线性多变量问题的平均情形复杂度

关于使用解析Korobov核进行多变量逼近的EC-(s,t)-弱可处理性注记

A note about EC-$(s,t)$-weak tractability of multivariate approximation with analytic Korobov kernels

本文研究了在最坏和平均情况下，使用解析Korobov核对[0,1]^d上连续函数进行多变量逼近的EC-(s,t)-弱可处理性条件，得出了不同参数范围下的必要和充分条件。

最小子采样秩1格用于多变量近似中的最优收敛速率

Minimal Subsampled Rank-1 Lattices for Multivariate Approximation with Optimal Convergence Rate

本文提出了一种最小子采样秩1格，用于多变量近似问题，实现了最优收敛速率和最小初始格大小。

可变解耦与有理函数的科莫戈罗夫超置定理

Variable decoupling and the Kolmogorov Superposition Theorem for rational functions

本文展示有理多变量函数的科莫戈罗夫超置定理可通过直观写出的单变量函数实现，无需计算解耦多变量有理函数变量。关键工具是多变量函数的洛温纳框架，应用于低复杂度多变量有理和多项式函数近似非有理函数。

多变量单调函数的蒙特卡洛近似难度

The Difficulty of Monte Carlo Approximation of Multivariate Monotone Functions

研究基于n次函数评估对多变量单调函数的L1逼近，发现确定性设置下存在维度诅咒，而随机设置下复杂度随√d增长，但小误差下最佳方法仍为确定性，且问题不弱可 tractable。

Asymptotic analysis in multivariate average case approximation with Gaussian kernels

We consider tensor product random fields $Y_d$, $d\in\mathbb{N}$, whose covariance funtions are Gaussian kernels. The average case approximation complexity $n^{Y_d}(\varepsilon)$ is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate $Y_d$, with relative $2$-average error not exceeding a given threshold $\varepsilon\in(0,1)$. We investigate the growth of $n^{Y_d}(\varepsilon)$ for arbitrary fixed $\varepsilon\in(0,1)$ and $d\to\infty$. Namely, we find criteria of boundedness for $n^{Y_d}(\varepsilon)$ on $d$ and of tending $n^{Y_d}(\varepsilon)\to…

Quantum Adaptive Search：一种用于多元函数全局优化的量子-经典混合算法

Quantum Adaptive Search: A Hybrid Quantum-Classical Algorithm for Global Optimization of Multivariate Functions

提出量子-经典混合算法QAGS，通过量子态复振幅映射动态缩小搜索空间并结合经典优化器局部优化，实现了多元函数全局优化的更高精度与更低时空复杂度。

Approximating smooth, multivariate functions on irregular domains

In this paper, we introduce a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in $d$ dimensions, where $d$ can be arbitrary. This method is simple, and relies only on orthogonal polynomials on a bounding tensor-product domain. In particular, the domain of the function need not be known in advance. When restricted to a subdomain, an orthonormal basis is no longer a basis, but a frame. Numerical computations with frames present potential difficulties, due to the near-linear dependence of the truncated approximation syst…

使用高斯核的多变量最差情形逼近的渐近分析

Asymptotic analysis in multivariate worst case approximation with Gaussian kernels

本文研究使用高斯核在最差情形下逼近多变量函数时信息复杂度随维度的渐近增长，分别在固定误差阈值和误差趋于零的情况下给出了渐近分析结果。

Near-optimal sampling strategies for multivariate function approximation on general domains

In this paper, we address the problem of approximating a multivariate function defined on a general domain in $d$ dimensions from sample points. We consider weighted least-squares approximation in an arbitrary finite-dimensional space $P$ from independent random samples taken according to a suitable measure. In general, least-squares approximations can be inaccurate and ill-conditioned when the number of sample points $M$ is close to $N = \dim(P)$. To counteract this, we introduce a novel method for sampling in general domains which leads to provably accurate and well-conditioned approximation…

On the power of standard information for tractability for $L_2$-approximation in the average case setting

We study multivariate approximation in the average case setting with the error measured in the weighted $L_2$ norm. We consider algorithms that use standard information $Λ^{\rm std}$ consisting of function values or general linear information $Λ^{\rm all}$ consisting of arbitrary continuous linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability for $Λ^{\rm std}$ and $Λ^{\rm all}$ for the absolute error criterion, and show that the power of $Λ^{\rm std}$ is the same as that of $Λ^{\rm all}$ for all notions of algebraic and exponential tr…