{"title":"Nesterov acceleration-based iterative method for backward problem of distributed-order time-fractional diffusion equation","authors":"Zhengqiang Zhang, Yuan-Xiang Zhang, Shimin Guo","doi":"10.1002/mma.10415","DOIUrl":null,"url":null,"abstract":"<p>This paper is concerned with the inverse problem of determining the initial value of the distributed-order time-fractional diffusion equation from the final time observation data, which arises in some ultra-slow diffusion phenomena in applied areas. Since the problem is ill-posed, we propose an iterated regularization method based on the Nesterov acceleration strategy to deal with it. Convergence rates for the regularized approximation solution are given under both the <i>a priori</i> and <i>a posteriori</i> regularization parameter choice rules. It is shown that the proposed method can always yield the order optimal convergence rates as long as the iteration parameter which appears in the Nesterov acceleration strategy is chosen large enough. In numerical aspect, the main advantage of the proposed method lies in its simplicity. Specifically, due to the Nesterov acceleration strategy, only a few number of iteration steps are required to obtain the approximation solution, and at each iteration step, we only need to numerically solve the standard initial-boundary value problem for the distributed-order time-fractional diffusion equation. Some numerical examples including one-dimensional and two-dimensional cases are presented to illustrate the validity and effectiveness of the proposed method.</p>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 2","pages":"1877-1900"},"PeriodicalIF":2.1000,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Methods in the Applied Sciences","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mma.10415","RegionNum":3,"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
This paper is concerned with the inverse problem of determining the initial value of the distributed-order time-fractional diffusion equation from the final time observation data, which arises in some ultra-slow diffusion phenomena in applied areas. Since the problem is ill-posed, we propose an iterated regularization method based on the Nesterov acceleration strategy to deal with it. Convergence rates for the regularized approximation solution are given under both the a priori and a posteriori regularization parameter choice rules. It is shown that the proposed method can always yield the order optimal convergence rates as long as the iteration parameter which appears in the Nesterov acceleration strategy is chosen large enough. In numerical aspect, the main advantage of the proposed method lies in its simplicity. Specifically, due to the Nesterov acceleration strategy, only a few number of iteration steps are required to obtain the approximation solution, and at each iteration step, we only need to numerically solve the standard initial-boundary value problem for the distributed-order time-fractional diffusion equation. Some numerical examples including one-dimensional and two-dimensional cases are presented to illustrate the validity and effectiveness of the proposed method.
期刊介绍:
Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome.
Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted.
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