同时反演源项和初值的抛物型反问题的条件稳定性和数值方法研究

The simultaneous reconstruction of source term and initial value in inverse parabolic problems is investigated in this project, including theoretical analysis(conditional stability) and numerical algorithm(regularization methods). Since these two problems are both ill-posed, and the reconstruction of initial value is severely ill-posed, our proposed problem is also ill-posed. For source terms of various kinds, we give the results of conditional stability for the source terms and initial value through discussing different definite conditions. For the numerical algorithm, we employ quasi-reversible regularization and mollification method to obtain the numerical solutions. Besides, we present several typical numerical examples in order to verify the validity of theoretical results and the efficiency of numerical methods.. For the studies on these problems, we can not only enrich the theory and algorithm of inverse problems, but also solve some practical problems by providing necessary theoretical basis and useful numerical methods, for example, the simultaneous determination of pollution source and initial concentration of the pollutant in water pollution.

本项目主要研究抛物型方程中同时反演源项和初值的反问题，包括理论分析（条件稳定性）和数值算法（正则化方法）。因为这两类问题都是经典的不适定问题，而初值的反演问题还是严重不适定的，所以，我们提出的反问题也是不适定的。对于不同形式的源项，通过讨论不同的定解条件，分别给出源项和初值的条件稳定性结果。对于数值算法，我们采用拟逆正则化方法，磨光化方法等来求解数值解。此外，还给出一些典型的数值算例，来验证我们得出的理论结果的正确性和数值方法的有效性。. 对于这些问题的研究，既可以丰富偏微分方程反问题的理论与算法，又可以对某些实际问题，如同时确定水污染中的污染源及污染物初始浓度等问题提供必要的理论依据和实用的数值方法。

同时反演初值和源项的逆热传导问题是近来研究的一个重点.本项目针对这类反问题进行了一系列研究：.1)对于无界区域(带型区域)的同时反演初值和源项的反问题，首先给出了唯一性结果；.2)利用唯一性结果的条件，分别采用拟逆正则化方法和数值微分方法对上述反问题给出了正则化解；.3)给出若干数值例子对上述问题进行验证，来显示我们方法的正确性和精确性..