ThomsonProblemSolver

This repo is a simple solver for n(n>=2) dimension Thomson Problem.

Requirements

GLOG(depends on GFlags)
OpenBLAS/Intel MKL
optional: CUDA/cuBLAS

Update

We optimized the implementation on CPU and fixed bugs on GPU. It works 20 times faster than the code before.

Usage

This solver is very easy to use.

//init solver with default parameters
//init electrons on the hypersphere(electrons number, dimension, device)
thomson::solver<double> tps = thomson::solver<double>(470, 3, -1);
tps.Random_Init_Electorns();
//set solving parameters
tps.set_base_lr_(1.0);
tps.set_max_iter_(100000);
tps.set_min_pe_error_(1e-6);
tps.set_display_interval_(20);
tps.set_fast_pe_calculation(true);
tps.set_snapshot_interval_(10000);
tps.set_lr_policy_(true);
//solving start
tps.Solve_Thomson_Problem(pls);

Speed Performance

float datatype cannot meet the accuracy requirements, all experiments are done on double.
As we know, the double-precision floating point capability on NVIDIA GTX series sucks, we got even worse results in samll cases.

Device	Number of Electrons	Dimension	Time Cost pre Iter
Intel i7-8700K	470	3	7.27ms
NVIDIA GTX 1060	470	3	7.84ms
Intel i7-8700K	10572	512	162816ms
NVIDIA GTX 1060	10572	512	82771ms
Intel Xeon E5-2673 v4	10572	512	193797ms
NVIDIA TESLA V100	10572	512	13812ms

Error Analysis

2 and 3 Dimension Case

We can check the results with the numerical solution(in 3 dimension) of the wiki page that we list above.

Number of Electrons	Theoretical Potential Energy	Solved Potential Energy	Number of Iterations
2	0.500000	0.500000	14
10	32.716949	32.717061	918
100	4448.350634	4448.415575	4586
470	104822.886324	104823.779129	32930
972	455651.080935	455653.838019	46433

High Dimension Case

We need present a proposition:

There are 2 randomly picked points on the $n(n\geq 2)$ dimension hyper-sphere of radius $R$. Let $L$ designate the Euclidean distance between these two points. The nature of the probability distribution of $L$ can be described as below:

$$\rho_{L,n}(L)=\frac{L^{n-2}}{C(n)R^{n-1}}[1-(\frac{L}{2R})^2]^{\frac{n-3}{2}}$$

The coefficients $C(n)$ are given as follows:

$$C(n)=\sqrt{\pi}\frac{\Gamma(\frac{n-1}{2})}{\Gamma(\frac{n}{2})}$$

Known Issues

~~The GPU code should be completely rewritten. Now, with irrational structural design, GPU code even slower than CPU :( ...~~
Some fast algorithms caused a certain degree of numerical precision error;

inlmouse / ThomsonProblemSolver