当已知具有n列的矩阵A被基矩阵B_1，.的线性组合很好地近似时，B_p，我们可以将A应用于随机向量并求解线性系以恢复此线性组合。同样的方法也可以用来恢复A^-1的近似值。一个基本的问题是这个线性系是否可逆和良态。本文证明了如果B_j的Gram矩阵是充分良态的，并且每个B_j都有高的数值秩，那么n {proportional} p log^2 n将保证线性系以高概率良态.我们的主要应用是探测具有光滑伪微分符号的线性算子，例如地震成像中的波动方程Hessian。我们数值证明，矩阵探测也可以产生良好的预条件，在可变介质中的反椭圆算子。
英文摘要：
When a matrix A with n columns is known to be well approximated by a linear combination of basis matrices B_1,..., B_p, we can apply A to a random vector and solve a linear system to recover this linear combination. The same technique can be used to recover an approximation to A^-1. A basic question is whether this linear system is invertible and well-conditioned. In this paper, we show that if the Gram matrix of the B_j's is sufficiently well-conditioned and each B_j has a high numerical rank, then n {proportional} p log^2 n will ensure that the linear system is well-conditioned with high probability. Our main application is probing linear operators with smooth pseudodifferential symbols such as the wave equation Hessian in seismic imaging. We demonstrate numerically that matrix probing can also produce good preconditioners for inverting elliptic operators in variable media.
On the conditioning of the matrix-matrix exponentiation
Polytechnic Institute of Coimbra/ISEC（科英布拉理工学院/ISEC）；Institute of Systems and Robotics, University of Coimbra（科英布拉大学系统机器人研究所）；Department of Mathematics, Robat Karim Branch, Islamic Azad University（伊斯兰阿扎德大学罗巴特卡里姆分校数学系）
Computing the Conditioning of the Components of a Linear Least Squares Solution
CERFACS（法国计算与实验流体力学研究中心）；University of Tennessee（田纳西大学）；Oak Ridge National Laboratory（橡树岭国家实验室）；University of Manchester（曼彻斯特大学）；Centre National d’Etudes Spatiales（法国国家空间研究中心）；University of Colorado at Denver and Health Sciences Center（科罗拉多大学丹佛分校健康科学中心）
Conditioning and backward error of block-symmetric block-tridiagonal linearizations of matrix polynomials
University of California, Santa Barbara（加州大学圣塔芭芭拉分校）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；Universidade de Lisboa（里斯本大学）；Faculdade de Economia do Porto（波尔图经济与管理学院）；Boston University（波士顿大学）
Matrix probing: a randomized preconditioner for the wave-equation Hessian
Department of Mathematics, MIT（麻省理工学院数学系）；Institute for Computational Mathematics and Engineering, Stanford（斯坦福大学计算数学与工程研究所）；Applied Mathematics Department, Université catholique de Louvain（天主教鲁汶大学应用数学系）；Total SA, Exploration & Production（道达尔公司勘探与生产部）；Courant Institute of Mathematical Sciences, NYU（纽约大学库朗数学科学研究所）
