Abstract The heat conduction process in composite medium can be modeled by a parabolic equation with discontinuous radiative coefficient. To detect the composite medium characterized by such a non-smooth coefficient from measurable information about the heat distribution, we consider a nonlinear inverse problem for parabolic equation, with the average measurement of temperature field in some time interval as the inversion input. We firstly establish the uniqueness for this nonlinear inverse problem, based on the property of the direct problem and the known uniqueness result for linear inverse source problem. To solve the inverse problem from a nonlinear operator equation, the differentiability and the tangential condition of this nonlinear map is analyzed. An iterative process called two-point gradient method is proposed by minimizing data-fit term and the penalty term alternatively, with rigorous convergence analysis in terms of the tangential condition. Numerical simulations are presented to illustrate the effectiveness of the proposed method.

中文翻译：

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利用扩散系统从非局域观测中恢复非光滑辐射系数

摘要 复合介质中的热传导过程可以用具有不连续辐射系数的抛物线方程建模。为了从有关热分布的可测量信息中检测具有这种非光滑系数特征的复合介质，我们考虑抛物线方程的非线性反问题，以一定时间间隔内温度场的平均测量值作为反演输入。我们首先根据直接问题的性质和线性逆源问题的已知唯一性结果，建立该非线性逆问题的唯一性。为了从非线性算子方程求解逆问题，分析了该非线性映射的可微性和切线条件。通过交替最小化数据拟合项和惩罚项，并根据切向条件进行严格的收敛分析，提出了一种称为两点梯度法的迭代过程。数值模拟被提出来说明所提出的方法的有效性。