This paper enhances the optimization method in [6] by using quasi-orthogonal vector bases and band matrices. Our approach employs the three-dimensional-variational (3D-Var) cost function to estimate posterior modes of error distributions. The proposed method works as follows: at each iteration, we estimate the 3D-Var cost function's negative gradient via a first-order Taylor approximation. This vector is then multiplied by random positive definite matrices to obtain a set of potential descent directions. We employ these directions to build an initial sub-space onto which partial analysis increments can be computed. Subsequently, we create a set of orthogonal directions to previous sub-spaces in a least-square sense; we search for additional analysis contributions onto the new directions via line-search optimization. Sub-space analysis increments are mapped back onto model spaces as they are found. We theoretically prove the convergence of our proposed optimization method. Experimental tests are performed by using the Lorenz-96 model. The results reveal that additional directions can improve the quality of analysis corrections during assimilation steps. Even more, as the sub-space dimension increases, the optimization method can converge faster to posterior modes of analysis distributions.

本文通过使用准正交向量基和带矩阵来增强[6]中的优化方法。我们的方法采用三维变分（3D-Var）成本函数来估计误差分布的后验模式。该方法的工作原理如下：在每次迭代中，我们通过一阶泰勒近似估计3D-Var成本函数的负梯度。然后将此向量乘以随机正定矩阵即可获得一组潜在的下降方向。我们采用这些方向来构建初始子空间，可以在其上计算部分分析增量。随后，我们在最小二乘意义上创建了一组与先前子空间正交的方向；我们通过线搜索优化来寻找对新方向的其他分析贡献。找到子空间分析增量后，会将它们映射回模型空间。我们从理论上证明了我们提出的优化方法的收敛性。通过使用Lorenz-96模型进行实验测试。结果表明，在同化步骤中，其他方向可以提高分析校正的质量。更重要的是，随着子空间维数的增加，优化方法可以更快地收敛到分析分布的后验模式。