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从终端数据识别生物传热过程中空间热源的反问题

铁旭玮 杨柳

铁旭玮, 杨柳. 从终端数据识别生物传热过程中空间热源的反问题[J]. 应用数学和力学, 2025, 46(7): 904-915. doi: 10.21656/1000-0887.450221
引用本文: 铁旭玮, 杨柳. 从终端数据识别生物传热过程中空间热源的反问题[J]. 应用数学和力学, 2025, 46(7): 904-915. doi: 10.21656/1000-0887.450221
TIE Xuwei, YANG Liu. The Inverse Problem of Identifying Spatial Heat Sources in Biological Heat Transfer Processes From Terminal Data[J]. Applied Mathematics and Mechanics, 2025, 46(7): 904-915. doi: 10.21656/1000-0887.450221
Citation: TIE Xuwei, YANG Liu. The Inverse Problem of Identifying Spatial Heat Sources in Biological Heat Transfer Processes From Terminal Data[J]. Applied Mathematics and Mechanics, 2025, 46(7): 904-915. doi: 10.21656/1000-0887.450221

从终端数据识别生物传热过程中空间热源的反问题

doi: 10.21656/1000-0887.450221
基金项目: 

国家自然科学基金 61663018

国家自然科学基金 11961042

甘肃省自然科学基金 22JR5RA341

详细信息
    作者简介:

    铁旭玮(2000—), 男, 硕士生(E-mail: happrtiewei@qq.com)

    通讯作者:

    杨柳(1977—), 女, 教授, 博士, 硕士生导师(通讯作者. E-mail: l_yang218@163.com)

  • 中图分类号: O414.11;O175.27

The Inverse Problem of Identifying Spatial Heat Sources in Biological Heat Transfer Processes From Terminal Data

  • 摘要: 研究探讨了重构生物体热波扩散模型中空间热源的反问题. 与传统的抛物型生物传热模型不同, 该文聚焦于更复杂、更实用的双曲型模型, 特别适用于生物医学工程应用. 首先, 基于最优控制理论, 将反问题转化为一个最优控制问题. 为了克服全变差函数不可微性带来的控制问题唯一最优解难题, 引入了一个精心设计的磨光全变差正则化项. 然后, 详细讨论了最优解的存在性及其必要条件. 随后, 在终端时刻较小的假设条件下, 利用Sobolev嵌入理论证明了最优解的唯一性和稳定性. 最后, 基于必要条件设计了一种梯度型优化算法, 并通过数值算例验证了算法的有效性.
  • 图  1  例1中不同情况下, 热源的重构

    Figure  1.  Reconstruction of heat sources under different conditions in example 1

    图  2  例2中不同情况下, 热源的重构

    Figure  2.  Reconstruction of heat sources under different conditions in example 2

    图  3  例3中不同情况下热源的重构

    Figure  3.  Reconstruction of heat sources under different conditions in example 3

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    [2] 孙学超, 刘少宝, 林敏, 等. 生物热-力-电生理耦合学[J]. 应用数学和力学, 2024, 45 (6): 651-669. doi: 10.21656/1000-0887.450079

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出版历程
  • 收稿日期:  2024-07-26
  • 修回日期:  2024-09-30
  • 网络出版日期:  2025-07-30
  • 刊出日期:  2025-07-01

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