An iterative method for nonsymmetric systems with multiple right-hand sides
SIAM Journal on Scientific Computing
Some large-scale matrix computation problems
Journal of Computational and Applied Mathematics - Special issue on TICAM symposium
Analysis of Projection Methods for Solving Linear Systems with Multiple Right-Hand Sides
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
A well-conditioned estimator for large-dimensional covariance matrices
Journal of Multivariate Analysis
A Comparative Study of Iterative Solvers Exploiting Spectral Information for SPD Systems
SIAM Journal on Scientific Computing
Convergence properties of some block Krylov subspace methods for multiple linear systems
Journal of Computational and Applied Mathematics
Recycling Krylov Subspaces for Sequences of Linear Systems
SIAM Journal on Scientific Computing
An estimator for the diagonal of a matrix
Applied Numerical Mathematics
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Low cost high performance uncertainty quantification
Proceedings of the 2nd Workshop on High Performance Computational Finance
Matrices, Moments and Quadrature with Applications
Matrices, Moments and Quadrature with Applications
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
SelInv---An Algorithm for Selected Inversion of a Sparse Symmetric Matrix
ACM Transactions on Mathematical Software (TOMS)
Journal of the ACM (JACM)
A block IDR(s) method for nonsymmetric linear systems with multiple right-hand sides
Journal of Computational and Applied Mathematics
Low-cost data uncertainty quantification
Concurrency and Computation: Practice & Experience
The Sparse Matrix Transform for Covariance Estimation and Analysis of High Dimensional Signals
IEEE Transactions on Image Processing
A Matrix-free Approach for Solving the Parametric Gaussian Process Maximum Likelihood Problem
SIAM Journal on Scientific Computing
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The subject of this work is accelerating data uncertainty quantification. In particular, we are interested in expediting the stochastic estimation of the diagonal of the inverse covariance (precision) matrix that holds a wealth of information concerning the quality of data collections, especially when the matrices are symmetric positive definite and dense. Schemes built on direct methods incur a prohibitive cubic cost. Recently proposed iterative methods can remedy this but the overall cost is raised again as the convergence of stochastic estimators can be slow. The motivation behind our approach stems from the fact that the computational bottleneck in stochastic estimation is the application of the precision matrix on a set of appropriately selected vectors. The proposed method combines block conjugate gradient with a block-seed approach for multiple right-hand sides, taking advantage of the nature of the right-hand sides and the fact that the diagonal is not sought to high accuracy. Our method is applicable if the matrix is only known implicitly and also produces a matrix-free diagonal preconditioner that can be applied to further accelerate the method. Numerical experiments confirm that the approach is promising and helps contain the overall cost of diagonal estimation as the number of samples grows.