Reduced Basis Methods: From Key Ingredients to 4DVar

12.07.2023, 10:15 - 11:15  –
Forschungsseminar Numerik/Angewandte Mathematik/Statistik

Martin Grepl (RWTH Aachen)

The reduced basis method is a certified model order reduction technique for the rapid and reliable solution of parametrized partial differential equations (PDEs), and it is especially suited for the many-query, real-time, and slim computing contexts. 
In the first part of this seminar, we recall the essential reduced basis ingredients for linear affine elliptic problems: (i) Galerkin projection onto a subspace spanned by solutions of the governing equation at N greedily selected points in parameter space, (ii) residual based a posteriori error estimation procedures to provide rigorous and sharp bounds for the error, (iii) offline-online computational procedures to decouple the generation and evaluation stage of the reduced basis method, and (iv) the greedy algorithm. We also briefly mention the extension to time-dependent (parabolic) problems.

In the second part, we present a certified RB approach to four dimensional variational data assimilation (4D-VAR). We consider the particular case in which the behaviour of the system is modelled by a parametrised parabolic partial differential equation where the initial condition and model parameters (e.g., material or geometric properties) are unknown, and where the model itself may be imperfect. We consider the standard strong-constraint as well as the weak-constraint 4DVar formulation. Since the model error is a distributed function in both space and time, the 4D-Var formulation generally leads to a large-scale optimization problem that must be solved for every given parameter instance. We introduce reduced basis spaces for the state, adjoint, initial condition, and model error. We then build upon results on RB methods for optimal control problems in order to derive a posteriori error estimates for RB approximations to solutions of the 4DVAR problem. Numerical tests are conducted to verify the validity of the proposed approach.

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