Tensor-Train Numerical Integration of Multivariate Functions with Singularities

具有奇异曲线的函数的多变量有理逼近

Multivariate rational approximation of functions with curves of singularities

本文研究了具有奇异曲线的函数的多变量有理逼近，通过将极点聚集从点推广到曲线，构造了多变量有理逼近，为处理复杂奇异性提供了新方法。

高阶近似非光滑多变量函数

High order approximation to non-smooth multivariate functions

本文提出一种高阶近似非光滑多变量函数的方法，针对非光滑函数在奇点附近低阶近似问题，通过设计修正项实现全域高阶近似。

High-Order Quadrature on Multi-Component Domains Implicitly Defined by Multivariate Polynomials

A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly defined geometry as the graph of an implicitly defined, multi-valued height function, and applies a dimension reduction approach needing only one-dimensional quadrature. In particular, we explore the use of Gauss-Legendre and tanh-sinh methods and demonstrate that the quadrature algorithm inherits their high-order convergence rates. Under the action of $h$-re…

关于本质奇点的收敛性

On Convergence to Essential Singularities

本文研究迭代方法在函数f上的收敛性，提出无需函数在极限点定义的收敛条件，扩展了Absil等人的结果，应用于分析发散序列及低秩张量近似研究。

Approximation and sampling of multivariate probability distributions in the tensor train decomposition

General multivariate distributions are notoriously expensive to sample from, particularly the high-dimensional posterior distributions in PDE-constrained inverse problems. This paper develops a sampler for arbitrary continuous multivariate distributions that is based on low-rank surrogates in the tensor-train format. We construct a tensor-train approximation to the target probability density function using the cross interpolation, which requires a small number of function evaluations. For sufficiently smooth distributions the storage required for the TT approximation is moderate, scaling linea…

用树张量网络压缩多变量函数

Compressing multivariate functions with tree tensor networks

本文提出用树张量网络（TTN）作为多变量连续函数的压缩格式，通过广义张量交叉插值算法构造初等函数和复杂函数，相比张量列（TT）更高效，并应用于多维非线性Fredholm方程求解。