Regularization matrices for discrete ill-posed problems in several space-dimensions
For researchers solving ill-posed problems in multiple dimensions, this provides a method to construct regularization matrices that facilitate standard form transformations, though the improvement is incremental.
The paper addresses the solution of large linear discrete ill-posed problems from Fredholm integral equations in several space-dimensions. It constructs penalty terms via a matrix-nearness problem to enable partial transformation to standard form in Tikhonov regularization, improving solution stability.
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, typically is contaminated by measurement error. Straightforward solution of these problems generally is 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 round-off 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.