{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:33:25Z","timestamp":1760146405234,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,10,29]],"date-time":"2024-10-29T00:00:00Z","timestamp":1730160000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key R&amp;D Program of China","doi-asserted-by":"publisher","award":["2023YFA1011303","12301477","12301479","42274166","JYTMS20230165","3132024202"],"award-info":[{"award-number":["2023YFA1011303","12301477","12301479","42274166","JYTMS20230165","3132024202"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2023YFA1011303","12301477","12301479","42274166","JYTMS20230165","3132024202"],"award-info":[{"award-number":["2023YFA1011303","12301477","12301479","42274166","JYTMS20230165","3132024202"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Fundamental Scientific Research Projects of Higher Education Institutions of Liaoning Provincial Department of Education","award":["2023YFA1011303","12301477","12301479","42274166","JYTMS20230165","3132024202"],"award-info":[{"award-number":["2023YFA1011303","12301477","12301479","42274166","JYTMS20230165","3132024202"]}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["2023YFA1011303","12301477","12301479","42274166","JYTMS20230165","3132024202"],"award-info":[{"award-number":["2023YFA1011303","12301477","12301479","42274166","JYTMS20230165","3132024202"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Optimization problems with PDE constraints are widely used in engineering and technical fields. In some practical applications, it is necessary to smooth the control variables and suppress their large fluctuations, especially at the boundary. Therefore, we propose an elliptic PDE-constrained optimization model with a control gradient penalty term. However, introducing this penalty term increases the complexity and difficulty of the problems. To solve the problems numerically, we adopt the strategy of \u201cFirst discretize, then optimize\u201d. First, the finite element method is employed to discretize the optimization problems. Then, a heterogeneous strategy is introduced to formulate the augmented Lagrangian function for the subproblems. Subsequently, we propose a three-block inexact heterogeneous alternating direction method of multipliers (three-block ihADMM). Theoretically, we provide a global convergence analysis of the three-block ihADMM algorithm and discuss the iteration complexity results. Numerical results are provided to demonstrate the efficiency of the proposed algorithm.<\/jats:p>","DOI":"10.3390\/axioms13110744","type":"journal-article","created":{"date-parts":[[2024,10,29]],"date-time":"2024-10-29T12:09:18Z","timestamp":1730203758000},"page":"744","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Three-Block Inexact Heterogeneous Alternating Direction Method of Multipliers for Elliptic PDE-Constrained Optimization Problems with a Control Gradient Penalty Term"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3353-3714","authenticated-orcid":false,"given":"Xiaotong","family":"Chen","sequence":"first","affiliation":[{"name":"School of Science, Dalian Maritime University, Dalian 116026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tongtong","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Science, Dalian Maritime University, Dalian 116026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4965-6874","authenticated-orcid":false,"given":"Xiaoliang","family":"Song","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"127634","DOI":"10.1016\/j.jmaa.2023.127634","article-title":"Stochastic collocation for optimal control problems with stochastic PDE constraints by meshless techniques","volume":"530","author":"Huang","year":"2024","journal-title":"J. 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