{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:27:49Z","timestamp":1787326069021,"version":"build-2736575974"},"reference-count":105,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/501100000275","name":"Leverhulme Trust","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000275","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000288","name":"Royal Society","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000288","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/V029428\/1"],"award-info":[{"award-number":["EP\/V029428\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/S026045\/1"],"award-info":[{"award-number":["EP\/S026045\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/T003553\/1"],"award-info":[{"award-number":["EP\/T003553\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/N014588\/1"],"award-info":[{"award-number":["EP\/N014588\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/T017961\/1"],"award-info":[{"award-number":["EP\/T017961\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100004440","name":"Wellcome Trust","doi-asserted-by":"publisher","award":["215733\/Z\/19\/Z"],"award-info":[{"award-number":["215733\/Z\/19\/Z"]}],"id":[{"id":"10.13039\/100004440","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100004440","name":"Wellcome Trust","doi-asserted-by":"publisher","award":["221633\/Z\/20\/Z"],"award-info":[{"award-number":["221633\/Z\/20\/Z"]}],"id":[{"id":"10.13039\/100004440","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010665","name":"H2020 Marie Sk\u0142odowska-Curie Actions","doi-asserted-by":"publisher","award":["777826 NoMADS"],"award-info":[{"award-number":["777826 NoMADS"]}],"id":[{"id":"10.13039\/100010665","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Cantab Capital Institute for the Mathematics of Information"},{"DOI":"10.13039\/100012338","name":"Alan Turing Institute","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100012338","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12090024"],"award-info":[{"award-number":["12090024"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Sino-German center","award":["M-0187"],"award-info":[{"award-number":["M-0187"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Imaging Sci."],"published-print":{"date-parts":[[2024,6,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Bayesian methods for solving inverse problems are a powerful alternative to classical methods since the Bayesian approach offers the ability to quantify the uncertainty in the solution. In recent years, data-driven techniques for solving inverse problems have also been remarkably successful, due to their superior representation ability. In this work, we incorporate data-based models into a class of Langevin-based sampling algorithms for Bayesian inference in imaging inverse problems. In particular, we introduce NF-ULA (normalizing flow-based unadjusted Langevin algorithm), which involves learning a normalizing flow (NF) as the image prior. We use NF to learn the prior because a tractable closed-form expression for the log prior enables the differentiation of it using autograd libraries. Our algorithm only requires a normalizing flow-based generative network, which can be pretrained independently of the considered inverse problem and the forward operator. We perform theoretical analysis by investigating the well-posedness and nonasymptotic convergence of the resulting NF-ULA algorithm. The efficacy of the proposed NF-ULA algorithm is demonstrated in various image restoration problems such as image deblurring, image inpainting, and limited-angle X-ray computed tomography reconstruction. NF-ULA is found to perform better than competing methods for severely ill-posed inverse problems.<\/jats:p>","DOI":"10.1137\/23m1581807","type":"journal-article","created":{"date-parts":[[2024,4,8]],"date-time":"2024-04-08T05:59:16Z","timestamp":1712555956000},"page":"820-860","source":"Crossref","is-referenced-by-count":9,"title":["NF-ULA: Normalizing Flow-Based Unadjusted Langevin Algorithm for Imaging Inverse Problems"],"prefix":"10.1137","volume":"17","author":[{"given":"Ziruo","family":"Cai","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Shanghai Jiao Tong University, China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4996-6079","authenticated-orcid":true,"given":"Junqi","family":"Tang","sequence":"additional","affiliation":[{"name":"School of Mathematics, University of Birmingham, Birmingham, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Subhadip","family":"Mukherjee","sequence":"additional","affiliation":[{"name":"Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology (IIT) Kharagpur, India."