Volker Grimm
Conjugated gradients on the normal equation (CGNE) is a popular method to regularise linear inverse problems. The idea of the method can be summarised as minimising the residuum over a suitable Krylov subspace. It is shown that using the same idea for the shift-and-invert rational Krylov subspace yields an order-optimal regularisation scheme.
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Numerical Stability of s-step Enlarged Krylov Subspace Conjugate Gradient Methods
本文提出s-step扩展Krylov子空间共轭梯度法，通过合并s次迭代提升收敛速度并减少通信开销，提出多种数值稳定的变体。
Krylov-subspace recycling via the POD-augmented conjugate-gradient method
本文提出一种新的Krylov子空间回收方法，用于高效求解右端变化的线性方程组。该方法基于目标导向的POD模型降阶思想，结合POD的最优性特性，提升子空间的适应性，通过混合直接/迭代三阶段方法提高求解效率。
Numerically Stable Recurrence Relations for the Communication Hiding Pipelined Conjugate Gradient Method
本文提出了一种数值稳定的流水线共轭梯度算法，通过将多项递推关系替换为两项递推关系，提高了算法的稳定性，同时保持了计算成本，从而在不同流水线长度下都能获得高精度解。
Preconditioning without a preconditioner: faster ridge-regression and Gaussian sampling with randomized block Krylov subspace methods
本文提出了一种无需显式构造预处理矩阵的随机块Krylov子空间方法，用于加速岭回归和高维高斯采样，理论证明其优于传统预处理方法。
Block Krylov methods have recently gained a lot of attraction. Due to their increased arithmetic intensity they offer a promising way to improve performance on modern hardware. Recently Frommer et al. presented a block Krylov framework that combines the advantages of block Krylov methods and data parallel methods. We review this framework and apply it on the Block Conjugate Gradients method,to solve linear systems with multiple right hand sides. In this course we consider challenges that occur on modern hardware, like a limited memory bandwidth, the use of SIMD instructions and the communicati…
Numerical analysis of the maximal attainable accuracy in communication hiding pipelined Conjugate Gradient methods
本文研究了流水线共轭梯度方法中局部舍入误差对可实现精度的影响，通过推导变量间隙表达式并进行数值实验，分析了误差传播对精度的效应。
Reduced Krylov Basis Methods for Parametric Partial Differential Equations
本文提出了一种基于预条件Krylov子空间方法的降维基方法，用于高效求解参数偏微分方程，显著降低计算成本。
Deflation and augmentation techniques in Krylov subspace methods for the solution of linear systems
本文综述了针对非Hermitian和Hermitian正定线性系统的Krylov子空间方法中的紧缩与增广加速收敛技术。
Flexibly Enlarged Conjugate Gradient Methods
本文针对扩展共轭梯度法及其s步版本内存需求高的问题，探究了多种在不大幅影响收敛性的前提下降低其内存开销的方案，这类方法原本通过扩展Krylov子空间实现了比经典CG更少的迭代次数。
Continuous analogues of Krylov methods for differential operators
本文提出连续Krylov方法求解二阶微分算子边界值问题，通过投影算子满足边界条件，并开发预处理算子保证有限次迭代得到近似解。
This paper deals with the definition and optimization of augmentation spaces for faster convergence of the conjugate gradient method in the resolution of sequences of linear systems. Using advanced convergence results from the literature, we present a procedure based on a selection of relevant approximations of the eigenspaces for extracting, selecting and reusing information from the Krylov subspaces generated by previous solutions in order to accelerate the current iteration. Assessments of the method are proposed in the cases of both linear and nonlinear structural problems.
