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Workshop on Advances on Scientific and Engineering Computing (I)

Overcoming Sign Problem via Particle Annihilation

Speaker

Sihong Shao(邵嗣烘) , Peking University

Time

26 Sep, 16:50 - 17:30

Abstract

The infamous numerical sign problem poses a fundamental obstacle to long-time stochastic Wigner simulations in high dimensional phase space. Although the existing particle annihilation via uniform mesh (PAUM) significantly alleviates the sign problem when dimensionality D < 5, the setting of regular grids gives rise to another challenge in data storage when D > 5 due to the curse of dimensionality. To this end, we developed an adaptive particle annihilation algorithm, termed sequential-clustering particle annihilation via discrepancy estimation (SPADE), which consists of adaptive clustering of particles via controlling their number-theoretic discrepancies and independent random matching in each group, and may learn the minimal amount of particles that can accurately capture the oscillating nature of the Wigner function. Both deterministic error bounds by the Koksma-Hlawka inequality and non-asymptotic random error bounds by concentration inequalities are proved to be affected by two factors. One factor measures the irregularity of point distributions and reflects their discrete nature. The other relies on the variation of test function and is influenced by the continuity. Only the latter implicitly depends on dimensionality D, implying that SPADE can be immune to the curse of dimensionality for a wide class of test functions. Combining SPADE with a recently proposed variance reduction technique via the stationary phase approximation (SPA), we make the first attempt to simulate the transitions of hydrogen energy levels in 6-D phase space, where the feasibility of PAUM with sample sizes about 109-1010 has also been explored as a comparison.