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Compared with random sampling, low-discrepancy sampling is more effective in covering the search space. However, the existing research cannot definitely state whether the impact of a low-discrepancy sample on particle swarm optimization (PSO) is positive or negative. Using Niderreiter's theorem, this study completes an error analysis of PSO, which reveals that the error bound of PSO at each iteration depends on the dispersion of the sample set in an expanded dimensional space. Based on this error analysis, an acceleration technique for PSO-type algorithms is proposed with low-discrepancy sampling in the expanded dimensional space. The acceleration technique can generate a low-discrepancy sample set with a smaller dispersion, compared with a random sampling, in the expanded dimensional space; it also reduces the error at each iteration, and hence improves the convergence speed. The acceleration technique is combined with the standard PSO and the comprehensive learning particle swarm optimization, and the performance of the improved algorithm is compared with the original algorithm. The experimental results show that the two improved algorithms have significantly faster convergence speed under the same accuracy requirement., Comment: 29 pages, 0 figures

In this paper, two new efficient algorithms for calculating the sign function of the large-scale sparse matrix are proposed by combining filtering algorithm with Newton method and Newton Schultz method respectively. Through the theoretical analysis of the error diffusion in the iterative process, we designed an adaptive filtering threshold, which can ensure that the filtering has little impact on the iterative process and the calculation result. Numerical experiments are consistent with our theoretical analysis, which shows that the computational efficiency of our method is much better than that of Newton method and Newton Schultz method, and the computational error is of the same order of magnitude as that of the two methods., Comment: 18 pages,12 figures

Heterogeneous comprehensive learning particle swarm optimization (HCLPSO) is a type of evolutionary algorithm with enhanced exploration and exploitation capabilities. The low-discrepancy sequence (LDS) is more uniform in covering the search space than random sequences. In this paper, making use of the good uniformity of LDS to improve HCLPSO is researched. Numerical experiments are performed to show that it is impossible to effectively improve the search ability of HCLPSO by only using LDS to generate the initial population. However, if we properly choose some random sequences from the HCLPSO velocities updating formula and replace them with the deterministic LDS, we can obtain a more efficient algorithm. Compared with the original HCLPSO under the same accuracy requirement, the HCLPSO updating the velocities with the deterministic LDS can significantly reduce the iterations required for finding the optimal solution, without decreasing the success rate., Comment: 29 pages, 5 figures

The objective of this research was to compute the principal matrix square root with sparse approximation. A new stable iterative scheme avoiding fully matrix inversion (SIAI) is provided. The analysis on the sparsity and error of the matrices involved during the iterative process is given. Based on the bandwidth and error analysis, a more efficient algorithm combining the SIAI with the filtering technique is proposed. The high computational efficiency and accuracy of the proposed method are demonstrated by computing the principal square roots of different matrices to reveal its applicability over the existing methods., Comment: 19 pages, 3 figures

The main interest of this paper is to study the relationship between the low-discrepancy sequence and the static solution to the multi-bodies problem in high-dimensional space. An assumption that the static solution to the multi-bodies problem is a low-discrepancy sequence is proposed. Considering the static solution to the multi-bodies problem corresponds to the minimum potential energy principle, we further assume that the distribution of the bodies is the most uniform when the potential energy is the smallest. To verify the proposed assumptions, a dynamical evolutionary model (DEM) based on the minimum potential energy is established to find out the static solution. The central difference algorithm is adopted to solve the DEM and an evolutionary iterative scheme is developed. The selection of the mass and the damping coefficient to ensure the convergence of the evolutionary iteration is discussed in detail. Based on the DEM, the relationship between the potential energy and the discrepancy during the evolutionary iteration process is studied. It is found that there is a significant positive correlation between them, which confirms the proposed assumptions. We also combine the DEM with the restarting technique to generate a series of low-discrepancy sequences. These sequences are unbiased and perform better than other low-discrepancy sequences in terms of the discrepancy, the potential energy, integrating eight test functions and computing the statistical moments for two practical stochastic problems. Numerical examples also show that the DEM can not only generate uniformly distributed sequences in cubes, but also in non-cubes.