四维变分资料同化（4D-Var）已成为一种重要方法，广泛应用于数值天气预报、海洋建模和气候预测领域。经典的无约束基于梯度的算法常陷入局部极小值，导致其数值性能对初始猜测高度敏感。本研究利用4D-Var问题的可分结构，提出了一种实用的交替方向乘子法（ADMM，alternating direction method of multipliers）变体，称为带正则化的线性化多块ADMM。与主要关注初始条件的经典一阶优化方法不同，我们的方法推导了整个动力系统的欧拉-拉格朗日方程，能够更全面、更有效地利用观测数据。当初始条件选择不佳时，arg min运算会将迭代引向观测数据，从而降低对初始猜测的敏感性。二次子问题进一步简化了求解过程，而并行结构则提升了计算效率，尤其是在使用现代硬件时。为验证我们的方法，我们利用Lorenz系统证明了其优越性能，即使在存在噪声观测数据的情况下也是如此。此外，我们展示了带正则化的线性化多块ADMM在求解粘性Burgers方程的4D-Var问题中的有效性，涵盖了有限差分、有限元和谱方法等多种数值方案。最后，我们在二维湍流场景下，特别是聚焦于涡量集中的情况，展示了噪声观测数据下的动力学恢复过程，凸显了我们的算法在处理复杂物理现象时的鲁棒性。
英文摘要：
The four-dimensional variational data assimilation (4D-Var) has emerged as an important methodology, widely used in numerical weather prediction, oceanographic modeling, and climate forecasting. Classical unconstrained gradient-based algorithms often struggle with local minima, making their numerical performance highly sensitive to the initial guess. In this study, we exploit the separable structure of the 4D-Var problem to propose a practical variant of the alternating direction method of multipliers (ADMM), referred to as the linearized multi-block ADMM with regularization. Unlike classical first-order optimization methods that primarily focus on initial conditions, our approach derives the Euler-Lagrange equation for the entire dynamical system, enabling more comprehensive and effective utilization of observational data. When the initial condition is poorly chosen, the arg min operation steers the iteration towards the observational data, thereby reducing sensitivity to the initial guess. The quadratic subproblems further simplify the solution process, while the parallel structure enhances computational efficiency, especially when utilizing modern hardware. To validate our approach, we demonstrate its superior performance using the Lorenz system, even in the presence of noisy observational data. Furthermore, we showcase the effectiveness of the linearized multi-block ADMM with regularization in solving the 4D-Var problems for the viscous Burgers' equation, across various numerical schemes, including finite difference, finite element, and spectral methods. Finally, we illustrate the recovery of dynamics under noisy observational data in a 2D turbulence scenario, particularly focusing on vorticity concentration, highlighting the robustness of our algorithm in handling complex physical phenomena.
Accelerated Stochastic ADMM with Variance Reduction
Computer Science Institute, Zhejiang University（浙江大学计算机学院）；Department of Respiratory Diseases, The First Affiliated Hospital, College of Medicine Zhejiang University（浙江大学医学院附属第一医院呼吸科）
We prove consistence, convergence and stability of the Domain Decomposition in space and time method of 4DVAR Data Assimilation problem. We introduce the condition number of the problem and validate the theoretical analysis through numerical results
In this paper, we develop a symmetric accelerated stochastic Alternating Direction Method of Multipliers (SAS-ADMM) for solving separable convex optimization problems with linear constraints. The objective function is the sum of a possibly nonsmooth convex function and an average function of many smooth convex functions. Our proposed algorithm combines both ideas of ADMM and the techniques of accelerated stochastic gradient methods possibly with variance reduction to solve the smooth subproblem. One main feature of SAS-ADMM is that its dual variable is symmetrically updated after each update o…
Accurate estimation of error covariances (both background and observation) is crucial for efficient observation compression approaches in data assimilation of large-scale dynamical problems. We propose a new combination of a covariance tuning algorithm with existing PCA-type data compression approaches, either observation- or information-based, with the aim of reducing the computational cost of real-time updating at each assimilation step. Relying on a local assumption of flow-independent error covariances, dynamical assimilation residuals are used to adjust the covariance in each assimilation…
Data assimilation is a method that combines observations (that is, real world data) of a state of a system with model output for that system in order to improve the estimate of the state of the system and thereby the model output. The model is usually represented by a discretised partial differential equation. The data assimilation problem can be formulated as a large scale Bayesian inverse problem. Based on this interpretation we will derive the most important variational and sequential data assimilation approaches, in particular three-dimensional and four-dimensional variational data assimil…
We investigate the techniques and ideas used in the convergence analysis of two proximal ADMM algorithms for solving convex optimization problems involving compositions with linear operators. Besides this, we formulate a variant of the ADMM algorithm that is able to handle convex optimization problems involving an additional smooth function in its objective, and which is evaluated through its gradient. Moreover, in each iteration we allow the use of variable metrics, while the investigations are carried out in the setting of infinite dimensional Hilbert spaces. This algorithmic scheme is inves…
We mathematically derive the sensitivity of data assimilation results for tsunami modelling, to perturbations in the observation operator. We consider results of variational data assimilation schemes on the one dimensional shallow water equations for (i) initial condition reconstruction, and (ii) bathymetry detection as presented in Kevlahan et al. (2019, 2020). We use variational methods to derive the Hessian of a cost function J representing the error between forecast solutions and observations. Using this Hessian representation and methods outlined by Shutyaev et al. (2017, 2018), we mathem…
Domain Decomposition of 4D-VAR Data Assimilation (DD-4DVAR) is made up of decomposition of the spate-time domain, solution of reduced forecast model and minimization of local 4D-VAR functionals. Relying on the existing software implementation of ROMS software, we describe main components of DD-4D VAR DA method, highlighting the topics that we will should address both on the mathematical problem underlying ROMS and the MPI-based code implementation of the ROMS-IS4DVAR formulation.