A rational conjugate gradient method for linear ill-conditioned problems
针对 Hilbert 空间中的线性病态算子方程，本文结合有理与普通 Krylov 空间，提出基于 Arnoldi/Lanczos 分解及短期递归的有理共轭梯度型正则化方法，并用数值算例验证其收敛与正则化效果。
Equivalence of Krylov Subspace Methods for Skew-Symmetric Linear Systems
本文证明了两种求解反对线性系统的Krylov子空间方法与共轭梯度法和Craig方法在精确算术下迭代结果等价，并展示了投影误差不增的性质。
A New Approach for Solving Singular Systems in Topology Optimization Using Krylov Subspace Methods
针对拓扑优化中刚度矩阵奇异导致设计参数消失的问题，提出使用共轭残量法和共轭梯度法仍能获得局部最优解，并证明CGM的收敛性。
Memory-efficient recycling of large Krylov-subspaces for sequences of Hermitian linear systems
本文提出一种高效回收Krylov子空间的方法，用于解决具有固定矩阵和变化右端的Hermitian线性系统序列，通过残差优化预处理共轭残差法实现更小的计算开销和存储需求。
Factorized Krylov subspace methods for solving large Sylvester equations
本文提出一种基于矩阵Krylov子空间方法重构的低秩框架，利用Sylvester算子数学性质和右端项低秩结构，实现高效求解大型Sylvester方程的因子化算法。
Pipelined, Flexible Krylov Subspace Methods
本文提出了一种流水线化和灵活的Krylov子空间方法，旨在通过重叠内积和范数计算与预处理器应用来隐藏网络延迟和同步开销，提升大规模系统和强标度极限下的性能。
Bayes meets Krylov: preconditioning CGLS for underdetermined systems
本文研究了在贝叶斯框架下如何利用先验信息通过右预处理改进CGLS方法，从而更紧密地连接贝叶斯推断与克罗内克子空间方法，解决欠定线性逆问题。
Scalable s-step Preconditioned Conjugate Gradient with Chebyshev Basis and Gauss-Seidel Gram Solve
本文提出一种结合切比雪夫稳定Krylov基和前向高斯-塞德尔迭代的s步预条件共轭梯度法，通过分析切比雪夫格拉姆矩阵的结构，提升格拉姆系统的稳定性，减少同步开销，实现高效可扩展的求解方案。
Minimal residual rational Krylov subspace method for sequences of shifted linear systems
本文提出了一种基于最小余数有理Krylov子空间的方法，用于高效求解移位线性系统序列，尤其在处理非对称复数移位问题时表现优异。
The abstract issue of 'Krylov solvability' is extensively discussed for the inverse problem $Af = g$ where $A$ is a (possibly unbounded) linear operator on an infinite-dimensional Hilbert space, and $g$ is a datum in the range of $A$. The question consists of whether the solution $f$ can be approximated in the Hilbert norm by finite linear combinations of $g, Ag, A^2 g, \dots$ , and whether solutions of this sort exist and are unique. After revisiting the known picture when $A$ is bounded, we study the general case of a densely defined and closed $A$. Intrinsic operator-theoretic mechanisms ar…
The Adaptive $s$-step Conjugate Gradient Method
本文提出自适应$s$-步共轭梯度法，通过动态调整块大小以在减少全局同步次数的同时保持精度，提升大规模并行计算中Krylov子空间方法的可靠性与效率。
The Deflated Conjugate Gradient Method: Convergence, Perturbation and Accuracy
本文研究Deflated共轭梯度法中精度要求对收敛性的影响，证明有效条件数渐近线性依赖于缩减子空间的扰动大小，并讨论缩减投影计算所需精度。
Some transpose-free CG-like solvers for nonsymmetric ill-posed problems
本文提出并分析了一类Krylov子空间方法，为求解非对称线性系统提供了高效替代方案。通过无转置Arnoldi算法获得矩阵转置的秩亏近似，并应用于形式右预处理系统，理论分析和数值测试表明其在多种情况下优于传统方法。
Augmented unprojected Krylov subspace methods
本文证明增广 Krylov 子空间方法的投影问题具有等价的非投影表述，据此推导出首个非投影增广 FOM 并证明其作为回收方法的有效性，同时展示了 R$^{3}$GMRES 和非投影增广短递归方法（如 AugCG）在特定条件下契合该框架。
Analyzing the effect of local rounding error propagation on the maximal attainable accuracy of the pipelined Conjugate Gradient method
本文研究了流水线共轭梯度方法中局部舍入误差对最大可实现精度的影响，提出自动残差替换策略以提升精度并保持并行性能。
