A Monte Carlo method is proposed for transforming high-dimensional potential energy surfaces evaluated on discrete grid points into a sum-of-products form, more precisely into a Canonical Polyadic Decomposition form. To this end, a modified existing ansatz based on the alternating least squares method is used, in which numerically exact integrals are replaced with Monte Carlo integrals. This largely reduces the numerical cost by avoiding the evaluation of the potential on all grid points and allows the treatment of surfaces with many degrees of freedom. Calculations on the 15D potential of the protonated water dimer (Zundel cation) in a sum-of-products form are presented and compared to the results obtained in a previous work [M. Schröder and H.-D. Meyer, J. Chem. Phys. 147, 064105 (2017)], where a sum-of-products form of the potential was obtained in the Tucker format.

中文翻译：

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使用蒙特卡洛方法将高维势能表面转换为规范的多元分解。

提出了一种蒙特卡洛方法，用于将在离散网格点上评估的高维势能面转换为乘积和形式，更精确地转换为规范多阿迪克分解形式。为此，使用了基于交替最小二乘法的修改后的现有ansatz，其中数值精确的积分被替换为Monte Carlo积分。通过避免对所有网格点上的电势进行评估，可以大大降低数值成本，并可以处理具有许多自由度的表面。给出了产品总和形式的质子化水二聚体（Zundel阳离子）的15D电势的计算结果，并将其与先前工作中获得的结果进行了比较[M. Schröder和H.-D. Meyer，化学杂志。物理 147，064105（2017）]，