二维热方程逆回溯问题的数值模拟

IF 0.8 Q2 MATHEMATICS Lobachevskii Journal of Mathematics Pub Date : 2024-08-28 DOI:10.1134/s1995080224602583
S. A. Kolesnik, E. M. Stifeev
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引用次数: 0

摘要

摘要 本文以具有第三类边界条件的二维热方程的初始条件恢复问题为例,提出了一种新的、独特的非线性条件逆问题数值求解方法。在这个问题中,初始条件是两个变量的未知函数,由实验温度值确定。所提出的方法基于参数识别法、隐式梯度下降法和 Tikhonov 正则化方法。已开发出一种算法和一个用于数值求解的软件包。与零阶方法相比,使用隐式梯度下降法可以更快地收敛(迭代次数)。获得并讨论了大量数值实验结果。对使用和不使用季霍诺夫正则函数的求解函数的行为以及正则参数的影响进行了分析。使用所提出的数值方法进行计算实验的结果表明,由于正则化参数的正确选择,所得结果的误差不会超过实验数据的误差。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

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Numerical Simulation of Inverse Retrospective Problems for a Two-Dimensional Heat Equation

Abstract

The paper proposes a new, unique method for numerically solving inverse problems for nonlinear conditions using the example of the problem of restoring the initial condition for a two-dimensional heat equation with boundary conditions of the third kind. In this problem, the initial condition is an unknown function of two variables, and is determined from experimental temperature values. The proposed method is based on using the parametric identification method, the implicit gradient descent method and Tikhonov’s regularization method. An algorithm and a software package for numerical solution have been developed. The use of the implicit gradient descent method allowed for faster convergence (number of iterations) compared to zero-order methods. Numerous results of numerical experiments have been obtained and discussed. An analysis of the behavior of solution functions with and without the use of Tikhonov’s regularizing functional along with the impact of the regularizing parameter has been carried out. The results of computational experiments using the proposed numerical method showed that the error in the results obtained does not exceed the error in the experimental data due to the correct choice of the regularizing parameter.

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来源期刊
CiteScore
1.50
自引率
42.90%
发文量
127
期刊介绍: Lobachevskii Journal of Mathematics is an international peer reviewed journal published in collaboration with the Russian Academy of Sciences and Kazan Federal University. The journal covers mathematical topics associated with the name of famous Russian mathematician Nikolai Lobachevsky (Lobachevskii). The journal publishes research articles on geometry and topology, algebra, complex analysis, functional analysis, differential equations and mathematical physics, probability theory and stochastic processes, computational mathematics, mathematical modeling, numerical methods and program complexes, computer science, optimal control, and theory of algorithms as well as applied mathematics. The journal welcomes manuscripts from all countries in the English language.
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