Recent results on the quasi-optimality principle

DOI10.1515/JIIP.2009.002zbMath1159.65051MaRDI QIDQ3612227

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Publication date: 3 March 2009

Published in: Journal of Inverse and Ill-posed Problems (Search for Journal in Brave)

zbMATH Keywords

convergence ratesill-posed problemlinear operator equationregularization methodsoracle inequalityquasi-optimality criterion

Mathematics Subject Classification ID

Numerical solutions to equations with linear operators (65J10) Numerical solutions of ill-posed problems in abstract spaces; regularization (65J20) Linear operators and ill-posed problems, regularization (47A52)

Related Items (11)

A stopping criterion for iterative regularization methods ⋮ Heuristic Parameter Choice in Tikhonov Method from Minimizers of the Quasi-Optimality Function ⋮ Adaptivity and Oracle Inequalities in Linear Statistical Inverse Problems: A (Numerical) Survey ⋮ A mollifier approach to regularize a Cauchy problem for the inhomogeneous Helmholtz equation ⋮ Combining approximate solutions for linear discrete ill-posed problems ⋮ Old and new parameter choice rules for discrete ill-posed problems ⋮ Comparison of parameter choices in regularization algorithms in case of different information about noise level ⋮ Comparing parameter choice methods for regularization of ill-posed problems ⋮ On Minimization Strategies for Choice of the Regularization Parameter in Ill-Posed Problems ⋮ Parameter Choices for Fast Harmonic Spline Approximation ⋮ A new heuristic parameter choice rule in Tikhonov regularization applied for Ritz approximation of an ill-posed problem

Cites Work

• What do we learn from the discrepancy principle?
• A fast ``Monte-Carlo cross-validation procedure for large least squares problems with noisy data
• Maximal Functions on Classical Lorentz Spaces and Hardy's Inequality with Weights for Nonincreasing Functions
• Robust generalized cross-validation for choosing the regularization parameter
• Some considerations concerning regularization and parameter choice algorithms
• Analysis of Profile Functions for General Linear Regularization Methods
• The convergence of a new heuristic parameter selection criterion for general regularization methods
• Remarks on choosing a regularization parameter using the quasi-optimality and ratio criterion
• Justification of the choice of regularization parameter according to quasi-optimality and quotient criteria
• Analysis of Discrete Ill-Posed Problems by Means of the L-Curve
• An iterative method for solving incorrectly posed problems
• Practical Approximate Solutions to Linear Operator Equations When the Data are Noisy
• The Weighted Hardy's Inequality for Nonincreasing Functions
• On Converse and Saturation Results for Tikhonov Regularization of Linear Ill-Posed Problems
• Geometry of linear ill-posed problems in variable Hilbert scales
• Convergence rates in the Prokhorov metric for assessing uncertainty in ill-posed problems
• Use of the regularization method in non-linear problems
• An Iteration Formula for Fredholm Integral Equations of the First Kind

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