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A non-iterative method for recovering the space-dependent source and the initial value simultaneously in a parabolic equation
Journal of Inverse and Ill-posed Problems ( IF 1.1 ) Pub Date : 2020-08-01 , DOI: 10.1515/jiip-2019-0017
Zewen Wang 1 , Shuli Chen 1 , Shufang Qiu 1 , Bin Wu 2
Affiliation  

Abstract This paper is concerned with the inverse problem for determining the space-dependent source and the initial value simultaneously in a parabolic equation from two over-specified measurements. By means of transforming information of the initial value into the source term and obtaining a combined source term, the parabolic equation problem is converted into a parabolic problem with homogeneous conditions. Then the considered inverse problem is formulated into a regularized minimization problem, which is implemented by the finite element method based on solving a sequence of well-posed direct problems. The uniqueness of inverse solutions are proved by the solvability of the corresponding variational problem, and the conditional stability as well as the convergence rate of regularized solutions are also provided. Then the error estimate of approximate regularization solutions is presented in the finite-dimensional space. The proposed method is a very fast non-iterative algorithm, and it can successfully solve the multi-dimensional inverse problem for recovering the space-dependent source and the initial value simultaneously. Numerical results of five examples including one- and two-dimensional cases show that the proposed method is efficient and robust with respect to data noise.

中文翻译:

抛物线方程中同时恢复空间相关源和初始值的一种非迭代方法

摘要 本文研究了在抛物线方程中从两个超指定测量值同时确定空间相关源和初始值的反问题。通过将初始值信息转化为源项,得到组合源项,将抛物线方程问题转化为齐次条件的抛物线问题。然后将所考虑的逆问题公式化为正则化的最小化问题,该问题通过基于求解一系列适定直接问题的有限元方法来实现。通过相应变分问题的可解性证明了逆解的唯一性,并给出了正则化解的条件稳定性和收敛速度。然后在有限维空间中给出近似正则化解的误差估计。该方法是一种速度非常快的非迭代算法,可以成功解决多维逆问题,同时恢复空间相关源和初始值。包括一维和二维情况在内的五个例子的数值结果表明,所提出的方法对于数据噪声是有效和鲁棒的。
更新日期:2020-08-01
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