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.
Highlights
High-dimensional integrals often occur in quantum mechanics [1], in statistics and probability, e.g. expectations with multivariate probability distributions [2], inverse problems with uncertainty [3], and many more
Bailey et al [85], using the values of Dd obtained by quasi Monte Carlo algorithm, were able to match the results of Nickel to 20 decimal digits
Based on our preliminary experiments with Quasi Monte Carlo (qMC), and assuming that the convergence rate ε ∼ Ne−v0al.7 will not deteriorate, we estimate that to reach the same accuracy with qMC we would need approximately 1013 years of calculations and 109 terawatt hours of energy — which exceeds the age of the Universe (≈ 1.3 · 1010 years) and annual world energy consumption (≈ 1.5 · 105 TWh in 2014) by three orders of magnitude
Summary
High-dimensional integrals often occur in quantum mechanics [1], in statistics and probability, e.g. expectations with multivariate probability distributions [2], inverse problems with uncertainty [3], and many more. A naïve approach, based on tensor product of one-dimensional quadrature rules, requires the total number of function evaluations N that grows exponentially with the problem dimension d, exceeding the possibilities of modern computers for d ≳ 10. Xd) on a tensor product n × · · · × n quadrature grid from a linear in d number of samples, which are adapted to f This adaptivity allows the proposed algorithm to locate important samples (e.g. areas of concentration of density) and reach faster convergence, compared to mainstream numerical methods, such as MC and qMC, where the positions of the samples are either not optimised, or are optimal for a wide class of functions. We briefly summarise the results of this paper and discuss some challenges and potential directions for future work
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