Strong Constraint 4D Variational (SC-4DVAR) is a data assimilation method that is widely used in climate and weather applications. SC-4DVAR involves solving a minimization problem to compute the maximum a posteriori estimate, which we tackle using the Gauss-Newton method. The computation of the descent direction is expensive since it involves the solution of a large-scale and potentially ill-conditioned linear system, solved using the preconditioned conjugate gradient (PCG) method. To address this cost, we efficiently construct scalable preconditioners using three different randomization techniques, which all rely on a certain low-rank structure involving the Gauss-Newton Hessian. The proposed techniques come with theoretical (probabilistic) guarantees on the condition number, and at the same time, are amenable to parallelization. We also develop an adaptive approach to estimate the sketch size and to choose between the reuse or recomputation of the preconditioner. We demonstrate the performance and effectiveness of our methodology on two representative model problems -- the Burgers and barotropic vorticity equation -- showing a drastic reduction in both the number of PCG iterations and the number of Gauss-Newton Hessian products (after including the preconditioner construction cost).

翻译：强约束四维变分数据同化（SC-4DVAR）是一种广泛应用于气候与天气预测的数据同化方法。该方法通过求解一个最小化问题来计算最大后验估计，我们采用高斯-牛顿法对此进行求解。由于涉及大规模且可能病态的线性系统（需通过预处理共轭梯度法求解），下降方向的计算成本高昂。为降低这一成本，我们利用三种不同的随机化技术高效构建可扩展的预处理算子，这些技术均依赖于高斯-牛顿海森矩阵的某种低秩结构。所提方法在条件数方面具有理论（概率性）保证，同时易于并行化。我们还开发了一种自适应方法来估计草图尺寸，并选择重用或重新计算预处理算子。我们通过两个具有代表性的模型问题——Burgers方程与正压涡度方程——验证了该方法的性能与有效性，结果表明，在考虑预处理算子构建成本后，PCG迭代次数及高斯-牛顿海森矩阵乘积次数均显著减少。