Multilevel preconditioning for sparse optimization of functionals with nonconvex fidelity terms
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Stephan Dahlke
,
Massimo Fornasier
Abstract
This paper is concerned with the development of numerical schemes for the minimization of functionals involving sparsity constraints and nonconvex fidelity terms. These functionals appear in a natural way in the context of Tikhonov regularization of nonlinear inverse problems with ℓ1 penalty terms. Our method of minimization is based on a generalized conditional gradient scheme. It is well known that these algorithms might converge quite slowly in practice. Therefore, we propose an acceleration which is based on a decreasing thresholding strategy. Its efficiency relies on certain spectral properties of the problem at hand. We show that under certain boundedness and contraction conditions the resulting algorithm is linearly convergent to a global minimizer and that the iteration is monotone with respect to the Tikhonov functional. We study important classes of operator equations to which our analysis can be applied. Moreover, we introduce a certain multilevel preconditioning strategy which in practice promotes the aforementioned spectral properties for problems where the nonlinearity is a perturbation of a linear operator.
Funding source: Deutsche Forschungsgemeinschaft (DFG)
Award Identifier / Grant number: DA 360/12-2
Funding source: LOEWE Center for Synthetic Microbiology (Synmikro), Marburg
© 2015 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- The interior transmission eigenvalue problem for a spherically-symmetric domain with anisotropic medium and a cavity
- Ill-conditioning versus ill-posedness for the boundary controllability of the heat equation
- Parameters identification in the mathematical model of immune competition cells
- Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations
- Model parameter estimation of linear time-invariant systems from combined data of forced and initial condition responses
- An inverse problem for the Vlasov–Poisson system
- An inexact Newton regularization in Banach spaces based on the nonstationary iterated Tikhonov method
- Multilevel preconditioning for sparse optimization of functionals with nonconvex fidelity terms
- The first solution of a long standing problem: Reconstruction formula for a 3-d phaseless inverse scattering problem for the Schrödinger equation
Artikel in diesem Heft
- Frontmatter
- The interior transmission eigenvalue problem for a spherically-symmetric domain with anisotropic medium and a cavity
- Ill-conditioning versus ill-posedness for the boundary controllability of the heat equation
- Parameters identification in the mathematical model of immune competition cells
- Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations
- Model parameter estimation of linear time-invariant systems from combined data of forced and initial condition responses
- An inverse problem for the Vlasov–Poisson system
- An inexact Newton regularization in Banach spaces based on the nonstationary iterated Tikhonov method
- Multilevel preconditioning for sparse optimization of functionals with nonconvex fidelity terms
- The first solution of a long standing problem: Reconstruction formula for a 3-d phaseless inverse scattering problem for the Schrödinger equation
