### Abstract

We propose a parallel version of the cross interpolation algorithm and apply it to calculate high-dimensional integrals motivated by Ising model in quantum physics. In contrast to mainstream approaches, such as Monte Carlo and quasi Monte Carlo, the samples calculated by our algorithm are neither random nor form a regular lattice. Instead we calculate the given function along individual dimensions (modes) and use these values to reconstruct its behaviour in the whole domain. The positions of the calculated univariate fibres are chosen adaptively for the given function. The required evaluations can be executed in parallel along each mode (variable) and over all modes.

To demonstrate the efficiency of the proposed method, we apply it to compute high-dimensional Ising susceptibility integrals, arising from asymptotic expansions for the spontaneous magnetisation in two-dimensional Ising model of ferromagnetism. We observe strong superlinear convergence of the proposed method, while the MC and qMC algorithms converge sublinearly. Using multiple precision arithmetic, we also observe exponential convergence of the proposed algorithm. Combining high-order convergence, almost perfect scalability up to hundreds of processes, and the same flexibility as MC and qMC, the proposed algorithm can be a new method of choice for problems involving high-dimensional integration, e.g. in statistics, probability, and quantum physics.

To demonstrate the efficiency of the proposed method, we apply it to compute high-dimensional Ising susceptibility integrals, arising from asymptotic expansions for the spontaneous magnetisation in two-dimensional Ising model of ferromagnetism. We observe strong superlinear convergence of the proposed method, while the MC and qMC algorithms converge sublinearly. Using multiple precision arithmetic, we also observe exponential convergence of the proposed algorithm. Combining high-order convergence, almost perfect scalability up to hundreds of processes, and the same flexibility as MC and qMC, the proposed algorithm can be a new method of choice for problems involving high-dimensional integration, e.g. in statistics, probability, and quantum physics.

Original language | English |
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Article number | 106869 |

Pages (from-to) | 1-15 |

Number of pages | 15 |

Journal | Computer Physics Communications |

Volume | 246 |

Early online date | 23 Aug 2019 |

DOIs | |

Publication status | Published - 1 Jan 2020 |

### Keywords

- high-dimensional integration
- high precision
- tensor train format
- cross interpolation
- ising integrals
- parallel algorithms

### Cite this

Dolgov, S., & Savostyanov, D. (2020). Parallel cross interpolation for high--precision calculation of high-dimensional integrals.

*Computer Physics Communications*,*246*, 1-15. [106869]. https://doi.org/10.1016/j.cpc.2019.106869