Parallel solution of saddle point systems with nested iterative solvers based on the Golub-Kahan Bidiagonalization

An iterative generalized Golub-Kahan algorithm for problems in structural mechanics

This paper studies the Craig variant of the Golub-Kahan bidiagonalization algorithm as an iterative solver for linear systems with saddle point structure. Such symmetric indefinite systems in 2x2 block form arise in many applications, but standard iterative solvers are often found to perform poorly on them and robust preconditioners may not be available. Specifically, such systems arise in structural mechanics, when a semidefinite finite element stiffness matrix is augmented with linear multi-point constraints via Lagrange multipliers. Engineers often use such multi-point constraints to introd…

用于反问题的新型灵活且不精确的 Golub-Kahan 算法

New flexible and inexact Golub-Kahan algorithms for inverse problems

提出一类基于灵活、不精确 Golub-Kahan 分解的 Krylov 子空间算法，通过自适应子空间投影迭代求解含一般数据保真度泛函的大规模线性反问题，并在成像实验中展现竞争力。

Weighted tensor Golub-Kahan-Tikhonov-type methods applied to image processing using a t-product

This paper discusses weighted tensor Golub-Kahan-type bidiagonalization processes using the t-product. This product was introduced in [M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp.~641--658]. A few steps of a bidiagonalization process with a weighted least squares norm are carried out to reduce a large-scale linear discrete ill-posed problem to a problem of small size. The weights are determined by symmetric positive definite (SPD) tensors. Tikhonov regularization is applied to the reduced problem. An algorithm for tensor…

面向大规模贝叶斯反问题的不精确广义Golub-Kahan方法

Inexact Generalized Golub-Kahan Methods for Large-Scale Bayesian Inverse Problems

本文针对正演算子不精确且先验协方差难以分解的大规模贝叶斯反问题，提出不精确广义Golub-Kahan分解与混合迭代投影方法，实现Tikhonov正则化参数自动选择，并通过模拟层析成像实验验证其稳定性和有效性。

用于广义鞍点系统的广义Golub-Kahan双对角化

Generalized Golub-Kahan bidiagonalization for generalized saddle point systems

该研究将广义Golub-Kahan双对角化方法推广到右下角块非零的广义鞍点系统，提出两类对应不同前导块性质的CRAIG求解算法，验证了其在计算效率或内存需求上的优势。

基于广义Golub-Kahan方法的高效采样方法用于大规模分层贝叶斯反问题

Efficient sampling approaches based on generalized Golub-Kahan methods for large-scale hierarchical Bayesian inverse problems

本文提出基于广义Golub-Kahan方法的高效采样方法，用于解决大规模分层贝叶斯反问题中的不确定性量化挑战，通过Metropolis-Hastings独立采样和低秩近似或预条件Lanczos方法进行有效采样。

高效通用的Golub-Kahan方法用于动态反问题

Efficient generalized Golub-Kahan based methods for dynamic inverse problems

本文提出基于广义Golub-Kahan bidiagonalization的高效迭代矩阵无关方法，用于解决动态反问题中的参数估计与不确定性评估，展示了方法在光电声学成像、去模糊和被动地震层析中的有效性。

基于爱因斯坦积的张量Golub-Kahan方法

Tensor Golub Kahan based on Einstein product

针对爱因斯坦积框架下的张量SVD逼近问题，提出基于张量Lanczos双对角化及Ritz增广重启的方法，验证了有效性并展示了在视频压缩、人脸识别中的应用。

Computation of generalized matrix functions with rational Krylov methods

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 …

用于反问题的随机化Krylov方法

Randomized Krylov methods for inverse problems

本文提出随机化Golub-Kahan方法及混合投影正则化框架，用于高效求解大规模矩形矩阵反问题，并通过图像去模糊和地震层析成像实验验证其有效性。

随机灵活的LSQR和LSMR及其在反问题中的应用

Randomized Flexible LSQR and LSMR with applications to inverse problems

本文提出了一种随机化的灵活LSQR和LSMR算法，用于解决大规模反问题，通过引入随机化技术提高计算效率并保持重建质量。

对称鞍点系统中非对角块的消去法

Deflation for the off-diagonal block in symmetric saddle point systems

针对对称鞍点系统非对角块奇异值分布不佳影响Krylov子空间法收敛的问题，提出基于广义Golub-Kahan双对角化椭圆奇异值分解向量的消去策略，扩展现有方阵消去框架，数值实验验证了其有效性。