[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85257-en":3,"doc-seo-85257-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85257,1374391974564,"Clementine","https://ap-avatar.wpscdn.com/avatar/14000253aa45c000a9e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779874745381141002",8,"Research & Report","Multilevel Preconditioning Strategies for Convex Optimization Methods in Image Deblurring","Multilevel preconditioning and proximal optimization are combined to accelerate convex optimization methods used in image deblurring. Proximal gradient schemes for regularized variational problems are enhanced through variable-metric and extrapolation ideas, while a multilevel framework is used to speed inertial and inexact forward–backward iterations. Numerical experiments on image deblurring validate that the resulting robust and consistent acceleration improves convergence speed versus standard approaches, for both standard and inexact forward-backward formulations.","arXiv :2607 . 10864v1 [math .NA] 12 Jul 2026  \nMultilevel Preconditioning Strategies for Convex Optimization Methods in Image Deblurring  \nStefano Aleotti 1 , Claudia Binda 1 , Marco Donatelli 1 , Rolf Krause2,3  \n1 Department of Science and High Technology, University of Insubria, Via Valleggio 11, Como, 22100, Italy.  \n2 AMCS, CEMSE, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Kingdom of Saudi Arabia.  \n3 Euler Institute and Faculty of Informatics, Universit`a della Svizzera italiana, Via Giuseppe Buffi 10, Lugano, 6900, Switzerland.  \nContributing authors: [stefano.aleotti@uninsubria.it](stefano.aleotti@uninsubria.it) ;  \n[cbinda2@uninsubria.it](cbinda2@uninsubria.it) ; [marco.donatelli@uninsubria.it](marco.donatelli@uninsubria.it) ;  \n[rolf.krause@kaust.edu.sa](rolf.krause@kaust.edu.sa) ;  \nAbstract  \nProximal gradient methods are widely used in imaging, and their speed of convergence can be accelerated by incorporating variable metrics and/or extrapolation steps. Recent works have shown that preconditioning strategies can significantly enhance this acceleration, in particular, for image deblurring problems. In parallel, a multilevel framework has been introduced to speed up inertial and inexact forward–backward schemes for image restoration problems.  \nIn this paper, we combine preconditioning and multilevel strategies to design a robust and consistent acceleration framework for both standard and inexact forward-backward schemes applied to regularized convex optimization problems. Numerical experiments in image deblurring confirm that our approach yields a substantial improvement in convergence speed compared to standard methods.  \nKeywords: Ill-posed problems, Multilevel convex optimization, Image deblurring, Preconditioning, Proximal gradient methods  \n1  \n1 Introduction  \nIn this manuscript, we focus on image deblurring problems described by the model equation  \nAu = bδ , (1)  \nwhere A ∈ Rs ×d represents the discretization of a space-invariant convolution operator, bδ ∈ Rs denotes the observed image corrupted by white Gaussian noise ηδ , and u ∈ Rdis the unknown two-dimensional image with d pixels. We assume that bδ satisfies  \nbδ = b + ηδ , kbδ − bk ≤ δ, (2)  \nwhere b denotes the unobserved noise-free data, k·k is the Euclidean norm, and δ > 0 is an upper bound on the noise level.  \nThe ill-posedness of the operator A, together with the presence of noise, requires the use of regularization in order to solve (1); see [1–3] . Regularization based on a standard variational approach consists of solving the optimization problem  \nargmin f(u) + g(Wu), (3)  \nu∈Rd  \nwhere f : Rd → R is convex and smooth, g : Rd′ → R ∪ {∞} is convex and possibly non-smooth, and W ∈ Rd′ ×d is a linear operator. Here, f denotes the data-fidelity term, depending on A and measuring the discrepancy between the observed data and the model, while the possibly non-differentiable term g◦W = g (W·) serves as a penalty that incorporates prior information about the reconstructed image. Traditionally, this penalty term is weighted by an explicit regularization parameter λ > 0 to balance the two terms. In this paper, in agreement with the white Gaussian noise assumption on the observed data, we consider the particular instance of the model problem (3) where  \nf (u) = 12 kAu − bδ k2 . (4)  \nA particularly effective way to compute approximate solutions of (3) is to employ proximal gradient methods [4–6], which are first-order iterative algorithms. These methods typically achieve mild-to-moderate accuracy at a relatively low cost per iteration, especially when the dimension d is large. Their basic iteration consists of alternating a gradient step on the differentiable part f with a proximal step on thenon-smooth term g ◦ W, namely  \nuk+1 = proxαg ◦W (uk − α∇f(uk )) , (5)  \nwhere α > 0 is the step-length parameter along the descent direction −∇f(uk ), and proxαg ◦W denotes the proximal operator associated with the non-smooth term αg◦W. Converg","cbCaiejmeWwraOCA","https://ap.wps.com/l/cbCaiejmeWwraOCA","pdf",2465373,1,39,"English","en",105,"# Introduction\n## Image deblurring model and regularized formulation\n## Proximal gradient methods and convergence assumptions\n## Drawbacks and acceleration directions\n## Variable metrics, extrapolation, and preconditioning","[{\"question\":\"What problem does the paper address in image deblurring?\",\"answer\":\"The paper focuses on deblurring formulated as a regularized convex optimization problem with a linear convolution operator and noisy observations modeled with white Gaussian noise bounds.\"},{\"question\":\"How are proximal gradient methods used in the proposed framework?\",\"answer\":\"They alternate gradient steps on the smooth data-fidelity term with proximal steps for the (possibly non-smooth) regularization term within standard and inexact forward–backward schemes.\"},{\"question\":\"What acceleration techniques are combined in this work?\",\"answer\":\"The approach integrates preconditioning strategies with a multilevel framework to accelerate both standard and inexact forward-backward iterations, improving convergence speed in numerical experiments.\"}]",1784202118,98,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"multilevel-preconditioning-strategies-for-convex-optimization-methods-in-image-deblurring","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/multilevel-preconditioning-strategies-for-convex-optimization-methods-in-image-deblurring/85257/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address in image deblurring?","Question",{"text":75,"@type":76},"The paper focuses on deblurring formulated as a regularized convex optimization problem with a linear convolution operator and noisy observations modeled with white Gaussian noise bounds.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are proximal gradient methods used in the proposed framework?",{"text":80,"@type":76},"They alternate gradient steps on the smooth data-fidelity term with proximal steps for the (possibly non-smooth) regularization term within standard and inexact forward–backward schemes.",{"name":82,"@type":73,"acceptedAnswer":83},"What acceleration techniques are combined in this work?",{"text":84,"@type":76},"The approach integrates preconditioning strategies with a multilevel framework to accelerate both standard and inexact forward-backward iterations, improving convergence speed in numerical 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