Adaptive Monte Carlo algorithm for Wigner kernel evaluation

Abstract

In this paper, we study numerically various approaches, namely an adaptive Monte Carlo algorithm, a particular rank-1 lattice algorithm based on generalized Fibonacci numbers and a Monte Carlo algorithm based on Latin hypercube sampling for computing multidimensional integrals. We compare the performance of the algorithms over three case studies—multidimensional integrals from Bayesian statistics, the so-called Genz test functions and the Wigner kernel—an important issue in quantum mechanics represented by multidimensional integrals. A comprehensive study and an analysis of the computational complexity of the algorithms under consideration has been presented. Adaptive strategy is well-established as an efficient and reliable tool for multidimensional integration of integrands functions with computational peculiarities like peaks. The presented adaptive Monte Carlo algorithm gives reliable results in computing the Wigner kernel by a stochastic approach that has significantly lower computational complexity than the existing deterministic approaches.

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Correspondence to Venelin Todorov.

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Venelin Todorov is supported by the Bulgarian National Science Fund under Project KP-06-PM32/4—2019 “Advanced Stochastic and Deterministic Approaches for Large-Scale Problems of Computational Mathematics” and by the National Scientific Program “Information and Communication Technologies for a Single Digital Market in Science, Education and Security (ICT in SES)”, contract No DO1-205/23.11.2018, financed by the Ministry of Education and Science in Bulgaria. Stoyan Dimitrov is supported by the Project KP-06-H22/3 financed by the National Scientific Fund of Bulgaria at the Ministry of Education and Science. The work is also partially supported by the Bulgarian National Science Fund under Project DN 12/5-2017. The authors are grateful to Dr. Jean Michel Sellier for the valuable discussions concerning quantum mechanics and especially Wigner function.

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Todorov, V., Dimov, I., Georgieva, R. et al. Adaptive Monte Carlo algorithm for Wigner kernel evaluation. Neural Comput & Applic 32, 9953–9964 (2020). https://doi.org/10.1007/s00521-019-04519-9

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Keywords

  • Multidimensional integration
  • Adaptive Monte Carlo algorithm
  • Fibonacci lattice sets
  • Latin hypercube sampling
  • Wigner kernel

Mathematics Subject Classification

  • 65C05
  • 65U05
  • 65F10
  • 65Y20