to beinvestigated.Second, in the linear case the valuetmay be arbitrary large, but in the nonlinear casethe choice of depends as on the small value of error of polynomial approximation, so onthe value of the Picard interval of the solution existence.As examples, some linear and nonlinear systems ODE by the Monte Carlo method aresolved. The simulation of branching Markov chains is used. The obtained solutions arecompared with the solutions found by the Runge-Kutta method.The error of the proposed methods is to be investigated in future, but the consideredexamples point of their perspective.The work is supported by RFBR,No17-01-00267-a.",
keywords = "Monte carlo method, ODE systems",
author = "Tovstik, {Tatiana Mikhailovna} and Ermakov, {Sergei Mikhailovich}",
year = "2019",
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language = "English",
pages = "90",
booktitle = "10th International Workshop on Simulation and Statistics",
publisher = "Universitat Salzburg",

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Tovstik, TM & Ermakov, SM 2019, The Monte Carlo Method for a Solution of ODE System. in 10th International Workshop on Simulation and Statistics: Workshop booklet. Universitat Salzburg, Salzburg, pp. 90, Salzburg, 2/09/19.
10th International Workshop on Simulation and Statistics: Workshop booklet. Salzburg : Universitat Salzburg, 2019. p. 90.

Research output

TY - GEN

T1 - The Monte Carlo Method for a Solution of ODE System

AU - Tovstik, Tatiana Mikhailovna

AU - Ermakov, Sergei Mikhailovich

PY - 2019/9

Y1 - 2019/9

N2 - The Monte Carlo method is an effective method for a solution of large dimension problems.An elaboration of methods of solution of Cauchu problems for large systems of ODE basedon a simulation of Markov processes is very interesting, but this problem is not enoughinvestigated now.Here this problem is discussed and some peculiarities are indicated. As a rule differentialequations are replaced by equivalent integral Volterra equations, and then the well knownNeumann-Ulam system is used (see Ermakov S.M., 2009).The appearing peculiarities are discussed. First, the linearization that complicate theMarkov process construction is to be used. The case of a polynomial nonlinearity is anexclusive one, because there exist algorithms based on the approximation of nonlinearfunctions by polynomials and on branching processes simulation. In the general case theinterval [0,t] is divided by subintervals of length and the approximation is performed ineach subinterval. Two types of errors, systematic and random, appear and they are to beinvestigated.Second, in the linear case the valuetmay be arbitrary large, but in the nonlinear casethe choice of depends as on the small value of error of polynomial approximation, so onthe value of the Picard interval of the solution existence.As examples, some linear and nonlinear systems ODE by the Monte Carlo method aresolved. The simulation of branching Markov chains is used. The obtained solutions arecompared with the solutions found by the Runge-Kutta method.The error of the proposed methods is to be investigated in future, but the consideredexamples point of their perspective.The work is supported by RFBR,No17-01-00267-a.

AB - The Monte Carlo method is an effective method for a solution of large dimension problems.An elaboration of methods of solution of Cauchu problems for large systems of ODE basedon a simulation of Markov processes is very interesting, but this problem is not enoughinvestigated now.Here this problem is discussed and some peculiarities are indicated. As a rule differentialequations are replaced by equivalent integral Volterra equations, and then the well knownNeumann-Ulam system is used (see Ermakov S.M., 2009).The appearing peculiarities are discussed. First, the linearization that complicate theMarkov process construction is to be used. The case of a polynomial nonlinearity is anexclusive one, because there exist algorithms based on the approximation of nonlinearfunctions by polynomials and on branching processes simulation. In the general case theinterval [0,t] is divided by subintervals of length and the approximation is performed ineach subinterval. Two types of errors, systematic and random, appear and they are to beinvestigated.Second, in the linear case the valuetmay be arbitrary large, but in the nonlinear casethe choice of depends as on the small value of error of polynomial approximation, so onthe value of the Picard interval of the solution existence.As examples, some linear and nonlinear systems ODE by the Monte Carlo method aresolved. The simulation of branching Markov chains is used. The obtained solutions arecompared with the solutions found by the Runge-Kutta method.The error of the proposed methods is to be investigated in future, but the consideredexamples point of their perspective.The work is supported by RFBR,No17-01-00267-a.

KW - Monte carlo method

KW - ODE systems

M3 - Conference contribution

SP - 90

BT - 10th International Workshop on Simulation and Statistics

PB - Universitat Salzburg

CY - Salzburg

ER -

Tovstik TM, Ermakov SM. The Monte Carlo Method for a Solution of ODE System. In 10th International Workshop on Simulation and Statistics: Workshop booklet. Salzburg: Universitat Salzburg. 2019. p. 90