We present a probabilistic analysis of two Krylov subspace methods for solving linear systems. We prove a central limit theorem for norms of the residual vectors that are produced by the conjugate gradient and MINRES algorithms when applied to a wide class of sample covariance matrices satisfying some standard moment conditions. The proof involves establishing a four moment theorem for the so-called spectral measure, implying, in particular, universality for the matrix produced by the Lanczos iteration. The central limit theorem then implies an almost-deterministic iteration count for the iter…
Randomized Krylov-Projected Iterated Tikhonov Regularization for Large-Scale Ill-posed Problems Under A Posteriori Stopping Rule
针对大规模线性不适定逆问题，提出RIGKT和RIAT随机迭代正则化框架，结合随机迭代Tikhonov正则化与Krylov子空间投影技术，纳入多种策略及后验停止规则，经分析和实验验证其能可靠重建结构特征。
For the large-scale linear discrete ill-posed problem $\min\|Ax-b\|$ or $Ax=b$ with $b$ contaminated by white noise, the Golub-Kahan bidiagonalization based LSQR method and its mathematically equivalent CGLS, the Conjugate Gradient (CG) method applied to $A^TAx=A^Tb$, are most commonly used. They have intrinsic regularizing effects, where the iteration number $k$ plays the role of regularization parameter. The long-standing fundamental question is: {\em Can LSQR and CGLS find 2-norm filtering best possible regularized solutions}? The author has given definitive answers to this question for sev…
Estimating the numerical range with a Krylov subspace
针对Krylov子空间估计矩阵数值范围的问题，提出仅依赖矩阵与子空间维数、特征基条件数的逼近质量估计方法，并给出几乎匹配的下界以验证论证紧致性。
The present paper is concerned with developing tensor iterative Krylov subspace methods to solve large multi-linear tensor equations. We use the well-known T-product for two tensors to define tensor global Arnoldi and tensor global Gloub-Kahan bidiagonalization algorithms. Furthermore, we illustrate how tensor-based global approaches can be exploited to solve ill-posed problems arising from recovering blurry multichannel (color) images and videos, using the so-called Tikhonov regularization technique, to provide computable approximate regularized solutions. We also review a generalized cross-v…
Krylov-Type Methods for Tensor Computations
本文提出几种推广矩阵Krylov方法的张量Krylov型过程，包括最小、最大和收缩张量积Krylov递归，并证明最小Krylov递归能在一定迭代次数内提取低秩张量的正确子空间，同时通过优化最小Krylov过程获得更好的张量逼近，数值实验展示了这些方法的性质。
Approximation Accuracy of the Krylov Subspaces for Linear Discrete Ill-Posed Problems
本文研究了Krylov子空间在处理线性离散病态问题时的近似精度，提出了一个通用的sinθ定理来估计Krylov子空间与主导右奇异子空间之间的2范数距离，并通过数值实验验证了结果的准确性。
One of the most computationally expensive steps of the low-rank ADI method for large-scale Lyapunov equations is the solution of a shifted linear system at each iteration. We propose the use of the extended Krylov subspace method for this task. In particular, we illustrate how a single approximation space can be constructed to solve all the shifted linear systems needed to achieve a prescribed accuracy in terms of Lyapunov residual norm. Moreover, we show how to fully merge the two iterative procedures in order to obtain a novel, efficient implementation of the low-rank ADI method, for an impo…
