Cacciapuoti, Rosalba (2022) Domain decomposition approaches for multiscale/multiphysics data assimilation. [Tesi di dottorato]

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Abstract

In this thesis, we present a Domain Decomposition (DD)-based parallel framework for solution of large scale variational (VAR) 3D, 4D Data Assimilation (DA) and Kalman Filter (KF) problems. The proposed approaches perform a decomposition of the whole physical domain, i.e. both along spatial and temporal directions; a reduction in space and time of both the Partial Differential Equations-based model and the DA functional; finally it uses a modified regularization cost functional describing restricted 3DVAR DA, 4DVAR DA and KF problems on DD (we call them DD-DA methods). Innovation of mainly lies in the introduction ab initio, i.e. on the numerical model – of a DD approach in space and time joining the idea of Schwarz’s method and Parallel in Time (PinT)–based approaches. We address DA problems where the observations are non uniformly distributed, general sparse and its distribution changes during the time window, consequently, we present Dynamic Domain Decomposition (DyDD) and Dynamic Domain Decomposition in Space and Time (DyDDST) methods to employ a load balancing scheme involving an adaptive and dynamic workload redistribution both along Space and Time directions for solving Data Assimilation problems. Main advantage of the DD–DA coupled with DyDD or DyDDST is to combine in the same theoretical framework model reduction, along the space and time directions, with filter localization, while providing a flexible, adaptive and dynamic decomposition. A validation analysis is performed discussing computational results on case studies relying on Shallow Water Equations and Constrained Least Square problem.

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