A Trust-Region Approach to the Regularization of Large-Scale Discrete Forms of Ill-Posed Problems

Abstract

We consider large-scale least squares problems where the coefficient matrix comes from the discretization of an operator in an ill-posed problem, and the right-hand side contains noise. Special techniques known as regularization methods are needed to treat these problems in order to control the effect of the noise on the solution. We pose the regularization problem as a quadratically constrained least squares problem. This formulation is equivalent to Tikhonov regularization, and we note that it is also a special case of the trust-region subproblem from optimization. We analyze the trust-region subproblem in the regularization case, and we consider the nontrivial extensions of a recently developed method for general large-scale subproblems that will allow us to handle this case. The method relies on matrix-vector products only, has low and fixed storage requirements, and can handle the singularities arising in ill-posed problems. We present numerical results on test problems, on an

Fractional Tikhonov regularization for linear discrete ill-posed problems

“…This type of problems is discussed in [3,8,10]. The following result sheds light on how Ax μ,α − b depends on α for solutions that satisfy (3.11).…”

Trust-region and other regularisations of linear least-squares problems

“…There is a long history of work on this topic [6,11,13,38,37,40,41] which we will review as we proceed.…”

“…It is straightforward to derive [11,40] usable optimality conditions for (1.2). Specifically, let λ ≥ 0 and define x(λ) so that…”

“…We resort in this case to iterative methods. We note that although we describe an approach using LSQR, there is at least one alternative based on a parametric eigenvalue formulation [40,41].…”

Global convergence of SSM for minimizing a quadratic over a sphere

“…Other approaches to (1) include the scheme of Golub and von Matt [4] based on a partial tridiagonalization of A using the Lanczos process, and the related implementation of Gould et al [6], as well as the parametric eigenvalue approaches of Sorensen [19] (further developed by Rojas in [17]), and the related scheme of Rendl and Wolkowicz [16]. Numerical comparisons between these approaches are given in [8].…”