Numerical Differentiation of Stationary Measures of Chaos
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Numerical Differentiation of Stationary Measures of Chaos


In this thesis we develop two algorithms,the non-intrusive shadowing and the fast linear response algorithms, for computing derivatives of SRB measures with respect to some parameters of the dynamical system, where SRB measures are fractal limiting stationary measures of chaotic systems. The accurate formula of such derivative is the linear response formula. The non-intrusive shadowing algorithm previously devised by the author is one of the fastest approximate algorithm for differentiating chaos. It restricts computations to the unstable subspace, whose dimension can be much lower than the system.

In this thesis, we first set the theoretical foundation of the shadowing method by showing that it accurately computes the shadowing contribution of the linear response formula, and it well approximates the entire linear response for some important cases, such as high-dimensional systems with low-dimensional unstable subspaces. Then we develop the finite difference version of the non-intrusive shadowing, which we demonstrate on a computational fluid problem with about a million of dimensions: this is one of the first time sensitivity analysis being performed on such complicated systems. Then we consider adjoint shadowing algorithms. We give the explicit formula of the adjoint operator, and the non-intrusive characterization by only adjoint solutions. This leads to the non-intrusive adjoint shadowing algorithm, which generalizes the traditional back-propagation method to chaos. We demonstrate non-intrusive adjoint shadowing in a high-dimensional computational fluids problem.

Finally, we devise the fast linear response algorithm, for accurately computing the other part of the linear response, which is called the unstable contribution. We derive the first computable expansion formula of the unstable divergence, a central object in the linear response theory for fractal attractors. Then we give a `fast' characterization of the expansion by renormalized second-order tangent equations, whose second derivative is taken in a modified shadowing direction, computed by the non-intrusive shadowing algorithm. The new characterization makes the algorithm efficient and robust: its main cost is solving $u$, the unstable dimension, many first-order and second-order tangent equations, and it does not compute oblique projections. Moreover, the algorithm works for chaos on Riemannian manifolds with any $u$; its convergence to the true derivative is proved for uniform hyperbolic systems. The algorithm is illustrated on an example which is difficult for previous methods. The procedure list is easy to understand and implement.

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