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In diffusion Monte Carlo methods, depending on the geometry continuous diffusion can be simulated in many ways such as walk-on-spheres (WOS), walk-on-planes (WOP), walk-on-rectangles (WOR) and so on. The diffusion ways are conformally the same satisfying the Laplace equation with the given boundary geometry. In this paper, using the WOP and the conformal map, we sample the WOS diffusion and show that the indirect sampling is more efficient than the direct WOS sampling. This signifies that fast diffusion Monte Carlo sampling via conformal map can be possible.

Monte Carlo methods use simulations of random events using random numbers [

Based on the probabilistic potential theory [

In diffusion Monte Carlo [

In this paper, using the WOP and the conformal map we sample the WOS diffusion and show that the indirect sampling is more efficient than the direct WOS sampling. This signifies that fast diffusion Monte Carlo sampling via conformal map can be possible.

In this section, we show how to do the conformal map between WOP and WOS. In

to σ ( ρ ) , that is, [ R c / R c 2 + ρ 2 ] 3 σ ( r , θ , ϕ ) . The inverting sphere with radius a corresponds to the WOS and the inverted plane to the WOP.

Using the method of image charge [

σ ( a , θ , ϕ ) = q ( a 2 − p 2 ) 4 π a ( a 2 − 2 a p cos ( θ ) + p 2 ) 3 / 2 . (1)

Also, induced charge distribution at the distance ρ = x 2 + y 2 on the xy-plane, the infinite grounded conducting plate, by a charge q located at ( 0,0, d ) is obtained as:

σ ( ρ ) = 1 2 π 1 d 2 + ρ 2 . (2)

In WOP algorithm, when the diffusion starts at the distance d from the WOP plane, the ρ sampling, the radial distance sampling from the center of WOP place, is simply,

ρ d = 1 − U 2 U 2 , U ∈ [ 0 , 1 ] . (3)

The induced charge distribution over the spherical surface and the one over the plate can be related by conformal map.

According to inversion transformation [

q ′ = R c a + p q , (4)

at the distance,

d = R c 2 a + p − 2 a , (5)

from the inverted plane. Also, from the other point of view of the charge distribution, the charge distributions, σ ( r ) (before the inversion) and σ ˜ ( r → ˜ ) (after the inversion) are related as follows:

σ ˜ ( r ˜ ) = ( R r ˜ ) 3 σ ( r ) , (6)

where R is radius of the inversion sphere, and r ˜ = | r ˜ | .

After sampling over the infinite plane (WOP) and converting over the spherical surface (WOS), we need to rotate the sampling point in regard to the point of the previous diffusion point. Here, we need the following Rodrigues’ rotation formula.

If v is a vector in R 3 and k is a unit vector describing an axis of rotation about which v rotates by an angle θ according to the right hand rule, the Rodrigues’ rotation formula gives the rotated vector v rot ,

v rot = v cos θ + ( k × v ) sin θ + k ( k ⋅ v ) ( 1 − cos θ ) . (7)

In this way, we can sample a diffusion point over the spherical surface (WOS sample).

In this section, we check that the sampling via conformal map works well and compare two samplings via operation counting and running time. Counting operations are good for computing performance. In this performance comparison, counting operations consist of numbers of addition/substraction, multiplication, division and special function.

In

The first table (

Algorithm | Addition/Substraction | Multiplication | Division | Special function |
---|---|---|---|---|

Sampling via conformal map | 7 | 9 | 5 | 13 |

Direct sampling | 20 | 32 | 14 | 17 |

Algorithm | CPU time per run (s) |
---|---|

Sampling via conformal map | 29.77 |

Direct sampling | 59.51 |

In Monte Carlo methods, sampling of random events using random numbers takes significant computing time. So, if we can sample fast, that can improve the performance of the Monte Carlo simulation. In diffusion Monte Carlo algorithms, fast diffusion simulation is essential. For fast diffusion simulation, we use many diffusion algorithms depending on the surrounding geometry like walk-on-spheres (WOS), walk-on-planes (WOP), walk-on-rectangles (WOR) and so on. Among them, WOP is the fastest sampling and for diffusion in free space WOS is the most-used algorithm.

In this paper, using the WOP and the conformal map we sample the WOS diffusion and show that the indirect sampling is more efficient than the direct WOS sampling. This signifies that fast diffusion Monte Carlo sampling via conformal map can be possible.

This work was supported by the National Research Foundation of Korea (NRF) grant (No. 2017R1E1A1A03070543) funded by the Korea government (Ministry of Science and Information & Communication Technology (ICT)). In addition, this work was supported by the GIST Research Institute (GRI) grant funded by GIST in 2019.

The authors declare no conflicts of interest regarding the publication of this paper.

Hwang, C.-O. and Do, M. (2020) Fast Diffusion Monte Carlo Sampling via Conformal Map. Applied Mathematics, 11, 35-41. https://doi.org/10.4236/am.2020.111004