23 research outputs found
Elastic-Net Regularization: Error estimates and Active Set Methods
This paper investigates theoretical properties and efficient numerical
algorithms for the so-called elastic-net regularization originating from
statistics, which enforces simultaneously l^1 and l^2 regularization. The
stability of the minimizer and its consistency are studied, and convergence
rates for both a priori and a posteriori parameter choice rules are
established. Two iterative numerical algorithms of active set type are
proposed, and their convergence properties are discussed. Numerical results are
presented to illustrate the features of the functional and algorithms
Greedy Solution of Ill-Posed Problems: Error Bounds and Exact Inversion
The orthogonal matching pursuit (OMP) is an algorithm to solve sparse
approximation problems. Sufficient conditions for exact recovery are known with
and without noise. In this paper we investigate the applicability of the OMP
for the solution of ill-posed inverse problems in general and in particular for
two deconvolution examples from mass spectrometry and digital holography
respectively.
In sparse approximation problems one often has to deal with the problem of
redundancy of a dictionary, i.e. the atoms are not linearly independent.
However, one expects them to be approximatively orthogonal and this is
quantified by the so-called incoherence. This idea cannot be transfered to
ill-posed inverse problems since here the atoms are typically far from
orthogonal: The ill-posedness of the operator causes that the correlation of
two distinct atoms probably gets huge, i.e. that two atoms can look much alike.
Therefore one needs conditions which take the structure of the problem into
account and work without the concept of coherence. In this paper we develop
results for exact recovery of the support of noisy signals. In the two examples
in mass spectrometry and digital holography we show that our results lead to
practically relevant estimates such that one may check a priori if the
experimental setup guarantees exact deconvolution with OMP. Especially in the
example from digital holography our analysis may be regarded as a first step to
calculate the resolution power of droplet holography
Optimal Convergence Rates for Tikhonov Regularization in Besov Scales
In this paper we deal with linear inverse problems and convergence rates for
Tikhonov regularization. We consider regularization in a scale of Banach
spaces, namely the scale of Besov spaces. We show that regularization in Banach
scales differs from regularization in Hilbert scales in the sense that it is
possible that stronger source conditions may lead to weaker convergence rates
and vive versa. Moreover, we present optimal source conditions for
regularization in Besov scales
Morozov's discrepancy principle for Tikhonov-type functionals with nonlinear operators
3D Reconstruction for Partial Data Electrical Impedance Tomography Using a Sparsity Prior
In electrical impedance tomography the electrical conductivity inside a
physical body is computed from electro-static boundary measurements. The focus
of this paper is to extend recent result for the 2D problem to 3D. Prior
information about the sparsity and spatial distribution of the conductivity is
used to improve reconstructions for the partial data problem with Cauchy data
measured only on a subset of the boundary. A sparsity prior is enforced using
the norm in the penalty term of a Tikhonov functional, and spatial
prior information is incorporated by applying a spatially distributed
regularization parameter. The optimization problem is solved numerically using
a generalized conditional gradient method with soft thresholding. Numerical
examples show the effectiveness of the suggested method even for the partial
data problem with measurements affected by noise.Comment: 10 pages, 3 figures. arXiv admin note: substantial text overlap with
arXiv:1405.655
Sparse Regularization with Penalty Term
We consider the stable approximation of sparse solutions to non-linear
operator equations by means of Tikhonov regularization with a subquadratic
penalty term. Imposing certain assumptions, which for a linear operator are
equivalent to the standard range condition, we derive the usual convergence
rate of the regularized solutions in dependence of the noise
level . Particular emphasis lies on the case, where the true solution
is known to have a sparse representation in a given basis. In this case, if the
differential of the operator satisfies a certain injectivity condition, we can
show that the actual convergence rate improves up to .Comment: 15 page
Compressive sensing principles and iterative sparse recovery for inverse and ill-posed problems
Sparsity and Compressed Sensing in Inverse Problems
This chapter is concerned with two important topics in the context of sparse recovery in inverse and ill-posed problems. In first part we elaborate condi-tions for exact recovery. In particular, we describe how both `1-minimization and matching pursuit methods can be used to regularize ill-posed problems and more-over, state conditions which guarantee exact recovery of the support in the sparse case. The focus of the second part is on the incomplete data scenario. We discuss ex-tensions of compressed sensing for specific infinite dimensional ill-posed measure-ment regimes. We are able to establish recovery error estimates when adequately relating the isometry constant of the sensing operator, the ill-posedness of the un-derlying model operator and the regularization parameter. Finally, we very briefly sketch how projected steepest descent iterations can be applied to retrieve the sparse solution