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7980-6901","authenticated-orcid":true,"given":"Jinglai","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics, University of Birmingham, Birmingham, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Carola-Bibiane","family":"Sch\u00f6nlieb","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2723-4451","authenticated-orcid":true,"given":"Xiaoqun","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, MOELSC and Institute of Natural Sciences, Shanghai Jiao Tong University, China."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,4,8]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.2018.2799231"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1109\/TCI.2017.2704439"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/acce5e"},{"key":"ref4","author":"Altekr\u00fcger F.","year":"2023","journal-title":"Trans. Mach. Learn. Res."},{"key":"ref5","first-page":"146","volume":"70","author":"Amos B.","year":"2017","journal-title":"Proc. Mach. Learn. Res. (PMLR)"},{"key":"ref6","unstructured":"L. Ardizzone, T. Bungert, F. Draxler, U. K\u00f6the, J. Kruse, R. Schmier, and P. Sorrenson, vislearn\/FrEIA, https:\/\/github.com\/vislearn\/FrEIA."},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492919000059"},{"key":"ref8","first-page":"399","volume":"119","author":"Asim M.","year":"2020","journal-title":"Proc. Mach. Learn. Res.\u00a0(PMLR)"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-9467-7"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492918000016"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.7551\/mitpress\/8579.001.0001"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.2017.1285773"},{"key":"ref13","first-page":"227","volume":"9","author":"Chambolle A.","year":"2010","journal-title":"Theoret. Found. Numer. Methods Sparse Recovery"},{"key":"ref14","author":"Cheng X.","year":"2019","journal-title":"J. Mach. Learn. Res."},{"key":"ref15","doi-asserted-by":"crossref","unstructured":"F. Coeurdoux, N. Dobigeon, and P. Chainais, Normalizing Flow Sampling with Langevin Dynamics in the Latent Space, preprint, arXiv:2305.12149, 2023.","DOI":"10.1007\/s10994-024-06623-x"},{"key":"ref16","doi-asserted-by":"crossref","unstructured":"F. Coeurdoux, N. Dobigeon, and P. Chainais, Plug-and-play split Gibbs sampler: Embedding deep generative priors in Bayesian inference, in International Conference on Learning Representations, 2024.","DOI":"10.1109\/TIP.2024.3404338"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-9569-8_10"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1137\/050626090"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1109\/MSP.2017.2765202"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1111\/rssb.12183"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2019.02.016"},{"key":"ref22","unstructured":"V. De Bortoli and A. Durmus, Convergence of Diffusions and Their Discretizations: From Continuous to Discrete Processes and Back, preprint, arXiv:1904.09808, 2019."},{"key":"ref23","first-page":"8780","volume":"34","author":"Dhariwal P.","year":"2021","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref24","unstructured":"L. Dinh, D. Krueger, and Y. Bengio, Nice: Non-linear independent components estimation, in International Conference on Learning Representations (Workshop), 2015."},{"key":"ref25","unstructured":"L. Dinh, J. Sohl-Dickstein, and S. Bengio, Density estimation using real NVP, in International Conference on Learning Representations, 2017."},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-97704-1"},{"key":"ref27","first-page":"2666","volume":"20","author":"Durmus A.","year":"2019","journal-title":"J. Mach. Learn. Res."},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1214\/16-AAP1238"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1137\/16M1108340"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1198\/jasa.2011.tm11181"},{"key":"ref31","first-page":"1776","volume":"134","author":"Erdogdu M. A.","year":"2021","journal-title":"Proc. Mach. Learn. Res. (PMLR)"},{"key":"ref32","doi-asserted-by":"publisher","DOI":"10.1201\/b14835"},{"key":"ref33","doi-asserted-by":"crossref","unstructured":"D. Gilton, G. Ongie, and R. Willett, Learned patch-based regularization for inverse problems in imaging, in 2019 IEEE 8th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), IEEE, Piscataway, NJ, 2019, pp. 211\u2013215.","DOI":"10.1109\/CAMSAP45676.2019.9022624"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1109\/TCI.2021.3118944"},{"key":"ref35","volume":"27","author":"Goodfellow I.","year":"2014","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1137\/21M1450604"},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/abe928"},{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2021.09.004"},{"key":"ref39","first-page":"6840","volume":"33","author":"Ho J.","year":"2020","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref40","first-page":"1303","volume":"14","author":"Hoffman M. D.","year":"2013","journal-title":"J. Mach. Learn. Res."