论文标题
使用Chebyshev插值的功能性Tucker近似
Functional Tucker approximation using Chebyshev interpolation
论文作者
论文摘要
这项工作涉及通过函数评估在张量产生域上定义的三变形函数。将张力的Chebyshev插值与低多线性等级的Tucker分解相结合，可以非常有效地计算和存储。现有的Chebfun3算法[Hashemi和Trefethen，Siam J. Sci。 Comput。，39（2017）]使用类似的格式，但近似的构造是通过所谓的slice-tucker分解间接进行的。结果，Chebfun3有时会不必要地使用许多功能评估，并且不会完全受益于塔克分解的潜力减少计算成本，有时会大幅度降低，有时会大幅度降低。我们提出了一种新型算法Chebfun3f，该算法利用单变量纤维而不是双变量切片来构建塔克分解。 Chebfun3f从几乎所有考虑的功能的功能评估数量（通常为75％），有时甚至超过98％的功能评估数量降低了近似的成本。
This work is concerned with approximating a trivariate function defined on a tensor-product domain via function evaluations. Combining tensorized Chebyshev interpolation with a Tucker decomposition of low multilinear rank yields function approximations that can be computed and stored very efficiently. The existing Chebfun3 algorithm [Hashemi and Trefethen, SIAM J. Sci. Comput., 39 (2017)]uses a similar format but the construction of the approximation proceeds indirectly, via a so called slice-Tucker decomposition. As a consequence, Chebfun3 sometimes uses unnecessarily many function evaluations and does not fully benefit from the potential of the Tucker decomposition to reduce, sometimes dramatically, the computational cost. We propose a novel algorithm Chebfun3F that utilizes univariate fibers instead of bivariate slices to construct the Tucker decomposition. Chebfun3F reduces the cost for the approximation in terms of the number of function evaluations for nearly all functions considered, typically by 75%, and sometimes by over 98%.
