Fast Diffusion Monte Carlo Sampling via Conformal Map

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.


Introduction
Monte Carlo methods use simulations of random events using random numbers [1] [2]. In the processes of Monte Carlo methods, the sampling of random events is the most crucial part and it takes most of the simulation time. Therefore, if we can save time in the sampling, it will expedite the Monte Carlo simulation.
Based on the probabilistic potential theory [3] [4], elliptic partial differential operators can be interpreted as diffusion motion so that via diffusion simulation we can solve some elliptic partial differential equations such as Laplace, Poisson equations and so on. Also, there is one-to-one correspondence between electrostatic problems and diffusion motion expectation one because both problems satisfy the same partial differential equations. Using the correspondence, Green's function first-passage algorithms have been developed.
In diffusion Monte Carlo [5] [6] [7] [8], we need simulate continuous diffusion via walk-on-spheres (WOS) [5], walk-on-planes (WOP) [9], walk-on-rectangles 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.

Conformal Map between WOP and WOS
In this section, we show how to do the conformal map between WOP and WOS. In The inverting sphere with radius a corresponds to the WOS and the inverted plane to the WOP.
Using the method of image charge [12], induced charge distribution on the spherical grounded conducting surface by a charge q inside of the sphere is given by (see Figure 1): Also, induced charge distribution at the distance 2 2 x y ρ = + on the xy-plane, the infinite grounded conducting plate, by a charge q located at ( ) 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,  , , The induced charge distribution over the spherical surface and the one over the plate can be related by conformal map. Figure 1 shows the conformal map.
We should note that sphere inversion and streographic projection are special cases of the conformal map.
According to inversion transformation [13] (see Figure 1), one of the conformal maps, the charge q inside of the sphere with radius a at a distance p from the center of the sphere is inverted to, , c R q q a p ′ = + (4) at the distance, 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 3 R 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 rot v ,  (7) In this way, we can sample a diffusion point over the spherical surface (WOS sample).

Results
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 Figure 2, we can see that the sampling via conformal map works well. The solid line is the analytic cumulative distribution over the spherical surface and the circles are the simulation sampling points combining the sampling on the infinite plane and the conformal map. The two sampling agrees well with each other.
The first table (Table 1) shows the comparison of the counting operations between the direct sampling and the indirect sampling via the conformal map.
We can reduce some computing operations. In real computation, Table 2 shows the overall efficiency of the indirect sampling. Usually, we run a Monte Carlo program over a week for a real data. In this regard, roughly speaking we can save a couple of days.

Conclusions
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.