{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T18:28:14Z","timestamp":1787509694702,"version":"build-2736575974"},"reference-count":56,"publisher":"American Mathematical Society (AMS)","issue":"360","license":[{"start":{"date-parts":[[2026,7,11]],"date-time":"2026-07-11T00:00:00Z","timestamp":1783728000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>A first-order optimization algorithm has been introduced by Liu, Liu, Osher, and Li [J. Comput. Phys. 500 (2024), 19 pp.] to solve time-implicit schemes of reaction-diffusion equations. In this research, we conduct theoretical studies on this first-order algorithm equipped with a quadratic regularization term. We provide sufficient conditions under which the proposed algorithm and its time-continuous limit converge exponentially fast to a desired time-implicit numerical solution. We show both theoretically and numerically that the convergence rate is independent of the grid size, which makes our method suitable for large scale problems. The efficiency of our algorithm has been verified via a series of numerical examples conducted on various types of reaction-diffusion equations. The choice of optimal hyperparameters as well as comparisons with some classical root-finding algorithms is also discussed in the numerical section.<\/p>","DOI":"10.1090\/mcom\/4097","type":"journal-article","created":{"date-parts":[[2025,7,11]],"date-time":"2025-07-11T17:32:19Z","timestamp":1752255139000},"page":"1775-1827","source":"Crossref","is-referenced-by-count":2,"title":["Numerical analysis of a first-order computational algorithm for reaction-diffusion equations via the primal-dual hybrid gradient method"],"prefix":"10.1090","volume":"95","author":[{"given":"Shu","family":"Liu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xinzhe","family":"Zuo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stanley","family":"Osher","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wuchen","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2025,7,11]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica 27 (1979), no. 6, 1085\u20131095.","DOI":"10.1016\/0001-6160(79)90196-2"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1137\/060649124","article-title":"On preconditioned iterative methods for Burgers equations","volume":"29","author":"Bai, Zhong-Zhi","year":"2007","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"J. W. Cahn, On spinodal decomposition, Acta Metallurgica 9 (1961), no. 9, 795\u2013801.","DOI":"10.1016\/0001-6160(61)90182-1"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1007\/s10208-021-09503-1","article-title":"Primal dual methods for Wasserstein gradient flows","volume":"22","author":"Carrillo, Jos\u00e9 A.","year":"2022","journal-title":"Found. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1615-3375","issn-type":"print"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"376","DOI":"10.1137\/23M1562068","article-title":"Structure preserving primal dual methods for gradient flows with nonlinear mobility transport distances","volume":"62","author":"Carrillo, Jos\u00e9 A.","year":"2024","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jcp.2013.03.036","article-title":"A new approach for the numerical solution of diffusion equations with variable and degenerate mobility","volume":"246","author":"Ceniceros, Hector D.","year":"2013","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"120","DOI":"10.1007\/s10851-010-0251-1","article-title":"A first-order primal-dual algorithm for convex problems with applications to imaging","volume":"40","author":"Chambolle, Antonin","year":"2011","journal-title":"J. Math. Imaging Vision","ISSN":"https:\/\/id.crossref.org\/issn\/0924-9907","issn-type":"print"},{"key":"8","unstructured":"L. Chen and J. Wei, Accelerated gradient and skew-symmetric splitting methods for a class of monotone operator equations, Preprint,  arXiv:2303.09009, 2023."},{"issue":"4","key":"9","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1515\/jnma-2022-0056","article-title":"Transformed primal-dual methods for nonlinear saddle point systems","volume":"31","author":"Chen, Long","year":"2023","journal-title":"J. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1570-2820","issn-type":"print"},{"key":"10","doi-asserted-by":"publisher","first-page":"Paper No. 110668, 29","DOI":"10.1016\/j.jcp.2021.110668","article-title":"Solving and learning nonlinear PDEs with Gaussian processes","volume":"447","author":"Chen, Yifan","year":"2021","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"11","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1016\/j.jcp.2013.09.049","article-title":"High accuracy solutions to energy gradient flows from material science models","volume":"257","author":"Christlieb, Andrew","year":"2014","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"4","key":"12","doi-asserted-by":"publisher","first-page":"947","DOI":"10.4208\/cicp.oa-2019-0006","article-title":"High accuracy benchmark problems for Allen-Cahn and Cahn-Hilliard dynamics","volume":"26","author":"Church, Jon Matteo","year":"2019","journal-title":"Commun. