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 severely and moderately ill-posed problems when the singular values of $A$ are simple. This paper extends the results to the multiple singular value case, and studies the approximation accuracy of Krylov subspaces, the quality of low rank approximations generated by Golub-Kahan bidiagonalization and the convergence properties of Ritz values. For the two kinds of problems, we prove that LSQR finds 2-norm filtering best possible regularized solutions at semi-convergence. Particularly, we consider some important and untouched issues on best, near best and general rank $k$ approximations to $A$ for the ill-posed problems with the singular values $σ_k=\mathcal{O}(k^{-α})$ with $α>0$, and the relationships between them and their nonzero singular values. Numerical experiments confirm our theory. The results on general rank $k$ approximations and the properties of their nonzero singular values apply to several Krylov solvers, including LSQR, CGME, MINRES, MR-II, GMRES and RRGMRES.
In the present work, we propose new tensor Krylov subspace method for ill posed linear tensor problems such as in color or video image restoration. Those methods are based on the tensor-tensor discrete cosine transform that gives fast tensor-tensor product computations. In particular, we will focus on the tensor discrete cosine versions of GMRES, Golub-Kahan bidiagonalisation and LSQR methods. The presented numerical tests show that the methods are very fast and give good accuracies when solving some linear tensor ill-posed problems.
Regularization Properties of the Krylov Iterative Solvers CGME and LSMR For Linear Discrete Ill-Posed Problems with an Application to Truncated Randomized SVDs
Prior recent work, devoted to the study of polynomial Krylov techniques for the approximation of the action of the matrix exponential ${\rm e}^{tA}v$, is extended to the case of associated $φ$-functions (which occur within the class of exponential integrators). In particular, a~posteriori error bounds and estimates, based on the notion of the defect (residual) of the Krylov approximation are considered. Computable error bounds and estimates are discussed and analyzed. This includes a new error bound which favorably compares to existing error bounds in specific cases. The accuracy of various er…
Block algorithms with augmented Rayleigh-Ritz projections for large-scale eigenpair computation
Beijing International Center for Mathematical Research, Peking University（北京大学数学科学中心）；Department of Computational and Applied Mathematics, Rice University（莱斯大学计算与应用数学系）
The use of block Krylov subspace methods for computing the solution to a sequence of shifted linear systems using subspace recycling was first proposed in [Soodhalter, SISC 2016], where a recycled shifted block GMRES algorithm (rsbGMRES) was proposed. Such methods use the equivalence of the shifted system to a Sylvester equation and exploit the shift invariance of the block Krylov subspace generated from the Sylvester operator. This avoids the need for initial residuals to span the same subspace and allows for a viable restarted Krylov subspace method with recycling for solving sequences of sh…
Krylov subspace methods are considered a standard tool to solve large systems of linear algebraic equations in many scientific disciplines such as image restoration or solving partial differential equations in mechanics of continuum. In the context of computer tomography however, the mostly used algebraic reconstruction techniques are based on classical iterative schemes. In this work we present software package that implements fully 3D cone beam projection operator and uses Krylov subspace methods, namely CGLS and LSQR to solve related tomographic reconstruction problems. It also implements b…
