This paper presents a solution for efficiently and accurately solving separable least squares problems with multiple datasets. These problems involve determining linear parameters that are specific to each dataset while ensuring that the nonlinear parameters remain consistent across all datasets. A well-established approach for solving such problems is the variable projection algorithm introduced by Golub and LeVeque, which effectively reduces a separable problem to its nonlinear component. However, this algorithm assumes that the datasets have equal sizes and identical auxiliary model parameters. This article is motivated by a real-world remote sensing application where these assumptions do not apply. Consequently, we propose a generalized algorithm that extends the original theory to overcome these limitations. The new algorithm has been implemented and tested using both synthetic and real satellite data for atmospheric carbon dioxide retrievals. It has also been compared to conventional state-of-the-art solvers, and its advantages are thoroughly discussed. The experimental results demonstrate that the proposed algorithm significantly outperforms all other methods in terms of computation time, while maintaining comparable accuracy and stability. Hence, this novel method can have a positive impact on future applications in remote sensing and could be valuable for other scientific fitting problems with similar properties.

翻译：本文提出了一种高效且精确求解多数据集可分离最小二乘问题的方案。这类问题需要确定每个数据集特有的线性参数，同时确保非线性参数在所有数据集中保持一致。解决此类问题的经典方法是Golub和LeVeque提出的变量投影算法，该算法能有效将可分离问题简化为其非线性分量。然而，该算法假设数据集具有相同规模且辅助模型参数完全一致。本文源于实际遥感应用场景，其中上述假设条件无法满足。为此，我们提出一种广义算法，在扩展原理论的基础上克服了这些局限性。该新算法已通过合成数据与卫星实测数据在大气二氧化碳反演任务中得到验证，并与现有顶尖求解器进行了比较，详细讨论了其优势。实验结果表明，所提算法在保持相当精度与稳定性的同时，计算效率显著优于其他所有方法。因此，这一新方法将对未来遥感应用产生积极影响，并有望适用于其他具有相似特性的科学拟合问题。