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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2008.05161 |
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| _version_ | 1866910399895437312 |
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| author | Xiong, Yunfeng Shao, Sihong |
| author_facet | Xiong, Yunfeng Shao, Sihong |
| contents | The infamous numerical sign problem poses a fundamental obstacle to particle-based stochastic Wigner simulations in high dimensional phase space. Although the existing particle annihilation via uniform mesh significantly alleviates the sign problem when dimensionality D $\le$ 4, the mesh size grows dramatically when D $\ge$ 6 due to the curse of dimensionality and consequently makes the annihilation very inefficient. In this paper, we propose an adaptive particle annihilation algorithm, termed Sequential-clustering Particle Annihilation via Discrepancy Estimation (SPADE), to overcome the sign problem. SPADE follows a divide-and-conquer strategy: Adaptive clustering of particles via controlling their number-theoretic discrepancies and independent random matching in each group, and it may learn the minimal amount of particles that can accurately capture the non-classicality of the Wigner function. Combining SPADE with the variance reduction technique based on the stationary phase approximation, we attempt to simulate the proton-electron couplings in 6-D and 12-D phase space. A thorough performance benchmark of SPADE is provided with the reference solutions in 6-D phase space produced by a characteristic-spectral-mixed scheme under a $73^3 \times 80^3$ uniform grid, which fully explores the limit of grid-based deterministic Wigner solvers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_05161 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Overcoming the numerical sign problem in the Wigner dynamics via adaptive particle annihilation Xiong, Yunfeng Shao, Sihong Computational Physics Probability 81S30, 60J85, 65C05, 62G09, 35Q40 The infamous numerical sign problem poses a fundamental obstacle to particle-based stochastic Wigner simulations in high dimensional phase space. Although the existing particle annihilation via uniform mesh significantly alleviates the sign problem when dimensionality D $\le$ 4, the mesh size grows dramatically when D $\ge$ 6 due to the curse of dimensionality and consequently makes the annihilation very inefficient. In this paper, we propose an adaptive particle annihilation algorithm, termed Sequential-clustering Particle Annihilation via Discrepancy Estimation (SPADE), to overcome the sign problem. SPADE follows a divide-and-conquer strategy: Adaptive clustering of particles via controlling their number-theoretic discrepancies and independent random matching in each group, and it may learn the minimal amount of particles that can accurately capture the non-classicality of the Wigner function. Combining SPADE with the variance reduction technique based on the stationary phase approximation, we attempt to simulate the proton-electron couplings in 6-D and 12-D phase space. A thorough performance benchmark of SPADE is provided with the reference solutions in 6-D phase space produced by a characteristic-spectral-mixed scheme under a $73^3 \times 80^3$ uniform grid, which fully explores the limit of grid-based deterministic Wigner solvers. |
| title | Overcoming the numerical sign problem in the Wigner dynamics via adaptive particle annihilation |
| topic | Computational Physics Probability 81S30, 60J85, 65C05, 62G09, 35Q40 |
| url | https://arxiv.org/abs/2008.05161 |