A novel inverse source problem concerning the determination of a term in the right-hand side of parabolic equations from boundary observation is investigated. The observation is given by an imprecise Dirichlet data on some part of the boundary. The unknown heat source is sought as a function depending on both space and time variables with an a priori information. The problem is reformulated as an optimal control problem with a Tikhonov regularization term. The gradient of the functional is derived via an adjoint problem. The space-time discretization approach is employed which allows the use of general space-time finite elements. The convergence of the approach is proved. Some numerical examples are presented for showing the efficiency of the approach.

中文翻译：

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从边界观测确定抛物线方程中源的时空有限元方法

研究了一种新的逆源问题，该问题涉及从边界观测确定抛物线方程右侧的项。观察是通过边界某些部分上不精确的Dirichlet数据给出的。根据具有先验信息的空间和时间变量来寻找未知热源作为函数。使用Tikhonov正则项将该问题重新表述为最优控制问题。函数的梯度是通过伴随问题导出的。采用了时空离散化方法，该方法允许使用一般的时空有限元。证明了该方法的收敛性。给出了一些数值示例，以说明该方法的效率。