{"title":"An MP-Newton method for computing nonlinear eigenpairs and its application for solving a semilinear Schrödinger equation","authors":"Xudong Yao","doi":"10.1016/j.cam.2024.116315","DOIUrl":null,"url":null,"abstract":"<div><div>ln Yao and Zhou (2008), a minimax method for computing nonlinear eigenpairs by calculating critical points of the Lagrange multiplier function is presented. But, the method is slow and can find limited amount of eigenpairs. In this paper, a new general characterization, orthogonal-max characterization, for critical points of the Lagrange multiplier function is suggested. An MP-Newton method for finding orthogonal-max type critical points is designed through analyzing how the minimax method works. The new method becomes fast and able to calculate more nonlinear eigenpairs. Numerical experiment confirms these two progresses. Also, the MP-Newton method inherits the advantages of the minimax method. A convergence result for the method is established. Finally, an application for solving a semilinear Schrödinger equation is discussed.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"457 ","pages":"Article 116315"},"PeriodicalIF":2.6000,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Computational and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0377042724005636","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
ln Yao and Zhou (2008), a minimax method for computing nonlinear eigenpairs by calculating critical points of the Lagrange multiplier function is presented. But, the method is slow and can find limited amount of eigenpairs. In this paper, a new general characterization, orthogonal-max characterization, for critical points of the Lagrange multiplier function is suggested. An MP-Newton method for finding orthogonal-max type critical points is designed through analyzing how the minimax method works. The new method becomes fast and able to calculate more nonlinear eigenpairs. Numerical experiment confirms these two progresses. Also, the MP-Newton method inherits the advantages of the minimax method. A convergence result for the method is established. Finally, an application for solving a semilinear Schrödinger equation is discussed.
期刊介绍：
The Journal of Computational and Applied Mathematics publishes original papers of high scientific value in all areas of computational and applied mathematics. The main interest of the Journal is in papers that describe and analyze new computational techniques for solving scientific or engineering problems. Also the improved analysis, including the effectiveness and applicability, of existing methods and algorithms is of importance. The computational efficiency (e.g. the convergence, stability, accuracy, ...) should be proved and illustrated by nontrivial numerical examples. Papers describing only variants of existing methods, without adding significant new computational properties are not of interest.
The audience consists of: applied mathematicians, numerical analysts, computational scientists and engineers.
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