In this paper, we propose different algorithms for the solution of a tensor linear discrete ill-posed problem arising in the application of the meshless method for solving PDEs in three-dimensional space using multiquadric radial basis functions. It is well known that the truncated singular value decomposition (TSVD) is the most common effective solver for ill-conditioned systems, but unfortunately the operation count for solving a linear system with the TSVD is computationally expensive for large-scale matrices. In the present work, we propose algorithms based on the use of the well known Ein…
We present a class of algorithms based on rational Krylov methods to compute the action of a generalized matrix function on a vector. These algorithms incorporate existing methods based on the Golub-Kahan bidiagonalization as a special case. By exploiting the quasiseparable structure of the projected matrices, we show that the basis vectors can be updated using a short recurrence, which can be seen as a generalization to the rational case of the Golub-Kahan bidiagonalization. We also prove error bounds that relate the error of these methods to uniform rational approximation. The effectiveness …
Exponential-Krylov methods for ordinary differential equations
本文提出了一种新的指数型积分器，通过单一低维克雷洛夫空间进行矩阵指数运算，适用于大规模ODE或半离散PDE系统，方法将时间离散化与克雷洛夫空间近似结合，无需监控线性系统误差。
ResQPASS: an algorithm for bounded variable linear least squares with asymptotic Krylov convergence
ResQPASS算法通过残差投影和活动集方法求解有界变量线性最小二乘问题，在无约束时退化为CG/LSQR，并具有渐近Krylov收敛性。
An inexact Newton-Krylov method for stochastic eigenvalue problems
本文提出了一种基于张量积结构的低秩近似牛顿方法，用于高效求解随机本征值问题，通过数值实验验证了该方法的有效性。
Convergence of the Iterative Rational Krylov Algorithm
本文针对状态空间对称系统，证明了迭代有理Krylov算法（IRKA）作为不动点迭代局部收敛于H2近似问题的局部极小值。
Regularization of linear inverse problems by rational Krylov methods
本文通过有理Krylov方法对线性反问题进行正则化，证明其在结合残差原理和大正则化参数时具有最优阶的正则化效果。
The conjugate gradient (CG) method is a classic Krylov subspace method for solving symmetric positive definite linear systems. We introduce an analogous semi-conjugate gradient (SCG) method for unsymmetric positive definite linear systems. Unlike CG, SCG requires the solution of a lower triangular linear system to produce each semi-conjugate direction. We prove that SCG is theoretically equivalent to the full orthogonalization method (FOM), which is based on the Arnoldi process and converges in a finite number of steps. Because SCG's triangular system increases in size each iteration, we study…
A general framework for Krylov ODE residuals with applications to randomized Krylov methods
针对非对称矩阵问题，提出一种基于微分方程残差的通用后验误差估计框架，统一并简化了现有结果，并扩展到随机化Arnoldi以外的多种Krylov方法，用于监控收敛并作为停止准则。
A Spectral Projection Preconditioner for Solving Ill Conditioned Linear Systems
本文提出基于谱投影的预条件器结合消去型Krylov子空间方法，用于求解病态线性系统，实验表明其收敛更快且解更精确。
Automatic smoothness detection of the resolvent Krylov subspace method for the approximation of $C_0$-semigroups
本文研究了残差Krylov子空间方法在$C_0$-半群近似中的自动平滑性检测问题，提出该方法在向量更光滑时收敛速度自动加快，无需用户了解理论，也不需修改方法以适应可能未知的向量平滑性，通过数值实验验证了这一结论。
Regularization Properties of the Krylov Iterative Solvers CGME and LSMR For Linear Discrete Ill-Posed Problems with an Application to Truncated Randomized SVDs
本文研究了CGME和LSMR求解线性离散病态问题的正则化性质，证明了CGME和LSMR的正则化解在精度上优于或等于LSQR，同时揭示了截断步骤对随机化SVD精度的影响。
