Stochastic methods for boundary value problems : numerics for high-dimensional PDEs and applications
This monograph is devoted to random walk based stochastic algorithms for solving high-dimensional boundary value problems of mathematical physics and chemistry.It includes Monte Carlo methods where the random walks live not only on the boundary, but also inside the domain.
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| Main Authors: | , |
|---|---|
| Format: | eBook Book |
| Language: | English |
| Published: |
Berlin
De Gruyter
2016
Walter de Gruyter GmbH |
| Edition: | 1 |
| Subjects: | |
| ISBN: | 3110479060, 9783110479065 |
| Online Access: | Get full text |
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Table of Contents:
- 6.1.2 Computing charge density -- 6.2 Stationary diffusion equation with absorption -- 6.3 Multiply connected domains -- 6.4 Stabilization method -- 6.5 Nonlinear Poisson equation -- 7 Splitting and survival probabilities in random walk methods and applications -- 7.1 Introduction -- 7.2 Survival probability for a sphere and an interval -- 7.3 The reciprocity theorem for particle collection in the general case of Robin boundary conditions -- 7.4 Splitting and survival probabilities -- 7.4.1 Splitting probabilities for a finite interval and nanowire growth simulation -- 7.4.2 Survival probability for a disc and the exterior of circular cylinder -- 7.4.3 Splitting probabilities for concentric spheres and annulus -- 7.5 Cathodoluminescence -- 7.5.1 The random WOS and hemispheres algorithm -- 7.6 Conclusion and discussion -- 8 A random WOS-based KMC method for electron-hole recombinations -- 8.1 Introduction -- 8.2 The mean field equations -- 8.3 Monte Carlo Algorithms -- 8.3.1 Random WOS for the diffusion simulation -- 8.3.2 Radiative and nonradiative recombination in the absence of diffusion -- 8.3.3 General case of radiative and nonradiative recombination in the presence of diffusion -- 8.4 Simulation results and comparison -- 8.5 Summary and conclusion -- 9 Monte Carlo methods for computing macromolecules properties and solving related problems -- 9.1 Diffusion-limited reaction rate and other integral parameters -- 9.1.1 Formulation of the problem -- 9.1.2 Capacitance calculations -- 9.2 Walk in subdomains and efficient simulation of Brownian motion exit points -- 9.3 Monte Carlo algorithms for boundary-value conditions containing the normal derivative -- 9.3.1 WOS algorithm for mixed boundary-value conditions -- 9.3.2 Mean-value relation at a point on the boundary -- 9.3.3 Construction of the algorithm and its convergence -- 9.4 Continuity BVP
- Intro -- Contents -- 1 Introduction -- 2 Random walk algorithms for solving integral equations -- 2.1 Conventional Monte Carlo scheme -- 2.2 Biased estimators -- 2.3 Linear-fractional transformations and their relations to iterative processes -- 2.4 Asymptotically unbiased estimators based on singular approximations -- 2.5 Integral equation of the first kind -- 3 Random walk-on-boundary algorithms for the Laplace equation -- 3.1 Newton potentials and boundary integral equations of the electrostatics -- 3.2 The interior Dirichlet problem and isotropic random walk-on-boundary process -- 3.3 Solution of the Neumann problem -- 3.4 Random estimators for the exterior Dirichlet problem -- 3.5 Third BVP and alternative methods of solving the Dirichlet problem -- 3.6 Inhomogeneous problems -- 3.7 Continuity BVP -- 3.7.1 Walk on boundary for the continuity problem -- 3.8 Calculation of the solution derivatives near the boundary -- 3.9 Normal derivative of a double-layer potential -- 4 Walk-on-boundary algorithms for the heat equation -- 4.1 Heat potentials and Volterra boundary integral equations -- 4.2 Nonstationary walk-on-boundary process -- 4.3 The Dirichlet problem -- 4.4 The Neumann problem -- 4.5 Third BVP -- 4.6 Unbiasedness and variance of the walk-on-boundary algorithms -- 4.7 The cost of the walk-on-boundary algorithms -- 4.8 Inhomogeneous heat equation -- 4.9 Calculation of derivatives on the boundary -- 5 Spatial problems of elasticity -- 5.1 Elastopotentials and systems of boundary integral equations of the elasticity theory -- 5.2 First BVP and estimators for singular integrals -- 5.3 Other BVPs for the Lamé equations and regular integral equations -- 6 Variants of the random walk on boundary for solving stationary potential problems -- 6.1 The Robin problem and the ergodic theorem -- 6.1.1 Monte Carlo estimator for computing capacitance
- 9.4.1 Monte Carlo method -- 9.4.2 Integral representation at a boundary point -- 9.4.3 Estimate for the boundary value -- 9.4.4 Construction of the algorithm and its convergence -- 9.5 Computing macromolecule energy -- 9.5.1 Mathematical model and computational results -- Bibliography
- Contents --
- 3. Random walk-on-boundary algorithms for the Laplace equation --
- 7. Splitting and survival probabilities in random walk methods and applications --
- Preface --
- 2. Random walk algorithms for solving integral equations --
- 1. Introduction --
- 9. Monte Carlo methods for computing macromolecules properties and solving related problems --
- 5. Spatial problems of elasticity --
- 8. A random WOS-based KMC method for electron–hole recombinations --
- 4. Walk-on-boundary algorithms for the heat equation --
- Frontmatter --
- 6. Variants of the random walk on boundary for solving stationary potential problems --
- Bibliography