We review a Tikhonov parameter criterion based on the search for local minima of the function µ (λ) = x(λ)y µ (λ), µ > 0 where x(λ) and y(λ) are the squared residual norm and the squared solution norm, respectively, proposed earlier by Regińska (1996, SIAM J. Sci. Comput. 3 740). As a consequence, we demonstrate that extreme points of µ (λ) are fixed points of a related function, and then propose a fixed-point algorithm for choosing the Tikhonov parameter. The algorithm constructs a regularization parameter associated with the corner of the L-curve in log-log scale, thus yielding solutions with accuracy comparable to that of the L-curve method but at a lower computational cost. The performance of the algorithm on representative discrete ill-posed problems is evaluated and compared with results obtained by the L-curve method, generalized cross-validation and another fixed-point algorithm from the literature.

Let W N = W N (z 1 , z 2 ,. .. , zn) be a rectangular Vandermonde matrix of order n × N, N ≥ n, with distinct nodes z j in the unit disk and z k−1 j as its (j, k) entry. Matrices of this type often arise in frequency estimation and system identification problems. In this paper, the conditioning of W N is analyzed and bounds for the spectral condition number κ 2 (W N) are derived. The bounds depend on n, N , and the separation of the nodes. By analyzing the behavior of the bounds as functions of N , we conclude that these matrices may become well conditioned, provided the nodes are close to the unit circle but not extremely close to each other and provided the number of columns of W N is large enough. The asymptotic behavior of both the conditioning itself and the bounds is analyzed and the theoretical results arising from this analysis verified by numerical examples.

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We describe an algorithm for large-scale discrete ill-posed problems, called GKB-FP, which combines the Golub-Kahan bidiagonalization algorithm with Tikhonov regularization in the generated Krylov subspace, with the regularization parameter for the projected problem being chosen by the fixed-point method by Bazán (Inverse Probl. 24(3), 2008). The fixed-point method selects as regularization parameter a fixed-point of the function r λ 2 / f λ 2 , where f λ is the regularized solution and r λ is the corresponding residual. GKB-FP determines the sought fixed-point by computing a finite sequence of fixed-points of functions rapproximates f λ in a k-dimensional Krylov subspace and r (k) λ is the corresponding residual. Based on this and provided the sought fixed-point is reached, we prove that the regularized solutions f (k) λ remain unchanged and therefore completely insensitive to the number of iterations. This and the performance of the method when applied to well-known test problems are illustrated numerically.

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We re-analyze a Tikhonov parameter choice rule devised by Regińska (1996 SIAM J. Sci. Comput. 3 740-49) and algorithmically realized through a fast fixed-point (FP) method by Bazán (2008 Inverse Problems 24 035001). The method determines a Tikhonov parameter associated with a point near the L-corner of the maximum curvature and at which the L-curve is locally convex. In practice, it works well when the L-curve presents an L-shaped form with distinctive vertical and horizontal parts, but failures may occur when there are several local convex corners. We derive a simple and computable condition which describes the regions where the L-curve is concave/convex, while providing insight into the choice of the regularization parameter through the L-curve method or FP. Based on this, we introduce variants of the FP algorithm capable of handling the parameter choice problem even in the case where the L-curve has several local corners. The theory is illustrated both graphically and numerically, and the performance of the variants on a difficult ill-posed problem is evaluated by comparing the results with those provided by the L-curve method, generalized cross-validation and the discrepancy principle.

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