Abstract We consider inverse problems of recovering a source term in initial boundary value problems for linear multidimensional partial differential equations (PDEs) of a general form. A universal stable method suitable for solving such inverse problems is proposed. The method allows one to obtain in the same way approximations to exact sources in different kinds of PDEs using various types of linear supplementary conditions specified with an error. The method is suitable for both spacewise dependent and time-dependent sources. The method consists in preliminary calculation of a special matrix introduced in the article, the matrix of the source inverse problem, and then inverting it using Tikhonov regularization. The matrix can be obtained by solving a number of initial boundary value problems in question with sources in the form of basis functions. Having spent some time for preliminary finding the matrix (for example, by finite element method with a sufficiently detailed grid), we can then use this matrix to quickly solve the inverse problem with various data. The same technique can be applied to solve inverse source problems in linear steady-state PDEs. We also propose an a posteriori error estimation method for the obtained approximate solution and give a numerical algorithm for such estimation. In addition, a relationship is established between the posterior estimate and the lower estimate for the optimal accuracy of solving the inverse problem. The proposed method of solving inverse source problems is illustrated by the numerical solution of model examples for one-dimensional and two-dimensional PDEs of different kinds with a posteriori error estimates.

中文翻译：

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线性偏微分方程中后验误差估计的源恢复

摘要 我们考虑在一般形式的线性多维偏微分方程 (PDE) 的初始边值问题中恢复源项的逆问题。提出了一种适用于解决此类逆问题的通用稳定方法。该方法允许使用各种类型的带有误差指定的线性补充条件，以相同的方式获得不同类型 PDE 中精确源的近似值。该方法适用于空间相关和时间相关的源。该方法包括初步计算文章中介绍的一个特殊矩阵，即源逆问题的矩阵，然后使用 Tikhonov 正则化对其求逆。该矩阵可以通过使用基函数形式的源求解许多有问题的初始边值问题来获得。花了一些时间初步找到矩阵（例如，通过具有足够详细网格的有限元方法），然后我们可以使用该矩阵快速解决各种数据的逆问题。可以应用相同的技术来解决线性稳态 PDE 中的逆源问题。我们还为获得的近似解提出了一种后验误差估计方法，并给出了这种估计的数值算法。此外，为了求解逆问题的最佳精度，在后验估计和下估计之间建立了关系。