An Introduction to Solving the Least-Squares Problem in Variational Data Assimilation
CERFACS（欧洲研究计算流体力学中心）；University of Potsdam（波茨坦大学）；School of Mathematical, Physical and Computational Sciences, University of Reading（雷丁大学数学、物理与计算科学学院）；National Centre for Earth Observation（国家地球观测中心）；Department of Mathematics and Statistics, University of Strathclyde（斯特拉斯克莱德大学数学与统计系）；Scientific Computing Department, Science & Technology Facilities Council, Rutherford Appleton Laboratory（卢瑟福阿普顿实验室科学计算部）；Eindhoven University of Technology（埃因霍温理工大学）
Virginia Tech（弗吉尼亚理工大学）；Oden Institute for Computational Engineering and Sciences, University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学研究所）
Data assimilation is the process of fusing information from imperfect computer simulations with noisy, sparse measurements of reality to obtain improved estimates of the state or parameters of a dynamical system of interest. The data assimilation procedures used in many geoscience applications, such as numerical weather forecasting, are variants of the our-dimensional variational (4D-Var) algorithm. The cost of solving the underlying 4D-Var optimization problem is dominated by the cost of repeated forward and adjoint model runs. This motivates substituting the evaluations of the physical model…
Data assimilation algorithms combine prior and observational information, weighted by their respective uncertainties, to obtain the most likely posterior of a dynamical system. In variational data assimilation the posterior is computed by solving a nonlinear least squares problem. Many numerical weather prediction (NWP) centres use full observation error covariance (OEC) weighting matrices, which can slow convergence of the data assimilation procedure. Previous work revealed the importance of the minimum eigenvalue of the OEC matrix for conditioning and convergence of the unpreconditioned data…
We develop innovative algorithms for solving the strong-constraint formulation of four-dimensional variational data assimilation in large-scale applications. We present a space-time decomposition approach that employs domain decomposition along both the spatial and temporal directions in the overlapping case and involves partitioning of both the solution and the operators. Starting from the global functional defined on the entire domain, we obtain a type of regularized local functionals on the set of subdomains providing the order reduction of both the predictive and the data assimilation mode…
A Revised Scheme to Compute Horizontal Covariances in an Oceanographic 3D-VAR Assimilation System
Centro Euro-Mediterraneo sui Cambiamenti Climatici（欧洲-地中海气候变化中心）；Department of Mathematics and Applications, University of Naples ”Federico II”（那不勒斯费德里科二世大学数学与应用系）
We consider the large-sparse symmetric linear systems of equations that arise in the solution of weak constraint four-dimensional variational data assimilation, a method of high interest for numerical weather prediction. These systems can be written as saddle point systems with a 3x3 block structure but block eliminations can be performed to reduce them to saddle point systems with a 2x2 block structure, or further to symmetric positive definite systems. In this paper, we analyse how sensitive the spectra of these matrices are to the number of observations of the underlying dynamical system. W…
In this paper we propose the use of a continuous data assimilation algorithm for miscible flow models in a porous medium. In the absence of initial conditions for the model, observed sparse measurements are used to generate an approximation to the true solution. Under certain assumption of the sparse measurements and their incorporation into the algorithm it can be shown that the resulting approximate solution converges to the true solution at an exponential rate as time progresses. Various numerical examples are considered in order to validate the suitability of the algorithm.
Variational data assimilation for the wave equation in heterogeneous media: Numerical investigation of stability
University College London（伦敦大学学院）；Inria Project-Team Makutu, Université de Pau et des Pays de l’Adour（法国国家信息与自动化研究所 Makutu 项目组，波城与阿杜尔地区大学）；Institute for Numerical and Applied Mathematics, University of Göttingen（哥廷根大学数值与应用数学研究所）
