Functional Random Walk on Spheres algorithm for biharmonic equation: optimization and error estimation

Sabelfeld, Karl K.; Shkarupa, E. V.

doi:10.1163/156939603322587461

Cited by 1 publication

(4 citation statements)

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“…Since recently, approaches have been developed for construction and optimization of the Monte Carlo algorithms for approximating the solution on the domain in whole [7] - [16]. These approaches are related with preliminary discretization of the problem (introduction of a grid), estimation of the solution at the grid nodes by the Monte Carlo method, and subsequent interpolation of the solution from the so-obtained approximate values at the nodes.…”

Section: Discrete-stochastic Numerical Methodsmentioning

confidence: 99%

“…We need to construct the optimal relation between the sample size N and the number of grid nodes M. For this purpose we have to carry out two steps as in works [9,11,12,15,16]:…”

Section: Construct a Grid Rmentioning

confidence: 99%

“…By taking into account the properties of the used estimators at the grid nodes (namely, independent, weakly dependent, or dependent-trials estimators), sufficiently accurate upper bounds for such maximums can be constructed [8,9,11,16]. Similar formula is obtained for algorithm with the intersection estimators if we replace H V byH V (ε).…”

Section: Upper Error Boundmentioning

confidence: 99%

“…In recent time the theory of discrete-stochastic numerical methods has been developed for different problems [7] - [16]. All necessary macroparameters of test particle flow (functionals of the distribution function) are computed during the trajectory simulation.…”

Section: Introductionmentioning

confidence: 99%

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The discrete-stochastic approaches to solving the linearized Boltzmann equation

Plotnikov¹,

Shkarupa²

2005

Monte Carlo Methods and Applications

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Abstracts -The test particle Monte Carlo method for solving the linearized Boltzmann equation is considered. The main idea of this work is the construction of the relations between the sample size and the number of grid nodes which guarantee the attainment of the given error level on the base of the theory of discrete-stochastic numerical methods. Three approaches to construction of the upper error bound of the method are suggested. The optimal (in the sense of the obtained upper error bounds) relations between the sample size and the number of grid nodes are constructed.

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Section: Discrete-stochastic Numerical Methodsmentioning

confidence: 99%

“…We need to construct the optimal relation between the sample size N and the number of grid nodes M. For this purpose we have to carry out two steps as in works [9,11,12,15,16]:…”

Section: Construct a Grid Rmentioning

confidence: 99%