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Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error
The ensemble Kalman filter and its variants have shown to be robust for data assimilation in high dimensional geophysical models, with localization, using ensembles of extremely small size relative to the model dimension. However, a reduced rank representation of the estimated covariance leaves a large dimensional complementary subspace unfiltered. Utilizing the dynamical properties of the filtration for the backward Lyapunov vectors, this paper explores a previously unexplained mechanism, providing a novel theoretical interpretation for the role of covariance inflation in ensemble-based Kalman filters. Our derivation of the forecast error evolution describes the dynamic upwelling of the unfiltered error from outside of the span of the anomalies into the filtered subspace. Analytical results for linear systems explicitly describe the mechanism for the upwelling, and the associated recursive Riccati equation for the forecast error, while nonlinear approximations are explored numerically
Fully adaptive structure-preserving hyper-reduction of parametric Hamiltonian systems
Model order reduction provides low-complexity high-fidelity surrogate models
that allow rapid and accurate solutions of parametric differential equations.
The development of reduced order models for parametric nonlinear Hamiltonian
systems is still challenged by several factors: (i) the geometric structure
encoding the physical properties of the dynamics; (ii) the slowly decaying
Kolmogorov -width of conservative dynamics; (iii) the gradient structure of
the nonlinear flow velocity; (iv) high variations in the numerical rank of the
state as a function of time and parameters. We propose to address these aspects
via a structure-preserving adaptive approach that combines symplectic dynamical
low-rank approximation with adaptive gradient-preserving hyper-reduction and
parameters sampling. Additionally, we propose to vary in time the dimensions of
both the reduced basis space and the hyper-reduction space by monitoring the
quality of the reduced solution via an error indicator related to the
projection error of the Hamiltonian vector field. The resulting adaptive
hyper-reduced models preserve the geometric structure of the Hamiltonian flow,
do not rely on prior information on the dynamics, and can be solved at a cost
that is linear in the dimension of the full order model and linear in the
number of test parameters. Numerical experiments demonstrate the improved
performances of the resulting fully adaptive models compared to the original
and reduced order models
Nonlinear model order reduction via Dynamic Mode Decomposition
We propose a new technique for obtaining reduced order models for nonlinear
dynamical systems. Specifically, we advocate the use of the recently developed
Dynamic Mode Decomposition (DMD), an equation-free method, to approximate the
nonlinear term. DMD is a spatio-temporal matrix decomposition of a data matrix
that correlates spatial features while simultaneously associating the activity
with periodic temporal behavior. With this decomposition, one can obtain a
fully reduced dimensional surrogate model and avoid the evaluation of the
nonlinear term in the online stage. This allows for an impressive speed up of
the computational cost, and, at the same time, accurate approximations of the
problem. We present a suite of numerical tests to illustrate our approach and
to show the effectiveness of the method in comparison to existing approaches
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