Projection Methods for Ill-Posed Problems Revisited
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Stefan Kindermann
Abstract
We consider the discretization of least-squares problems for linear ill-posed operator equations in Hilbert spaces. The main subject of this article concerns conditions for convergence of the associated discretized minimum-norm least-squares solution to the exact solution using exact attainable data. The two cases of global convergence (convergence for all exact solutions) or local convergence (convergence for a specific exact solution) are investigated. We review the existing results and prove new equivalent conditions when the discretized solution always converges to the exact solution. An important tool is to recognize the discrete solution operator as an oblique projection. Hence, global convergence can be characterized by certain subspaces having uniformly bounded angles. We furthermore derive practically useful conditions when this holds and put them into the context of known results. For local convergence, we generalize results on the characterization of weak or strong convergence and state some new sufficient conditions. We furthermore provide an example of a bounded sequence of discretized solutions which does not converge at all, not even weakly.
The author would like to thank Andreas Neubauer for useful discussions and for providing the counterexample in Theorem 4.1.
© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Vertex-Based Compatible Discrete Operator Schemes on Polyhedral Meshes for Advection-Diffusion Equations
- Reliable Averaging for the Primal Variable in the Courant FEM and Hierarchical Error Estimators on Red-Refined Meshes
- Monotone Finite Difference Schemes for Quasilinear Parabolic Problems with Mixed Boundary Conditions
- The Laguerre Collocation Method for Third Kind Integral Equations on Unbounded Domains
- Projection Methods for Ill-Posed Problems Revisited
- Explicit Constants in Poincaré-Type Inequalities for Simplicial Domains and Application to A Posteriori Estimates
- Regularization by Aggregation of Global and Local Data on the Sphere
- A Note on Lower Bounds of Eigenvalues of Fourth Order Elliptic Operators on Local Quasi-Uniform Grids
- Numerical Analysis and Computation of a Type of IMEX Method for the Time-Dependent Natural Convection Problem
- The Valuation of Carbon Bonds Linked with Carbon Price
Articles in the same Issue
- Frontmatter
- Vertex-Based Compatible Discrete Operator Schemes on Polyhedral Meshes for Advection-Diffusion Equations
- Reliable Averaging for the Primal Variable in the Courant FEM and Hierarchical Error Estimators on Red-Refined Meshes
- Monotone Finite Difference Schemes for Quasilinear Parabolic Problems with Mixed Boundary Conditions
- The Laguerre Collocation Method for Third Kind Integral Equations on Unbounded Domains
- Projection Methods for Ill-Posed Problems Revisited
- Explicit Constants in Poincaré-Type Inequalities for Simplicial Domains and Application to A Posteriori Estimates
- Regularization by Aggregation of Global and Local Data on the Sphere
- A Note on Lower Bounds of Eigenvalues of Fourth Order Elliptic Operators on Local Quasi-Uniform Grids
- Numerical Analysis and Computation of a Type of IMEX Method for the Time-Dependent Natural Convection Problem
- The Valuation of Carbon Bonds Linked with Carbon Price
