Elliptic Mathematical Programs with Equilibrium Constraints in Function Space: Optimality Conditions and Numerical Realization

Hintermüller, Michael; Laurain, Antoine; Löbhard, Caroline; Rautenberg, Carlos N.; Surowiec, Thomas M.

doi:10.1007/978-3-319-05083-6_9

Cited by 5 publications

(2 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The resulting problems present several difficulties due to the frequent non-smoothness of the lower-level problem (think of TV regularization), which, in general, makes it impossible to verify Karush–Kuhn–Tucker constraint qualification conditions. This issue has led to the development of alternative analytical approaches in order to obtain first-order necessary optimality conditions (Bonnans and Tiba 1991, De los Reyes 2011, Hintermüller, Laurain, Löbhard, Rautenberg and Surowiec 2014). The bilevel problems under consideration are also related to generalized mathematical programs with equilibrium constraints in function spaces (Luo, Pang and Ralph 1996, Outrata 2000).…”

Section: Learning In Functional Analytic Regularizationmentioning

confidence: 99%

Solving inverse problems using data-driven models

et al. 2019

View full text Add to dashboard Cite

Recent research in inverse problems seeks to develop a mathematically coherent foundation for combining data-driven models, and in particular those based on deep learning, with domain-specific knowledge contained in physical–analytical models. The focus is on solving ill-posed inverse problems that are at the core of many challenging applications in the natural sciences, medicine and life sciences, as well as in engineering and industrial applications. This survey paper aims to give an account of some of the main contributions in data-driven inverse problems.

show abstract

Section: Learning In Functional Analytic Regularizationmentioning

confidence: 99%

Solving inverse problems using data-driven models

et al. 2019

View full text Add to dashboard Cite

show abstract

“…The resulting problems are difficult to treat due to the non-smooth structure of the lower level problem, which makes it impossible to verify standard constraint qualification conditions for Karush-Kuhn-Tucker (KKT) systems. Therefore, in order to obtain characterising first-order necessary optimality conditions, alternative analytical approaches have emerged, in particular regularisation techniques [4,20,28]. We consider such an approach here and study the related regularised problem in depth.…”

Section: Introductionmentioning

confidence: 98%

Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

2016

View full text Add to dashboard Cite

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 combined quasiNewton/semismooth 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.

show abstract

Convergent Regularization in Inverse Problems and Linear Plug-and-Play Denoisers

Hauptmann,

Mukherjee,

Schönlieb

et al. 2024

Found Comput Math

View full text Add to dashboard Cite

Regularization is necessary when solving inverse problems to ensure the well-posedness of the solution map. Additionally, it is desired that the chosen regularization strategy is convergent in the sense that the solution map converges to a solution of the noise-free operator equation. This provides an important guarantee that stable solutions can be computed for all noise levels and that solutions satisfy the operator equation in the limit of vanishing noise. In recent years, reconstructions in inverse problems are increasingly approached from a data-driven perspective. Despite empirical success, the majority of data-driven approaches do not provide a convergent regularization strategy. One such popular example is given by iterative plug-and-play (PnP) denoising using off-the-shelf image denoisers. These usually provide only convergence of the PnP iterates to a fixed point, under suitable regularity assumptions on the denoiser, rather than convergence of the method as a regularization technique, thatis under vanishing noise and regularization strength. This paper serves two purposes: first, we provide an overview of the classical regularization theory in inverse problems and survey a few notable recent data-driven methods that are provably convergent regularization schemes. We then continue to discuss PnP algorithms and their established convergence guarantees. Subsequently, we consider PnP algorithms with learned linear denoisers and propose a novel spectral filtering technique of the denoiser to control the strength of regularization. Further, by relating the implicit regularization of the denoiser to an explicit regularization functional, we are the first to rigorously show that PnP with a learned linear denoiser leads to a convergent regularization scheme. The theoretical analysis is corroborated by numerical experiments for the classical inverse problem of tomographic image reconstruction.

show abstract

Elliptic Mathematical Programs with Equilibrium Constraints in Function Space: Optimality Conditions and Numerical Realization

Cited by 5 publications

References 40 publications

Solving inverse problems using data-driven models

Solving inverse problems using data-driven models

Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

Convergent Regularization in Inverse Problems and Linear Plug-and-Play Denoisers

Contact Info

Product

Resources

About