},{"key":"ref41","doi-asserted-by":"publisher","DOI":"10.1137\/17M1135694"},{"key":"ref42","unstructured":"S. Hurault, A. Leclaire, and N. Papadakis, Gradient step denoiser for convergent plug-and-play, in International Conference on Learning Representations, https:\/\/openreview.net\/forum?id=fPhKeld3Okz, 2022."},{"key":"ref43","volume-title":"Stochastic Differential Equations and Diffusion Processes","author":"Ikeda N.","year":"2014"},{"key":"ref44","first-page":"4673","volume":"119","author":"Jaini P.","year":"2020","journal-title":"Proc. Mach. Learn. Res. (PMLR)"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2017.2713099"},{"key":"ref46","series-title":"Appl. Math. Sci. 160","volume-title":"Statistical and Computational Inverse Problems","author":"Kaipio J.","year":"2006"},{"key":"ref47","doi-asserted-by":"publisher","DOI":"10.1109\/MSP.2022.3199595"},{"key":"ref48","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0949-2"},{"key":"ref49","doi-asserted-by":"crossref","unstructured":"T. Karras, S. Laine, and T. Aila, A style-based generator architecture for generative adversarial networks, in Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition, IEEE Computer Society, Los Alamitos, CA,\u00a02019, pp. 4401\u20134410.","DOI":"10.1109\/CVPR.2019.00453"},{"key":"ref50","first-page":"10236","volume":"31","author":"Kingma D. P.","year":"2018","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref51","first-page":"4743","volume":"29","author":"Kingma D. P.","year":"2016","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref52","unstructured":"D. P. Kingma and M. Welling, Auto-encoding variational bayes, in International Conference on Learning Representations, 2014."},{"key":"ref53","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2021.3124086"},{"key":"ref54","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2020.2992934"},{"key":"ref55","doi-asserted-by":"publisher","DOI":"10.1137\/19M1247176"},{"key":"ref56","doi-asserted-by":"publisher","DOI":"10.1137\/21M1406349"},{"key":"ref57","doi-asserted-by":"crossref","unstructured":"R. Laumont, V. D. Bortoli, A. Almansa, J. Delon, A. Durmus, and M. Pereyra, Supplementary materials: Bayesian imaging using plug & play priors: When Langevin meets Tweedie, https:\/\/epubs.siam.org\/doi\/suppl\/10.1137\/21M1406349\/suppl_file\/M140634_01.pdf, 2022.","DOI":"10.1137\/21M1406349"},{"key":"ref58","doi-asserted-by":"publisher","DOI":"10.1214\/23-AAP1935"},{"key":"ref59","doi-asserted-by":"crossref","unstructured":"V. Lempitsky, A. Vedaldi, and D. Ulyanov, Deep image prior, in 2018 IEEE\/CVF Conference on Computer Vision and Pattern Recognition, IEEE Computer Society, Los Alamitos, CA, 2018, pp. 9446\u20139454.","DOI":"10.1109\/CVPR.2018.00984"},{"key":"ref60","doi-asserted-by":"publisher","DOI":"10.1038\/s41597-021-00893-z"},{"key":"ref61","first-page":"2378","volume":"29","author":"Liu Q.","year":"2016","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref62","doi-asserted-by":"publisher","DOI":"10.1137\/120902276"},{"key":"ref63","first-page":"8516","volume":"31","author":"Lunz S.","year":"2018","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref64","doi-asserted-by":"publisher","DOI":"10.1007\/s11009-020-09809-7"},{"key":"ref65","doi-asserted-by":"publisher","DOI":"10.1214\/19-AAP1535"},{"key":"ref66","doi-asserted-by":"publisher","DOI":"10.2307\/1427522"},{"key":"ref67","doi-asserted-by":"publisher","DOI":"10.1109\/MSP.2020.3016905"},{"key":"ref68","first-page":"1","volume":"23","author":"Mou W.","year":"2022","journal-title":"J. Mach. Learn. Res."},{"key":"ref69","first-page":"21413","volume":"34","author":"Mukherjee S.","year":"2021","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref70","unstructured":"S. Mukherjee, S. Dittmer, Z. Shumaylov, S. Lunz, O. \u00d6ktem, and C.B. Sch\u00f6nlieb, Learned Convex Regularizers for Inverse Problems, preprint, arXiv:2008.02839, 2020."},{"key":"ref71","volume":"5","author":"Neal R.","year":"1992","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref72","doi-asserted-by":"publisher","DOI":"10.1109\/JSAIT.2020.2991563"},{"key":"ref73","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2021.3115428"},{"key":"ref74","first-page":"1","volume":"22","author":"Papamakarios G.","year":"2021","journal-title":"J. Mach. Learn. Res."