Regularization of ill-posed problems in Banach spaces: convergence rates

By Elena Resmerita. 2005

173 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a8fd4a35fa95cdf5e6e3cf6455f2f8471d205d73

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Citations in the corpus

Iterative methods for nonlinear ill-posed problems in Banach spaces: convergence and applications to parameter identification problems (2009, background): “A general treatise of quantitative aspects of regularizations for ill-posed problems in Banach spaces is [20].”
An analysis of electrical impedance tomography with applications to Tikhonov regularization (2012, methodology): “We study the linearized and fully nonlinear models separately, by capitalizing on recent progress on nonsmooth regularization [5, 8, 24, 28, 34, 35, 46, 50].”
On Solving the Ill-Conditioned System Ax=b: General-Purpose Conditioners Obtained From the Boundary-Collocation Solution of the Laplace Equation, Using Trefftz Expansions With Multiple Length Scales (2009, methodology): “…customary to use a “regularization" method to solve this sort of ill-posed problem [Kunisch and Zou (1998); Wang and Xiao (2001); Xie and Zou (2002); Resmerita (2005)], where a suitable regularization parameter is used to suppress the bias in the computed solution, by seeking a better balance of…”
Linear Convergence of Iterative Soft-Thresholding (2007, unknown): “A common approach is to regularize by minimizing a Tikhonov functional [7, 15, 28].”
An iterative regularization method for the solution of the split feasibility problem in Banach spaces (2008, background): “For the projections onto affine subspaces it would be preferable to have something like convergence rates, possibly in dependence of source conditions (see also [35, 44]).”
Nonlinear iterative methods for linear ill-posed problems in Banach spaces (2006, unknown): “[1, 2, 6, 4, 17 , 5, 15]: � p(x, y) := 1 p � y� p − 1 p � x� p − inf{� ξ, y − x� : ξ ∈ Jp(x)} ,x , y∈ X. (2.11)”
An extension of the variational inequality approach for obtaining convergence rates in regularization of nonlinear ill-posed problems (2010, background): “MSC2000 subject classification: 47J06, 65J20, 47A52, 49N45 Keywords: Ill-posed problems, Tikhonov regularization, convergence rates, variational inequalities, source conditions, distance function, structure of nonlinearity, generalized Young inequality.”
Modern regularization methods for inverse problems (2018, background): “[319, 191]), or large noise that is not necessarily in V (cf.”
Novel Algorithms Based on the Conjugate Gradient Method for Inverting Ill-Conditioned Matrices, and a New Regularization Method to Solve Ill-Posed Linear Systems (2010, methodology): “E-mail: liucs@ntu.edu.tw 2 Center for Aerospace Research & Education, University of California, Irvine”
On the interplay of source conditions and variational inequalities for nonlinear ill-posed problems (2010, result): “For recent results we refer to the monograph [29] and in an exemplary manner to the papers [3, 4, 10, 12, 13, 14, 19, 21, 22, 26, 27, 28, 30, 31, 33] as well as to the thesis [24].”
Regularization with non-convex separable constraints (2009, methodology): “Starting from total variation regularization [37] and maximum entropy regularization [16, 18] the focus has moved to general Tikhonov functionals with convex constraints [1,9,23,25,29,32,33] with special cases like Besov-space constraints [28] and sparsity constraints [10, 14, 21, 26, 27, 30].”
REGULARIZED D-BAR METHOD FOR THE INVERSE CONDUCTIVITY PROBLEM (2009, background): “Iterative regularization of both linear and nonlinear inverse problems and convergence rates are discussed in Hilbert space setting in [9, 33, 35, 65, 66] and in Banach space setting in [34, 44, 45, 48, 76, 77, 78].”
Convergence rates and source conditions for Tikhonov regularization with sparsity constraints (2008, result): “In this case previous results give convergence rates in the Bregman distance only [7, 15, 19, 20].”
Necessary and sufficient conditions for linear convergence of ℓ1‐regularization (2011, result): “In addition, we compare these results with general convergence rates results with respect to the Bregman distance (see [6, 23, 24, 25, 26, 28, 29, 31, 32, 34]).”