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/1815-2406","issn-type":"print"},{"issue":"3","key":"13","doi-asserted-by":"publisher","first-page":"1314","DOI":"10.1137\/16M1080859","article-title":"Primal-dual extragradient methods for nonlinear nonsmooth PDE-constrained optimization","volume":"27","author":"Clason, Christian","year":"2017","journal-title":"SIAM J. Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/1052-6234","issn-type":"print"},{"key":"14","doi-asserted-by":"publisher","first-page":"297","DOI":"10.2307\/2003354","article-title":"An algorithm for the machine calculation of complex Fourier series","volume":"19","author":"Cooley, James W.","year":"1965","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"15","unstructured":"D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, Unpublished article, 6, 1998."},{"issue":"6","key":"16","first-page":"557","article-title":"Optimum damping design for an abstract wave equation","volume":"32","author":"Fahroo, Fariba","year":"1996","journal-title":"Kybernetika (Prague)","ISSN":"https:\/\/id.crossref.org\/issn\/0023-5954","issn-type":"print"},{"key":"17","unstructured":"C. Finn, P. Abbeel, and S.  Levine, Model-agnostic meta-learning for fast adaptation of deep networks, International Conference on Machine Learning, PMLR, 2017, pp. 1126\u20131135."},{"key":"18","doi-asserted-by":"crossref","unstructured":"P. J. Flory, Thermodynamics of high polymer solutions, J. Chem. Phys. 10 (1942), no. 1, 51\u201361.","DOI":"10.1063\/1.1723621"},{"key":"19","doi-asserted-by":"publisher","first-page":"Paper No. 112375, 30","DOI":"10.1016\/j.jcp.2023.112375","article-title":"High order spatial discretization for variational time implicit schemes: Wasserstein gradient flows and reaction-diffusion systems","volume":"491","author":"Fu, Guosheng","year":"2023","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"20","doi-asserted-by":"publisher","first-page":"Paper No. 112994, 28","DOI":"10.1016\/j.jcp.2024.112994","article-title":"Generalized optimal transport and mean field control problems for reaction-diffusion systems with high-order finite element computation","volume":"508","author":"Fu, Guosheng","year":"2024","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"21","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1016\/j.jcp.2017.10.028","article-title":"Convex splitting method for the calculation of transition states of energy functional","volume":"353","author":"Gu, Shuting","year":"2018","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"34","key":"22","doi-asserted-by":"publisher","first-page":"8505","DOI":"10.1073\/pnas.1718942115","article-title":"Solving high-dimensional partial differential equations using deep learning","volume":"115","author":"Han, Jiequn","year":"2018","journal-title":"Proc. Natl. Acad. Sci. USA","ISSN":"https:\/\/id.crossref.org\/issn\/0027-8424","issn-type":"print"},{"issue":"4","key":"23","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1007\/s40304-017-0117-6","article-title":"Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations","volume":"5","author":"E, Weinan","year":"2017","journal-title":"Commun. Math. Stat.","ISSN":"https:\/\/id.crossref.org\/issn\/2194-6701","issn-type":"print"},{"issue":"344","key":"24","doi-asserted-by":"publisher","first-page":"2515","DOI":"10.1090\/mcom\/3843","article-title":"A linear second-order maximum bound principle-preserving BDF scheme for the Allen-Cahn equation with a general mobility","volume":"92","author":"Hou, Dianming","year":"2023","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"25","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-09017-6","volume-title":"Numerical solution of time-dependent advection-diffusion-reaction equations","volume":"33","author":"Hundsdorfer, Willem","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/3540034404"},{"issue":"3","key":"26","doi-asserted-by":"publisher","first-page":"1100","DOI":"10.1137\/18M118640X","article-title":"Solving large-scale optimization problems with a convergence rate independent of grid size","volume":"57","author":"Jacobs, Matt","year":"2019","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"27","doi-asserted-by":"crossref","unstructured":"A. M. Jokisaari, P. W. Voorhees, J. E. Guyer, J. Warren, and