Golub-Kahan iterative bidiagonalization represents the core algorithm in several regularization methods for solving large linear noise-polluted ill-posed problems. We consider a general noise setting and derive explicit relations between (noise contaminated) bidiagonalization vectors and the residuals of bidiagonalization-based regularization methods LSQR, LSMR, and CRAIG. For LSQR and LSMR residuals we prove that the coefficients of the linear combination of the computed bidiagonalization vectors reflect the amount of propagated noise in each of these vectors. For CRAIG the residual is only a multiple of a particular bidiagonalization vector. We show how its size indicates the regularization effect in each iteration by expressing the CRAIG solution as the exact solution to a modified compatible problem. Validity of the results for larger two-dimensional problems and influence of the loss of orthogonality is also discussed.

The paper focuses on numerical solution of parametrized diffusion equations with scalar parameterdependent coefficient function by the stochastic (spectral) Galerkin method. We study preconditioning of the related discretized problems using preconditioners obtained by modifying the stochastic part of the partial differential equation. We present a simple but general approach for obtaining two-sided bounds to the spectrum of the resulting matrices, based on a particular splitting of the discretized operator. Using this tool and considering the stochastic approximation space formed by classical orthogonal polynomials, we obtain new spectral bounds depending solely on the properties of the coefficient function and the type of the approximation polynomials for several classes of block-diagonal preconditioners. These bounds are guaranteed and applicable to various distributions of parameters. Moreover, the conditions on the parameter-dependent coefficient function are only local, and therefore less restrictive than those usually assumed in the literature.

It is well-established that any non-increasing convergence curve is possible for GM-RES and a family of pairs (A, b) can be constructed for which GMRES exhibits a given convergence curve with A having arbitrary spectrum. No analog of this result has been established for block GMRES, wherein multiple right-hand sides are considered. By reframing the problem as a single linear system over a ring of square matrices, we develop convergence results for block Arnoldi and block GMRES. In particular, we show what convergence behavior is admissible for block GMRES and how the matrices and right-hand sides producing any admissible behavior can be constructed. Moreover, we show that the convergence of the block Arnoldi method for eigenvalue approximation can be almost fully independent of the convergence of block GMRES for the same coefficient matrix and the same starting vectors.

This paper focuses on efficient computational approaches to compute approximate solutions of a linear inverse problem that is contaminated with mixed Poisson-Gaussian noise, and when there are additional outliers in the measured data. The Poisson-Gaussian noise leads to a weighted minimization problem, with solution-dependent weights. To address outliers, the standard least squares fit-to-data metric is replaced by the Talwar robust regression function. Convexity, regularization parameter selection schemes, and incorporation of non-negative constraints are investigated. A projected Newton algorithm is used to solve the resulting constrained optimization problem, and a preconditioner is proposed to accelerate conjugate gradient Hessian solves. Numerical experiments on problems from image deblurring illustrate the effectiveness of the methods.

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