Parallel framework for dynamic domain decomposition of data assimilation problems: a case study on Kalman Filter algorithm

Abstract

We focus on Partial Differential Equation (PDE)-based Data Assimilation problems (DA) solved by means of variational approaches and Kalman filter algorithm. Recently, we presented a Domain Decomposition framework (we call it DD-DA, for short) performing a decomposition of the whole physical domain along space and time directions, and joining the idea of Schwarz's methods and parallel in time approaches. For effective parallelization of DD-DA algorithms, the computational load assigned to subdomains must be equally distributed. Usually computational cost is proportional to the amount of data entities assigned to partitions. Good quality partitioning also requires the volume of communication during calculation to be kept at its minimum. In order to deal with DD-DA problems where the observations are nonuniformly distributed and general sparse, in the present work we employ a parallel load balancing algorithm based on adaptive and dynamic defining of boundaries of DD-which is aimed to balance workload according to data location. We call it DyDD. As the numerical model underlying DA problems arising from the so-called discretize-then-optimize approach is the constrained least square model (CLS), we will use CLS as a reference state estimation problem and we validate DyDD on different scenarios.

Scalability analysis of a two level domain decomposition approach in space and time solving data assimilation models

“…In this way it leads to an “adaptive composition of local solvers” in which, following a tree configuration manner, its customization varies from sub‐problem to sub‐problem. Finally, it is worth to note that the proposed approach is non‐intrusive, allowing the incremental transition of existing software (as for instance, the Regional Ocean Modelling System‐ROMS) 19 . As depicted in Figure 1, the proposed approach consists in decomposing the domain of computation $$$ \Omega \times \Delta $$$ into subdomains in space and time and solving reduced forecast models and local 4DVAR DA problems as described in Reference 2.…”

Space-Time Decomposition of Kalman Filter

“…Among them, it makes it possible to apply deep-learning techniques to develop consistency constraints which will ensure that the solutions are physically meaningful even at the boundary of the small domains in the output of the local models [5,16,31]. Another possible extension could be the employment of a dynamic load balancing scheme based on adaptive and dynamic redefining of initial decomposition [7,10]. Specifically, in order to optimally choose the domain decomposition configuration, the partitioning into subdomains must satisfy certain conditions.…”

“…Good quality partitioning also requires the volume of communication during calculation to be kept at its minimum. In [7,10] the authors employed a dynamic load balancing scheme based on adaptive and dynamic redefining of initial decomposition, aimed to balance load between processors according to data location. In particular, the authors focused on the introduction of a dynamic redefining of initial DD in order to deal with problems where the observations are non uniformly distributed and general sparse.…”

“…Summarizing, we partition initial domain along space dimension and time steps, then we extend each subdomain to overlap its neighbors by an amount; partitioning can be adapted according to the availability of measurements and data [10]. Accord-ingly, we decompose forecasting model both in space and time.…”