O. G. Heinonen, Benchmark problems for numerical implementations of phase field models, Comput. Mater. Sci. 126 (2017), 139\u2013151.","DOI":"10.1016\/j.commatsci.2016.09.022"},{"key":"28","isbn-type":"print","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1142\/9789814415309_0005","article-title":"Models of pattern formation in first-order phase transitions","author":"Langer, J. S.","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/9971978423"},{"issue":"6","key":"29","doi-asserted-by":"publisher","first-page":"1591","DOI":"10.1016\/j.camwa.2010.06.041","article-title":"An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation","volume":"60","author":"Li, Yibao","year":"2010","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"key":"30","unstructured":"T. Lin, C. Jin, and M. Jordan, On gradient descent ascent for nonconvex-concave minimax problems, International Conference on Machine Learning, PMLR, 2020, pp. 6083\u20136093."},{"key":"31","doi-asserted-by":"publisher","first-page":"Paper No. 110253, 22","DOI":"10.1016\/j.jcp.2021.110253","article-title":"A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance","volume":"436","author":"Liu, Chun","year":"2021","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"4","key":"32","doi-asserted-by":"publisher","first-page":"A2276--A2301","DOI":"10.1137\/21M1444825","article-title":"A second-order accurate, operator splitting scheme for reaction-diffusion systems in an energetic variational formulation","volume":"44","author":"Liu, Chun","year":"2022","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"2","key":"33","doi-asserted-by":"publisher","first-page":"781","DOI":"10.1137\/21M1421283","article-title":"Convergence analysis of the variational operator splitting scheme for a reaction-diffusion system with detailed balance","volume":"60","author":"Liu, Chun","year":"2022","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"34","doi-asserted-by":"publisher","first-page":"109566","DOI":"10.1016\/j.jcp.2020.109566","article-title":"On Lagrangian schemes for porous medium type generalized diffusion equations: a discrete energetic variational approach","volume":"417","author":"Liu, Chun","year":"2020","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"35","doi-asserted-by":"publisher","first-page":"Paper No. 112753, 19","DOI":"10.1016\/j.jcp.2024.112753","article-title":"A first-order computational algorithm for reaction-diffusion type equations via primal-dual hybrid gradient method","volume":"500","author":"Liu, Shu","year":"2024","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"36","doi-asserted-by":"publisher","first-page":"Paper No. 111654, 16","DOI":"10.1016\/j.jcp.2022.111654","article-title":"A primal-dual approach for solving conservation laws with implicit in time approximations","volume":"472","author":"Liu, Siting","year":"2023","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"37","unstructured":"T. Meng, W. Hao, S. Liu, S. J. Osher, and W. Li, Primal-dual hybrid gradient algorithms for computing time-implicit Hamilton-Jacobi equations, Preprint,  arXiv:2310.01605, 2023."},{"issue":"2","key":"38","doi-asserted-by":"publisher","first-page":"334","DOI":"10.1006\/jcph.1994.1105","article-title":"Motion of multiple functions: a level set approach","volume":"112","author":"Merriman, Barry","year":"1994","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"39","series-title":"Interdisciplinary Applied Mathematics","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/b98869","volume-title":"Mathematical biology. II","volume":"18","author":"Murray, J. D.","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0387952284","edition":"3"},{"key":"40","doi-asserted-by":"crossref","unstructured":"J. E. Pearson, Complex patterns in a simple system, Science 261 (1993), no. 5118, 189\u2013192.","DOI":"10.1126\/science.261.5118.189"},{"key":"41","doi-asserted-by":"publisher","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","article-title":"Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations","volume":"378","author":"Raissi, M.","year":"2019","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"3","key":"42","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1016\/0022-5193(79)90042-0","article-title":"Simple chemical reaction systems with limit cycle behaviour","volume":"81","author":"Schnakenberg, J.","year":"1979","journal-title":"J. Theoret. Biol.