The FEAST eigenvalue algorithm is a subspace iteration algorithm that uses contour integration in the complex plane to obtain the eigenvectors of a matrix for the eigenvalues that are located in any user-defined search interval. By computing small numbers of eigenvalues in specific regions of the complex plane, FEAST is able to naturally parallelize the solution of eigenvalue problems by solving for multiple eigenpairs simultaneously. The traditional FEAST algorithm is implemented by directly solving collections of shifted linear systems of equations; in this paper, we describe a variation of …
An algorithm for constructing a $J$-orthogonal basis of the extended Krylov subspace $\mathcal{K}_{r,s}=\operatorname{range}\{u,Hu, H^2u,$ $ \ldots, $ $H^{2r-1}u, H^{-1}u, H^{-2}u, \ldots, H^{-2s}u\},$ where $H \in \mathbb{R}^{2n \times 2n}$ is a large (and sparse) Hamiltonian matrix is derived (for $r = s+1$ or $r=s$). Surprisingly, this allows for short recurrences involving at most five previously generated basis vectors. Projecting $H$ onto the subspace $\mathcal{K}_{r,s}$ yields a small Hamiltonian matrix. The resulting HEKS algorithm may be used in order to approximate $f(H)u$ where $f$ …
Speeding up an unsteady flow simulation by adaptive BDDC and Krylov subspace recycling
本文通过自适应BDDC和Krylov子空间回收技术，显著加速了不稳流模拟中大规模线性方程组的求解，节省了超过40%的计算时间。
Some Results on Regularization of LSQR and CGLS for Large-Scale Discrete Ill-Posed Problems
本文研究了LSQR和CGLS在大规模离散病态问题中的正则化特性，建立了Krylov子空间与主导右奇异向量子空间的距离界限，指出严重和中等病态问题中Krylov子空间能捕捉主导奇异向量，但轻微病态问题则不然，并提出混合LSQR方法适用于轻微病态问题。
Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods
针对大型稀疏半正定矩阵的von Neumann熵近似计算问题，提出结合多项式与有理Krylov子空间算法，并利用随机化迹估计器和图着色探测技术，通过误差界与启发式规则实现高效计算，数值实验验证了方法的有效性。
Randomized Tensor Krylov Subspace Methods via Sketched Einstein Product with Applications to Image and Video Restoration
本文提出了一种基于刻痕爱因斯坦乘积的随机张量Krylov子空间方法，用于高效求解图像和视频恢复中的大规模多线性系统。
H-CMRH: a novel inner product free hybrid Krylov method for large-scale inverse problems
本文提出无内积混合Krylov方法H-CMRH，用于大规模反问题，在稳定半收敛上媲美混合GMRES，但计算与存储开销更低。
Sketched and Truncated Polynomial Krylov Subspace Methods: Matrix Sylvester Equations
本文针对矩阵Sylvester方程提出一种新的草图化与截断的多项式Krylov子空间方法，以解决投影问题适定性、误差估计、残差范数估计和显式基缺失等困难，并通过数值实验验证其在计算时间和存储需求上的优势。
Krylov and core transformation algorithms for an inverse eigenvalue problem to compute recurrences of multiple orthogonal polynomials
本文提出两种算法用于计算多重正交多项式递推系数，基于Krylov子空间和高斯消元，并通过实验验证其准确性和稳定性。
Optimizing network robustness via Krylov subspaces
本文提出结合 Krylov 子空间、贪心策略与内点法/L-BFGS 的新方法，通过有界修改部分边权来最大化提升或降低复杂网络鲁棒性，并在合成与真实网络上验证其优于现有技术。
Krylov subspace residual and restarting for certain second order differential equations
本文提出了基于二阶ODE残差概念和“残差-时间重启”Krylov子空间框架的算法，用于高效求解大型振荡二阶常微分方程系统，并通过Faber和Chebyshev级数分析及数值实验验证了其高效性。
An Arnoldi-based approach to polynomial and rational least squares problems
本文针对高次多项式与有理最小二乘问题中系数矩阵病态导致精度下降的问题，提出基于Arnoldi正交化的Krylov子空间方法，并通过算例验证了其有效性。
Preconditioned Krylov solvers for kernel regression
本文研究了核回归中求解密集线性系统的问题，提出了一种新型灵活预条件器，提升收敛速度并利用快速核矩阵向量乘法。
Flexible inner-product free Krylov methods for inverse problems
本文提出面向反问题的新型灵活无内积 Krylov 方法及随机化版本，支持迭代相关预处理，并通过多类变分正则化实验验证其性能。
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