We consider the inverse source problem in the parabolic equation, where the unknown source possesses the semi-discrete formulation. Theoretically, we prove that the flux data from any nonempty open subset of the boundary can uniquely determine the semi-discrete source. This means the observed area can be extremely small, and that is why we call the data as sparse boundary data. For the numerical reconstruction, we formulate the problem from the Bayesian sequential prediction perspective and conduct the numerical examples which estimate the space-time-dependent source state by state. To better demonstrate the performance of the method, we solve two common multiscale problems from two models with a long sequence of the source. The numerical results illustrate that the inversion is accurate and efficient.

翻译：我们考虑了抛物线等式中的反源问题, 未知来源拥有半分立的配方。 从理论上讲, 我们证明来自边界中任何非空开放子集的通量数据能够独有地决定半分立源。 这意味着观测到的区域可能是极小的, 这就是为什么我们把数据称为稀疏的边界数据。 在数字重建中, 我们从巴伊西亚的顺序预测角度来分析问题, 并用数字示例来按州来估计取决于空间的时间源状态。 为了更好地显示方法的性能, 我们从两个模型中解决了两个共同的多尺度问题, 两个模型的源序列很长。 数字结果显示, 反向数据是准确和高效的 。