PBDW - Parameterised-Background Data-Weak formulation
Contents
PBDW - Parameterised-Background Data-Weak formulation#
The Parameterised-Background Data-Weak (PBDW) was introduced in [17] as a practical algorithm to general variational data assimilation
It has been shown in [25] that this can be written as a weak formulation to be later converted into a linear system of small dimension. The state estimation can be written as a linear combination in the following way
in which the first summation represents the correction term related to the measurements, whereas the latter is the part arising from the reduced basis approximation of the snapshots space. The coefficients are the solution of the following linear system
provided the following definitions: let
The algorithm is implemented in OpenFOAM, applied to scalar fields only, the details of the implemented version of the formulation can be found in [21].
There are 2 folders containing the offline and online phase of the algorithm.
ScalarPBDW_Offline
ScalarPBDW_Online
The PBDW is a general framework to combine data and mathematical models approximated through reduced basis techniques, hence it can accomodate different techniques to generate the basis functions and the basis sensors.
In this work, the default option is given by the couple WeakGreedy+SGREEDY, alternatevely the greedy procedure of GEIM is used.
WeakGreedy algorithm#
The rationale behind this algorithm is quite similar to the GEIM one, and the main difference between the two stands in the generation of the basis functions.
The first iteration starts by selecting the generating function
and the correspondent basis function as
Then the main loop, where
where
SGREEDY algorithm#
This algorithm maximizes the inf-sup constant

Theorem 1 (Inf-Sup theorem)
The inf-sup constant
where the matrices are defined as