Application of uniformity and hyper uniformity in sciences

2019 Fiscal Year Final Research Report

• Project/Area Number: 26310211
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 特設分野
• Research Field: Mathematical Sciences in Search of New Cooperation
• Research Institution: Hiroshima University
• Principal Investigator: Matsumoto Makoto 広島大学, 理学研究科, 教授 (70231602)
• Project Period (FY): 2014-07-18 – 2020-03-31
• Keywords: 準モンテカルロ法

Outline of Final Research Achievements

Monte Carlo integration of a function f on a hypercube means to generate uniformly random sample points on the hypercube, and then take the average of the function evaluated at these points as a numerical approximation of the integration of f. Quasi-Monte Carlo integration is to choose hyper-uniform sample points to make the integration error smaller. Widely known point sets uses the t-value as a figure of merit. In this study we introduced "WAFOM with parameters" as a new figure of merit, and obtained hyper-uniform point sets according to this index. It is experimentally shown that the new point sets perform better than existing methods, for relatively smooth integrands

Free Research Field

代数学

Academic Significance and Societal Importance of the Research Achievements

モンテカルロ積分・準モンテカルロ積分は、高い次元の空間上の積分を行う際に有用であり、工学・科学・金融など幅広い応用を持つ。モンテカルロ積分法では、次元に無関係に誤差をサンプルポイント数の-0.5乗で小さくできるが、この誤差収束の遅さが問題となることも多い。本研究では、滑らかな関数に対しては誤差収束をNの-1乗以上に効率化する点集合の構成法を与えた。また、滑らかでない関数に対しても従来広く使われているlow discrepancy点集合と同程度の性能を持つことを実験的に示した。