The inverse problem of simultaneously determining, i.e., identifying and reconstructing, the space-dependent reaction coefficient and source term component from time-integral temperature measurements is investigated. This corresponds to thermal applications in which the heat is generated from a source depending linearly on the temperature, but with unknown space-dependent coefficients. For the resulting nonlinear inverse problem, we first prove the existence of solution based on the Schauder fixed point theorem. Then, under certain additional conditions, the solution is also proved to be unique. For the numerical reconstruction of solution, the problem is reformulated as a least-squares minimisation whose Fréchet gradients with respect to the two unknowns are derived in terms of the solution of an adjoint problem. The conjugate gradient method (CGM) to calculate the numerical solution is developed, and its convergence is proved from the Lipschitz continuity of these gradients. Three numerical examples for one- and two-dimensional inverse problems are illustrated to reveal the accuracy and stability of the solutions applying the CGM regularised by the discrepancy principle when noisy data are inverted.

中文翻译：

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同时识别和重建空间相关反应系数和源项

研究了从时间积分温度测量中同时确定（即识别和重建）空间相关反应系数和源项分量的反问题。这对应于热应用，在热应用中，热量是从源产生的，线性地取决于温度，但是具有未知的空间相关系数。对于由此产生的非线性逆问题，我们首先证明基于Schauder不动点定理的解的存在性。然后，在某些附加条件下，该解决方案也被证明是唯一的。对于解的数值重构，将问题重新构造为最小二乘最小化，其最小二乘相对于两个未知数的Fréchet梯度是根据伴随问题的解导出的。提出了计算数值解的共轭梯度法（CGM），并从这些梯度的Lipschitz连续性证明了其收敛性。举例说明了一维和二维反问题的三个数值示例，以揭示当噪声数据反转时，应用由差异原理调整的CGM的解决方案的准确性和稳定性。