Dedicated to Claude Brezinski and Sebastiano Seatzu on the Occasion of Their 70th Birthdays.Abstract. Tikhonov regularization replaces a linear discrete ill-posed problem by a penalized least-squares problem, whose solution is less sensitive to errors in the data and round-off errors introduced during the solution process. The penalty term is defined by a regularization matrix and a regularization parameter. The latter generally has to be determined during the solution process. This requires repeated solution of the penalized least-squares problem. It is therefore attractive to transform the least-squares problem to simpler form before solution. The present paper describes a transformation of the penalized least-squares problem to simpler form that is faster to compute than available transformations in the situation when the regularization matrix has linearly dependent columns and no exploitable structure. Properties of this kind of regularization matrices are discussed and their performance is illustrated.

Abstract. The generalized singular value decomposition (GSVD) often is used to solve Tikhonov regularization problems with a regularization matrix without exploitable structure. This paper describes how the standard methods for the computation of the GSVD of a matrix pair can be simplified in the context of Tikhonov regularization. Also, other regularization methods, including truncated GSVD, are considered. We compare the computational efforts required by the simplified GSVD method and the A-weighted generalized inverse introduced by Eldén.

Abstract. The generalized singular value decomposition (GSVD) of a pair of matrices expresses each matrix as a product of an orthogonal, a diagonal, and a nonsingular matrix. The nonsingular matrix, which we denote by X T , is the same in both products. Available software for computing the GSVD scales the diagonal matrices and X T so that the squares of corresponding diagonal entries sum to one. This paper proposes a scaling that seeks to minimize the condition number of X T . The rescaled GSVD gives rise to new truncated GSVD methods, one of which is well suited for the solution of linear discrete ill-posed problems. Numerical examples show this new truncated GSVD method to be competitive with the standard truncated GSVD method as well as with Tikhonov regularization with regard to the quality of the computed approximate solution.

Summary Many applications in science and engineering require the solution of large linear discrete ill‐posed problems that are obtained by the discretization of a Fredholm integral equation of the first kind in several space dimensions. The matrix that defines these problems is very ill conditioned and generally numerically singular, and the right‐hand side, which represents measured data, is typically contaminated by measurement error. Straightforward solution of these problems is generally not meaningful due to severe error propagation. Tikhonov regularization seeks to alleviate this difficulty by replacing the given linear discrete ill‐posed problem by a penalized least‐squares problem, whose solution is less sensitive to the error in the right‐hand side and to roundoff errors introduced during the computations. This paper discusses the construction of penalty terms that are determined by solving a matrix nearness problem. These penalty terms allow partial transformation to standard form of Tikhonov regularization problems that stem from the discretization of integral equations on a cube in several space dimensions.

Dedicated to Ken Hayami on the occasion of his 60th birthday.Abstract. Circulant preconditioners are commonly used to accelerate the rate of convergence of iterative methods when solving linear systems of equations with a Toeplitz matrix. Block extensions that can be applied when the system has a block Toeplitz matrix with Toeplitz blocks also have been developed. This paper is concerned with preconditioning of linear systems of equations with a symmetric block Toeplitz matrix with symmetric Toeplitz blocks that stem from the discretization of a linear ill-posed problem. The right-hand side of the linear systems represents available data and is assumed to be contaminated by error. These kinds of linear systems arise, e.g., in image deblurring problems. It is important that the preconditioner does not affect the invariant subspace associated with the smallest eigenvalues of the block Toeplitz matrix to avoid severe propagation of the error in the right-hand side. A perturbation result indicates how the dimension of the subspace associated with the smallest eigenvalues should be chosen and allows the determination of a suitable preconditioner when an estimate of the error in the right-hand side is available. This estimate also is used to decide how many iterations to carry out by a minimum residual iterative method. Applications to image restoration are presented.