arXiv subjects

J. C. De los Reyes

Publications and source records attributed to J. C. De los Reyes.

5 recordsLinked to original sources

arXiv2021

Bilevel parameter learning for nonlocal image denoising models

We propose a bilevel optimization approach for the estimation of parameters in nonlocal image denoising models. The parameters we consider are both the fidelity weight and weights within the kernel of the nonlocal operator. In both cases we investigate the differentiability of the solution operator in function spaces and derive a first order optimality system that characterizes local minima. For the numerical solution of the problems, we use a second-order trust-region algorithm in combination with a finite element discretization of the nonlocal denoising models and we introduce a computational strategy for the solution of the resulting dense linear systems. Several experiments illustrate the applicability and effectiveness of our approach.

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arXiv2018

Total Generalized Variation Regularization in Variational Data Assimilation for Burgers' Equation

We propose a second-order total generalized variation (TGV) regularization for the reconstruction of the initial condition in variational data assimilation problems. After showing the equivalence between TGV regularization and the Bayesian method for the MAP estimator, we focus on the detailed study of the inviscid Burgers' data assimilation problem. Due to the difficult structure of the governing hyperbolic conservation law, we consider a discretize-then-optimize approach and derive first-order optimality conditions for the problem. For the numerical solution, we propose a globalized reduced Newton-type method and prove convergence of the algorithm to stationary points. The paper finishes with some numerical experiments where among others, the performance of TGV-regularization compared to the TV-regularization is tested.

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arXiv2016

Learning optimal spatially-dependent regularization parameters in total variation image restoration

We consider a bilevel optimization approach in function space for the choice of spatially dependent regularization parameters in TV image restoration models. First- and second-order optimality conditions for the bilevel problem are studied, when the spatially-dependent parameter belongs to the Sobolev space $H^1(Ω)$. A combined Schwarz domain decomposition-semismooth Newton method is proposed for the solution of the full optimality system and local superlinear convergence of the semismooth Newton method is analyzed. Exhaustive numerical computations are finally carried out to show the suitability of the approach.

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arXiv2016

Second-order orthant-based methods with enriched Hessian information for sparse $\ell_1$-optimization

We present a second order algorithm, based on orthantwise directions, for solving optimization problems involving the sparsity enhancing $\ell_1$-norm. The main idea of our method consists in modifying the descent orthantwise directions by using second order information both of the regular term and (in weak sense) of the $\ell_1$-norm. The weak second order information behind the $\ell_1$-term is incorporated via a partial Huber regularization. One of the main features of our algorithm consists in a faster identification of the active set. We also prove that a reduced version of our method is equivalent to a semismooth Newton algorithm applied to the optimality condition, under a specific choice of the algorithm parameters. We present several computational experiments to show the efficiency of our approach compared to other state-of-the-art algorithms.

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arXiv2015

Bilevel parameter learning for higher-order total variation regularisation models

We consider a bilevel optimisation approach for parameter learning in higher-order total variation image reconstruction models. Apart from the least squares cost functional, naturally used in bilevel learning, we propose and analyse an alternative cost, based on a Huber regularised TV-seminorm. Differentiability properties of the solution operator are verified and a first-order optimality system is derived. Based on the adjoint information, a quasi-Newton algorithm is proposed for the numerical solution of the bilevel problems. Numerical experiments are carried out to show the suitability of our approach and the improved performance of the new cost functional. Thanks to the bilevel optimisation framework, also a detailed comparison between TGV$^2$ and ICTV is carried out, showing the advantages and shortcomings of both regularisers, depending on the structure of the processed images and their noise level.

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