Minimal Polynomial Extrapolation (MPE) and Reduced Rank Extrapolation (RRE) are two polynomial methods used for accelerating the convergence of sequences of vectors $\{{x}_m\}$. They are applied successfully in conjunction with fixed-point iterative schemes in the solution of large and sparse systems of linear and nonlinear equations in different disciplines of science and engineering. Both methods produce approximations $s_k$ to the limit or antilimit of $\{{x}_m\}$ that are of the form ${s}_k=\sum^k_{i=0}γ_i{x}_i$ with $\sum^k_{i=0}γ_i=1$, for some scalars $γ_i$. The way the two methods are derived suggests that they might, somehow, be related to each other; this has not been explored so far, however. In this work, we tackle this issue and show that the vectors $s_{k}^\text{MPE}$ and $s_k^\text{RRE}$ produced by the two methods are related in more than one way, and independently of the way the $x_m$ are generated. One of our results states that RRE stagnates, in the sense that ${s}_k^\text{RRE}={s}_{k-1}^\text{RRE}$, if and only if ${s}_{k}^\text{MPE}$ does not exist. Another result states that, when ${s}_{k}^\text{MPE}$ exists, there holds $$μ_k{s}_k^\text{RRE} = μ_{k-1}{s}_{k-1}^\text{RRE} + ν_k{s}_{k}^\text{MPE} \quad \text{with} \quad μ_k = μ_{k-1} + ν_k,$$ for some positive scalars $μ_k$, $μ_{k-1}$, and $ν_k$ that depend only on ${s}_k^\text{RRE}$, ${s}_{k-1}^\text{RRE}$, and ${s}_{k}^\text{MPE}$, respectively. Our results are valid when MPE and RRE are defined in any weighted inner product and the norm induced by it. They also contain as special cases the known results pertaining to the connection between the method of Arnoldi and the method of generalized minimal residuals, two important Krylov subspace methods for solving nonsingular linear systems.
Reduced rank extrapolation for multi-term Sylvester equations
Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；Eindhoven University of Technology（埃因霍温理工大学）；Leipzig University of Applied Sciences（莱比锡应用科学大学）
Generalizing Reduced Rank Extrapolation to Low-Rank Matrix Sequences
Eindhoven University of Technology（埃因霍温理工大学）；Leipzig University of Applied Sciences（莱比锡应用科技大学）；Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）
Reduced Higher Order SVD: ubiquitous rank-reduction method in tensor-based scientific computing
张量数值方法，基于秩结构的张量表示的$d$变量函数和运营商，旨在提供$O(dn)$复杂性的数值计算$n^{\otimes d }$网格相反的$O(n^d)$缩放传统的基于网格的方法。然而，多个张量操作可能导致目标数据的张量秩（秩的诅咒）的巨大增加，使得计算变得棘手。因此，在张量计算中最重要的步骤之一是鲁棒的和有效的降秩过程，在多维算子和函数演算中的各种张量变换的过程中，应该多次执行。在文献[33]中引入的基于降阶高阶奇异值分解（RHOSVD）的降秩方法在张量数值方法的发展中起了重要作用。在这里，我们简要地调查RHOSVD方法的要点，然后集中在一些新的理论和计算方面的RHOSVD证明，这种降秩技术构成的基本成分，在张量计算的现实生活中的问题。特别地，给出了RHOSVD的稳定性分析。我们介绍了张量的多重线性代数表示的范围分离（RS）张量格式。这使得RHOSVD降秩技术可以应用于具有许多奇异性的非规则函数数据，例如，生物分子建模中集体多粒子相互作用势的秩结构计算，以及复杂的复合径向函数。给出了RHOSVD在散乱数据建模中应用的新的理论和数值结果。RHOSVD被证明是一种有效的降秩技术，在许多应用中，从多粒子系统的数值处理到PDE约束控制问题的数值解。
Structure-Preserving Reduced-Order Modeling via Low-Rank Transport Signatures
Duke University（杜克大学）；University of Washington（华盛顿大学）；Rensselaer Polytechnic Institute（伦斯勒理工学院）；University of Pennsylvania（宾夕法尼亚大学）；Cornell University（康奈尔大学）；Georgia Institute of Technology（佐治亚理工学院）
Proximal Gradient-based Low Rank Tensor Decomposition for State Dependent Riccati Equation
Institute of Mathematical Science, ShanghaiTech University（上海科技大学数学科学学院）；Department of Mathematics, University of Alabama at Birmingham（阿拉巴马大学伯明翰分校数学系）
