Inverse problems involve making inference about unknown parameters of a physical process using observational data. This paper investigates an important class of inverse problems -- the estimation of the initial condition of a spatio-temporal advection-diffusion process using spatially sparse data streams. Three spatial sampling schemes are considered, including irregular, non-uniform and shifted uniform sampling. The irregular sampling scheme is the general scenario, while computationally efficient solutions are available in the spectral domain for non-uniform and shifted uniform sampling. For each sampling scheme, the inverse problem is formulated as a regularized convex optimization problem that minimizes the distance between forward model outputs and observations. The optimization problem is solved by the Alternating Direction Method of Multipliers algorithm, which also handles the situation when a linear inequality constraint (e.g., non-negativity) is imposed on the model output. Numerical examples are presented, code is made available on GitHub, and discussions are provided to generate some useful insights of the proposed inverse modeling approaches.

翻译：逆问题涉及利用观测数据推断物理过程未知参数的过程。本文研究了一类重要的逆问题——利用空间稀疏数据流估计时空平流-扩散过程的初始条件。考虑三种空间采样方案：不规则采样、非均匀采样和偏移均匀采样。不规则采样是一般场景，而非均匀采样和偏移均匀采样可在谱域中实现高效计算。针对每种采样方案，将逆问题建模为正则化凸优化问题，最小化前向模型输出与观测值之间的距离。该优化问题通过交替方向乘子法算法求解，该方法也能处理对模型输出施加线性不等式约束（如非负性）的情形。文中给出了数值示例，在GitHub上提供了代码，并通过讨论为所提出的逆建模方法提供有用见解。