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-5193","issn-type":"print"},{"issue":"3","key":"43","doi-asserted-by":"publisher","first-page":"474","DOI":"10.1137\/17M1150153","article-title":"A new class of efficient and robust energy stable schemes for gradient flows","volume":"61","author":"Shen, Jie","year":"2019","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"issue":"4","key":"44","doi-asserted-by":"publisher","first-page":"1669","DOI":"10.3934\/dcds.2010.28.1669","article-title":"Numerical approximations of Allen-Cahn and Cahn-Hilliard equations","volume":"28","author":"Shen, Jie","year":"2010","journal-title":"Discrete Contin. Dyn. Syst.","ISSN":"https:\/\/id.crossref.org\/issn\/1078-0947","issn-type":"print"},{"key":"45","doi-asserted-by":"crossref","unstructured":"M. B. Short, P. Jeffrey Brantingham, A. L. Bertozzi, and G. E. Tita, Dissipation and displacement of hotspots in reaction-diffusion models of crime, Proceedings of the National Academy of Sciences, 107(9):3961\u20133965, 2010.","DOI":"10.1073\/pnas.0910921107"},{"key":"46","doi-asserted-by":"crossref","unstructured":"G. Strang, A proposal for Toeplitz matrix calculations, Stud. Appl. Math. 74 (1986), no. 2, 171\u2013176.","DOI":"10.1002\/sapm1986742171"},{"issue":"1","key":"47","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1137\/S0036144598336745","article-title":"The discrete cosine transform","volume":"41","author":"Strang, Gilbert","year":"1999","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"issue":"5","key":"48","doi-asserted-by":"publisher","first-page":"055012","DOI":"10.1088\/0266-5611\/30\/5\/055012","article-title":"A primal-dual hybrid gradient method for nonlinear operators with applications to MRI","volume":"30","author":"Valkonen, Tuomo","year":"2014","journal-title":"Inverse Problems","ISSN":"https:\/\/id.crossref.org\/issn\/0266-5611","issn-type":"print"},{"issue":"3","key":"49","doi-asserted-by":"publisher","first-page":"930","DOI":"10.4208\/cicp.oa-2020-0086","article-title":"Solving Allen-Cahn and Cahn-Hilliard equations using the adaptive physics informed neural networks","volume":"29","author":"Wight, Colby L.","year":"2021","journal-title":"Commun. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/1815-2406","issn-type":"print"},{"key":"50","doi-asserted-by":"publisher","first-page":"826","DOI":"10.1016\/j.cma.2018.09.017","article-title":"On the stability and accuracy of partially and fully implicit schemes for phase field modeling","volume":"345","author":"Xu, Jinchao","year":"2019","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"5","key":"51","doi-asserted-by":"publisher","first-page":"693","DOI":"10.4208\/ijnam2023-1030","article-title":"Numerical approximations of the Allen-Cahn-Ohta-Kawasaki equation with modified physics-informed neural networks (PINNs)","volume":"20","author":"Xu, Jingjing","year":"2023","journal-title":"Int. J. Numer. Anal. Model.","ISSN":"https:\/\/id.crossref.org\/issn\/1705-5105","issn-type":"print"},{"key":"52","doi-asserted-by":"publisher","first-page":"294","DOI":"10.1016\/j.jcp.2016.09.029","article-title":"Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends","volume":"327","author":"Yang, Xiaofeng","year":"2016","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"1-3","key":"53","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1007\/s10915-008-9218-4","article-title":"Application of discontinuous Galerkin methods for reaction-diffusion systems in developmental biology","volume":"40","author":"Zhu, Jianfeng","year":"2009","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"key":"54","unstructured":"M. Zhu and T. Chan, An efficient primal-dual hybrid gradient algorithm for total variation image restoration, UCLA CAM Report, 34:8\u201334, 2008."},{"key":"55","unstructured":"X. Zuo, Z. Chen, H. Yao, Y. Cao, and Q. Gu, Understanding train-validation split in meta-learning with neural networks, The Eleventh International Conference on Learning Representations, 2022."},{"issue":"2","key":"56","doi-asserted-by":"publisher","first-page":"467","DOI":"10.4310\/amsa.2024.v9.n2.a7","article-title":"Primal-dual damping algorithms for optimization","volume":"9","author":"Zuo, Xinzhe","year":"2024","journal-title":"Ann. Math. Sci. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/2380-288X","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-360\/S0025-5718-2025-04097-3\/S0025-5718-2025-04097-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:04:40Z","timestamp":1776837880000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-360\/S0025-5718-2025-04097-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,11]]},"references-count":56,"journal-issue":{"issue":"360","published-print":{"date-parts":[[2026,7]]}},"alternative-id":["S0025-5718-2025-04097-3"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/4097","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2025,7,11]]}}}