Parallel cross interpolation for high-precision calculation of high-dimensional integrals
Parallel cross interpolation for high-precision calculation of high-dimensional integrals
复制标题
DOI：
10.1016/j.cpc.2019.106869
复制
发表时间：
2020-01-01
影响因子：
6.3
通讯作者：
Savostyanov, Dmitry
中科院分区：
文献类型：
--
作者：
Dolgov, Sergey;Savostyanov, Dmitry
我们提出了一种并行的交叉内插算法，并将其应用于量子物理中伊辛模型激发的高维积分的计算。与蒙特卡罗和准蒙特卡罗等主流方法相比，我们的算法计算的样本既不是随机的，也不形成规则的晶格。相反，我们沿着各个维度(模式)计算给定的函数，并使用这些值来重建其在整个域中的行为。对于给定的函数，计算的单变量纤维的位置被自适应地选择。所需的计算可以沿着每个模式(变量)和所有模式并行执行。为了证明该方法的有效性，我们将其应用于计算高维Ising磁化率积分，该积分源于二维Ising铁磁模型中自发磁化的渐近展开。我们观察到了该方法的强超线性收敛，而MC和QMC算法则是次线性收敛的。使用多精度算法，我们还观察到所提出的算法的指数收敛。该算法结合了高阶收敛、几乎完美的可扩展到数百个过程以及与MC和QMC相同的灵活性，可以为涉及高维积分的问题(如统计、概率和量子物理)提供一种新的选择方法。(C)2019年提交人。爱思唯尔出版公司(Elsevier B.V.)
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. (C) 2019 The Authors. Published by Elsevier B.V.