A scalable space-time domain decomposition approach for solving large-scale nonlinear regularized inverse ill-posed problems in 4D variational data assimilation

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 models. We analyze the algorithm convergence and its performance in terms of reduction of time complexity and algorithmic scalability. The numerical experiments are carried out on the shallow water equation on the sphere according to the setup available at the Ocean Synthesis/Reanalysis Directory provided by Hamburg University.Comment: Received: 10 March 2020 / Revised: 29 November 2021 / Accepted: 7 March 202

A scalable approach for Variational Data Assimilation

Data assimilation (DA) is a methodology for combining mathematical models simulating complex systems (the background knowledge) and measurements (the reality or observational data) in order to improve the estimate of the system state (the forecast). The DA is an inverse and ill posed problem usually used to handle a huge amount of data, so, it is a large and computationally expensive problem. Here we focus on scalable methods that makes DA applications feasible for a huge number of background data and observations. We present a scalable algorithm for solving variational DA which is highly parallel. We provide a mathematical formalization of this approach and we also study the performance of the resulted algorith

Toward a fully parallel multigrid in time algorithm in PETSc environment: A case study in ocean models

We consider linear systems that arise from the discretization of evolutionary models. Typically, solution algorithms are based on a time-stepping approach, solving for one time step after the other. Parallelism is limited to the spatial dimension only. Because time is sequential in nature, the idea of simultaneously solving along time steps is not intuitive. One approach to achieve parallelism in time direction is MGRIT algorithm [7], based on multigrid reduction (MGR) techniques. Here we refer to this approach as MGR-1D. Other kind of approach is the space-time multigrid, where time is simply another dimension in the grid. Analougsly, we refer to this approach as MGR-4D. In this work, motivated by the need of maximizing the availability of new algorithms to climate science, we propose a new parallel approach that mixes both the MGR-1D idea and classical space multigrid methods. We refer to it as the MGR3D+1 approach. Moreover, we discuss their implementation in the high performance scientific library PETSc, as starting point to develope more efficient and scalable algorithms in ocean models

On the problem-decomposition of scalable 4D-Var Data Assimilation models

We present an innovative approach for solving Four Dimensional Variational Data Assimilation (4D-VAR DA) problems. The approach we consider starts from a decomposition of the physical domain; it uses a partitioning of the solution and a modified regularization functional describing the 4D-VAR DA problem on the decomposition. We provide a mathematical formulation of the model and we perform a feasibility analysis in terms of computational cost and of algorithmic scalability. We use the scale-up factor which measure the performance gain in terms of time complexity reduction. We verify the reliability of the approach on a consistent test case (the Shallow Water Equations)

FWPT filtering properties

A simple test case shows that effective restoration algorithms can be employed for image restoration by using a suitable preconditioner together with a regularization scheme. The filtering properties of fast wavelet transforms are investigated in order to improve: the image stored by PCG; the image restored by PCG and the regularized technique; the image restored by conjugate gradient (CG) method and the regularization technique; and the numerical performances of image restoration software tool