Multi-parameter regularization and its numerical realization

Abstract

In this paper we propose and analyse a choice of parameters in the multi-parameter regularization of Tikhonov type. A modified discrepancy principle is presented within the multi-parameter regularization framework. An order optimal error bound is obtained under the standard smoothness assumptions. We also propose a numerical realization of the multi-parameter discrepancy principle based on the model function approximation. Numerical experiments on a series of test problems support theoretical results. Finally we show how the proposed approach can be successfully implemented in Laplacian Regularized Least Squares for learning from labeled and unlabeled examples.

A Golub–Kahan-Type Reduction Method for Matrix Pairs

“…Brezinski et al [3] consider this problem for small to moderately sized problems. More recent treatments are provided by Gazzola and Novati [9] and Lu and Pereverzyev [18]. With x = V y as before, and applying the decompositions (2.6), we get similarly as for (3.1) that (3.8) minimized over R(V ) is equivalent to the reduced minimization problem…”

“…Methods for determining suitable regularization parameters for this minimization problem are discussed in [2,3,9,18].…”

Multidirectional Subspace Expansion for One-Parameter and Multiparameter Tikhonov Regularization

“…In either case, operator splitting is a straightforward approach, but does not necessarily satisfy the discrepancy principle exactly. Lu and Pereverzyev [19] and later Fornasier et al [5] rewrite the constrained minimization problem as a differential equation and approximate…”

A modified quasi-boundary value method for an abstract ill-posed biparabolic problem

“…More precisely, we propose an improved modi ed quasi-boundary-value method with two parameters α > and r ≥ , where the parameter α is introduced to lter the high frequencies, and the second parameter r to include the regularity of the solution of the original problem. The advantage of the multi-parameter regularization is such that it gives more freedom in attaining order optimal accuracy [22][23][24][25][26][27][28][29].…”