In this paper, we introduce a mesh-free two-level hybrid Tucker tensor format for approximation of multivariate functions, which combines the product Chebyshev interpolation with the ALS-based Tucker decomposition of the tensor of Chebyshev coefficients. It allows to avoid the expenses of the rank-structured approximation of function-related tensors defined on large spacial grids, while benefiting from the Tucker decomposition of the rather small core tensor of Chebyshev coefficients. This leads to nearly optimal Tucker rank parameters which are close to the results for well established Tucker-ALS algorithm applied to the large grid-based tensors. These rank parameters inherited from the Tucker-ALS decomposition of the coefficient tensor can be much less than the polynomial degrees of the initial Chebyshev interpolant via function independent basis set. Furthermore, the tensor product Chebyshev polynomials discretized on a tensor grid leads to a low-rank two-level orthogonal algebraic Tucker tensor that approximates the initial function with controllable accuracy. It is shown that our techniques could be gainfully applied to the long-range part of the electrostatic potential of multi-particle systems approximated in the range-separated tensor format. Error and complexity estimates of the proposed methods are presented. We demonstrate the efficiency of the suggested method numerically on examples of the long-range components of multi-particle interaction potentials generated by 3D Newton kernel for large bio-molecule systems and lattice-type compounds.

翻译：本文提出了一种用于多元函数逼近的无网格双层混合塔克张量格式，该格式将乘积型切比雪夫插值与基于ALS的切比雪夫系数张量塔克分解相结合。该方法避免了在大规模空间网格上定义函数相关张量时进行秩结构化逼近的高昂计算代价，同时受益于对规模较小的切比雪夫系数核心张量进行塔克分解。由此得到的塔克秩参数近乎最优，与成熟的Tucker-ALS算法应用于大型网格张量所得结果相近。从系数张量的Tucker-ALS分解继承的这些秩参数，可远小于通过函数无关基组构建的初始切比雪夫插值器的多项式次数。此外，在张量网格上离散化的张量积切比雪夫多项式会生成一个低秩双层正交代数塔克张量，能以可控精度逼近初始函数。研究表明，我们的技术可有效应用于多粒子系统静电势的长程部分，该部分在范围分离张量格式中进行逼近。文中给出了所提方法的误差与复杂度估计。我们通过三维牛顿核生成的大型生物分子系统及晶格型化合物的多粒子相互作用势长程分量算例，数值验证了该方法的有效性。