Abstract. Tikhonov regularization for projected solutions of large-scale ill-posed problems is considered. The Golub-Kahan iterative bidiagonalization is used to project the problem onto a subspace and regularization then applied to find a subspace approximation to the full problem. Determination of the regularization parameter for the projected problem by unbiased predictive risk estimation, generalized cross validation and discrepancy principle techniques is investigated. It is shown that the regularized parameter obtained by the unbiased predictive risk estimator can provide a good estimate for that to be used for a full problem which is moderately to severely ill-posed. A similar analysis provides the weight parameter for the weighted generalized cross validation such that the approach is also useful in these cases, and also explains why the generalized cross validation without weighting is not always useful. All results are independent of whether systems are over or underdetermined. Numerical simulations for standard one dimensional test problems and two dimensional data, for both image restoration and tomographic image reconstruction, support the analysis and validate the techniques. The size of the projected problem is found using an extension of a noise revealing function for the projected problem Hnȇtynková, Plesinger, and Strakos, [BIT Numerical Mathematics 49 (2009), 4 pp. 669-696.]. Furthermore, an iteratively reweighted regularization approach for edge preserving regularization is extended for projected systems, providing stabilization of the solutions of the projected systems and reducing dependence on the determination of the size of the projected subspace.
Abstract.We discuss the solution of numerically ill-posed overdetermined systems of equations using Tikhonov a-priori-based regularization. When the noise distribution on the measured data is available to appropriately weight the fidelity term, and the regularization is assumed to be weighted by inverse covariance information on the model parameters, the underlying cost functional becomes a random variable that follows a χ 2 distribution. The regularization parameter can then be found so that the optimal cost functional has this property. Under this premise a scalar Newton root-finding algorithm for obtaining the regularization parameter is presented. The algorithm, which uses the singular value decomposition of the system matrix is found to be very efficient for parameter estimation, requiring on average about 10 Newton steps. Additionally, the theory and algorithm apply for Generalized Tikhonov regularization using the generalized singular value decomposition. The performance of the Newton algorithm is contrasted with standard techniques, including the L-curve, generalized cross validation and unbiased predictive risk estimation. This χ 2 -curve Newton method of parameter estimation is seen to be robust and cost effective in comparison to other methods, when white or colored noise information on the measured data is incorporated.
Tikhonov regularization, least squares, regularization parameterAMS classification scheme numbers: 15A09, 15A29, 65F22, 62F15, 62G08
Submitted to: Inverse Problems, 24 October 2008This is an author-produced, peer-reviewed version of this article. The final, definitive version of this document can be found online at Inverse Problems, published by Institute of Physics. Copyright restrictions may apply.
Sparse inversion of gravity data based on L 1 -norm regularization is discussed. An iteratively reweighted least squares algorithm is used to solve the problem. At each iteration the solution of a linear system of equations and the determination of a suitable regularization parameter are considered. The LSQR iteration is used to project the system of equations onto a smaller subspace that inherits the ill-conditioning of the full space problem. We show that the gravity kernel is only mildly to moderately ill-conditioned. Thus, while the dominant spectrum of the projected problem accurately approximates the dominant spectrum of the full space problem, the entire spectrum of the projected problem inherits the ill-conditioning of the full problem. Consequently, determining the regularization parameter based on the entire spectrum of the projected problem necessarily over compensates for the non-dominant portion of the spectrum and leads to inaccurate approximations for the full-space solution. In contrast, finding the regularization parameter using a truncated singular space of the projected operator is efficient and effective. Simulations for synthetic examples with noise demonstrate the approach using the method of unbiased predictive risk estimation for the truncated projected spectrum. The method is arXiv:1601.00114v3 [physics.geo-ph] 19 Jun 2017 2 S. Vatankhah, R. A. Renaut, V. E. Ardestani used on gravity data from the Mobrun ore body, northeast of Noranda, Quebec, Canada.The 3-D reconstructed model is in agreement with known drill-hole information.
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