Abstract
We propose a parallel version of the cross interpolation algorithm and apply it to calculate highdimensional 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 highdimensional Ising susceptibility integrals, arising from asymptotic expansions for the spontaneous magnetisation in twodimensional 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 highorder 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 highdimensional integration, e.g. in statistics, probability, and quantum physics.
To demonstrate the efficiency of the proposed method, we apply it to compute highdimensional Ising susceptibility integrals, arising from asymptotic expansions for the spontaneous magnetisation in twodimensional 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 highorder 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 highdimensional integration, e.g. in statistics, probability, and quantum physics.
Original language  English 

Article number  106869 
Pages (fromto)  115 
Number of pages  15 
Journal  Computer Physics Communications 
Volume  246 
Early online date  23 Aug 2019 
DOIs  
Publication status  Published  1 Jan 2020 
Keywords
 highdimensional integration
 high precision
 tensor train format
 cross interpolation
 ising integrals
 parallel algorithms
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