On the Features of Numerical Solution of Coefficient Inverse Problems for Nonlinear Equations of the Reaction–Diffusion–Advection Type with Data of Various Types

Pub Date : 2024-02-26 DOI:10.1134/s0012266123120133

D. V. Lukyanenko, R. L. Argun, A. A. Borzunov, A. V. Gorbachev, V. D. Shinkarev, M. A. Shishlenin, A. G. Yagola

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Abstract

The paper discusses the features of constructing numerical schemes for solving coefficient inverse problems for nonlinear partial differential equations of the reaction–diffusion–advection type with data of various types. As input data for the inverse problem, we consider (1) data at the final moment of time, (2) data at the spatial boundary of a domain, (3) data at the position of the reaction front. To solve the inverse problem in all formulations, the gradient method of minimizing the target functional is used. In this case, when constructing numerical minimization schemes, both an approach based on discretization of the analytical expression for the gradient of the functional and an approach based on differentiating the discrete approximation of the functional to be minimized are considered. Features of the practical implementation of these approaches are demonstrated by the example of solving the inverse problem of reconstructing the linear gain coefficient in a nonlinear Burgers-type equation.

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论各类数据的反应-扩散-对流型非线性方程系数逆问题数值解法的特点

摘要本文讨论了用各种类型的数据求解反应-扩散-对流型非线性偏微分方程系数逆问题的数值方案的构建特点。作为逆问题的输入数据，我们考虑了(1) 最后时刻的数据，(2) 域空间边界的数据，(3) 反应前沿位置的数据。为了解决所有公式中的逆问题，我们采用了梯度法使目标函数最小化。在这种情况下，在构建数值最小化方案时，既要考虑基于函数梯度分析表达式离散化的方法，也要考虑基于微分待最小化函数离散近似值的方法。通过解决在非线性布尔格斯型方程中重建线性增益系数的逆问题的例子，展示了这些方法实际应用的特点。

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