Convergence rates and source conditions for Tikhonov regularization with sparsity constraints
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D. A. Lorenz
Abstract
This paper addresses the regularization by sparsity constraints by means of weighted ℓp penalties for 0 ≤ p ≤ 2. For 1 ≤ p ≤ 2 special attention is payed to convergence rates in norm and to source conditions. As main results it is proven that one gets a convergence rate of 
in the 2-norm for 1 < p ≤ 2 and in the 1-norm for p = 1 as soon as the unknown solution is sparse. The case p = 1 needs a special technique where not only Bregman distances but also a so-called Bregman-Taylor distance has to be employed.
For p < 1 only preliminary results are shown. These results indicate that, different from p ≥ 1, the regularizing properties depend on the interplay of the operator and the basis of sparsity. A counterexample for p = 0 shows that regularization need not to happen.
© de Gruyter 2008
Articles in the same Issue
- Simultaneous identification of independent parameters in elliptic equations — numerical studies
- Modulus of continuity of Nemytskii operators with application to the problem of option pricing
- Convergence rates and source conditions for Tikhonov regularization with sparsity constraints
- Metric and Bregman projections onto affine subspaces and their computation via sequential subspace optimization methods
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- Chemnitz Symposium on Inverse Problems Chemnitz, Germany, September 27–28, 2007
Articles in the same Issue
- Simultaneous identification of independent parameters in elliptic equations — numerical studies
- Modulus of continuity of Nemytskii operators with application to the problem of option pricing
- Convergence rates and source conditions for Tikhonov regularization with sparsity constraints
- Metric and Bregman projections onto affine subspaces and their computation via sequential subspace optimization methods
- Regularization of linear ill-posed problems with noisy right hand side and noisy operator
- Chemnitz Symposium on Inverse Problems Chemnitz, Germany, September 27–28, 2007