},{"key":"ref75","first-page":"2335","volume":"30","author":"Papamakarios G.","year":"2017","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref76","doi-asserted-by":"publisher","DOI":"10.1007\/s11222-015-9567-4"},{"key":"ref77","doi-asserted-by":"publisher","DOI":"10.1137\/20M1387961"},{"key":"ref78","doi-asserted-by":"crossref","unstructured":"J. Prost, A. Houdard, A. Almansa, and N. Papadakis, Learning local regularization for variational image restoration, in Scale Space and Variational Methods in Computer Vision: 8th International Conference, SSVM 2021, Virtual Event, Proceedings, Springer, Cham, Switzerland, 2021, pp. 358\u2013370.","DOI":"10.1007\/978-3-030-75549-2_29"},{"key":"ref79","first-page":"1530","volume":"37","author":"Rezende D.","year":"2015","journal-title":"Proc. Mach. Learn. Res. (PMLR)"},{"key":"ref80","doi-asserted-by":"publisher","DOI":"10.2307\/3318418"},{"key":"ref81","doi-asserted-by":"publisher","DOI":"10.1137\/16M1102884"},{"key":"ref82","unstructured":"M. Ronchetti, Torchradon: Fast Differentiable Routines for Computed Tomography, preprint, arXiv:2009.14788, 2020."},{"key":"ref83","doi-asserted-by":"crossref","unstructured":"O. Ronneberger, P. Fischer, and T. Brox, U-net: Convolutional networks for biomedical image segmentation, in Medical Image Computing and Computer-Assisted Intervention\u2013MICCAI 2015: 18th International Conference, Munich, Germany, 2015, Proceedings, Part III 18, Springer, Cham, Switzerland, 2015, pp. 234\u2013241.","DOI":"10.1007\/978-3-319-24574-4_28"},{"key":"ref84","first-page":"5546","volume":"35","author":"Ryu E.","year":"2019","journal-title":"Proc. Mach. Learn. Res. (PMLR)"},{"key":"ref85","first-page":"6653","volume":"32","author":"Salim A.","year":"2019","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref86","doi-asserted-by":"crossref","first-page":"10766","DOI":"10.52202\/068431-0782","volume":"35","author":"Salmona A.","year":"2022","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref87","unstructured":"Y. Song, L. Shen, L. Xing, and S. Ermon, Solving inverse problems in medical imaging with score-based generative models, in International Conference on Learning Representations, 2022."},{"key":"ref88","unstructured":"Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole, Score-based generative modeling through stochastic differential equations, in International Conference on Learning Representations, 2021."},{"key":"ref89","doi-asserted-by":"publisher","DOI":"10.1088\/1361-6420\/ab6f43"},{"key":"ref90","doi-asserted-by":"publisher","DOI":"10.1109\/TCI.2016.2599778"},{"key":"ref91","series-title":"Elem. Non-local Data Interactions Found. Appl.","volume-title":"Generalized Normalizing Flows via Markov Chains","author":"Steidl G.","year":"2022"},{"key":"ref92","doi-asserted-by":"publisher","DOI":"10.1023\/A:1010086427957"},{"key":"ref93","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492910000061"},{"key":"ref94","doi-asserted-by":"crossref","unstructured":"H. Y. Tan, S. Mukherjee, J. Tang, and C.B. Sch\u00f6nlieb, Provably Convergent Plug-and-Play Quasi-Newton Methods, preprint, arXiv:2303.07271, 2023.","DOI":"10.1137\/23M157185X"},{"key":"ref95","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898717921"},{"key":"ref96","doi-asserted-by":"crossref","unstructured":"S. V. Venkatakrishnan, C. A. Bouman, and B. Wohlberg, Plug-and-play priors for model based reconstruction, in 2013 IEEE Global Conference on Signal and Information Processing, IEEE, Piscataway, NJ, 2013, pp. 945\u2013948.","DOI":"10.1109\/GlobalSIP.2013.6737048"},{"key":"ref97","first-page":"1054","volume":"189","author":"Verine A.","year":"2022","journal-title":"Proc. Mach. Learn. Res. (PMLR)"},{"key":"ref98","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-71050-9"},{"key":"ref99","first-page":"11146","volume":"139","author":"Whang J.","year":"2021","journal-title":"Proc. Mach. Learn. Res. (PMLR)"},{"key":"ref100","first-page":"5933","volume":"33","author":"Wu H.","year":"2020","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref101","first-page":"105","volume":"56","author":"Yang L.","year":"2023","journal-title":"ACM Comput. Surv."},{"key":"ref102","first-page":"2481","volume":"21","author":"Yu G.","year":"2011","journal-title":"IEEE Trans. Image Process."},{"key":"ref103","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2021.3088914"},{"key":"ref104","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2017.2662206"},{"key":"ref105","doi-asserted-by":"crossref","unstructured":"D. Zoran and Y. Weiss, From learning models of natural image patches to whole image restoration, in 2011 International Conference on Computer Vision, IEEE, Piscataway, NJ, 2011, pp. 479\u2013486.","DOI":"10.1109\/ICCV.2011.6126278"}],"container-title":["SIAM Journal on Imaging Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/23M1581807","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:26:24Z","timestamp":1787322384000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/23M1581807"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,8]]},"references-count":105,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,6,30]]}},"alternative-id":["10.1137\/23M1581807"],"URL":"https:\/\/doi.org\/10.1137\/23m1581807","relation":{},"ISSN":["1936-4954"],"issn-type":[{"value":"1936-4954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,4,8]]}}}