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The independent set perturbation method for efficient computation of sensitivities with applications to data assimilation and a finite element shallow water model. (English) Zbl 1391.76322
Summary: An adjoint model for a 2D Galerkin/Petrov-Galerkin finite element (FE) shallow water (S-W) model is developed using the Independent Set Perturbation (ISP) sensitivity analysis. Its performance in a full 4-D Var setup with a limited area shallow water equations model is assessed by comparing with the adjoint model derived by the automatic differentiation approach (TAMC), where it is used for optimising the initial conditions. It is shown that the ISP sensitivity analysis provides a very simple approach of forming the adjoint code/gradients/differentiation of discrete forward models (even complex governing equations, discretization methods and non-linear parameterizations) and is realised using a graph colouring approach combined with a perturbation method. Importantly, the adjoint is automatically updated as the forward code continues to be developed. In the test cases, it is shown that the adjoint model using the ISP sensitivity analysis can achieve the accuracy of traditional adjoint models derived by the automatic differentiation method (TAMC). Further comparison shows that the CPU time required for running the adjoint model using the ISP sensitivity analysis is much less than that required for the automatic differentiation derived adjoint model since the ISP derived adjoint CPU time scales linearly with the problem size.
The ISP sensitivity analysis is further applied to a highly non-linear Petrov-Galerkin FE model. The perturbation size used in deriving the tangent linear model with the ISP sensitivity analysis method is then optimised and the resulting approach used to assimilate both sparse (more realistic) and dense observational data for optimising the initial conditions. A simple first order formula is developed to calculate the perturbation size for each variable, at each node and time level. By applying the ISP sensitivity method to an intermediate complexity model (a shallow water model) this paper outlines steps towards applying the approach to data assimilation (DA) problems involving realistic complex models.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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