The application of optimal control techniques to the Shadowing approach for time averaged sensitivity analysis of chaotic systems

Abstract

Sensitivity analysis of non-linear systems is performed by finding solutions to the tangent/adjoint equations when the sensitivity of a function of interest is required with respect to single/multiple parameter(s). For chaotic systems, the simulation of the tangent/adjoint equations produces an unstable solution which leads to inaccurate sensitivities. There have been two main methods developed in the literature to overcome this limitation. The first are Shadowing methods, where their main feature is to find a tangent solution which remains bounded and shadows the original non-linear trajectory. This is achieved through modification of the initial condition to the tangent equation typically through a minimisation routine. These methods are typically computationally expensive whilst producing accurate sensitivity values. The second are the computationally cheap but potentially inaccurate numerical viscosity (NV) methods. NV methods stabilise the tangent/adjoint solution through incorporation of an additional control term into the tangent/adjoint equation that dissipates excess energy. Additional work is required when determining and tuning this additional control term, which leads to these methods being potentially inaccurate. The benefit of the NV method is that only one non-linear solution and one tangent/adjoint solution is required making them computationally cheap. There is, currently, a large gap in understanding of how these two methods relate and it is this question that this thesis attempts to answer. This thesis proposes a novel method to answer this question through introducing a generic control term into the tangent equation and finding its structure using optimal control techniques. This is referred to as Optimal Control Shadowing and abbreviated to OCS. The optimal control approach utilises a control term, similar to the control term in the NV methods, and finds its structure using a minimisation procedure, similar to the minimisation procedures used in the Shadowing approaches. Therefore, OCS can be seen as bridging the Shadowing and NV methods. Both tangent and adjoint forms of OCS have been developed for single and multiple parameters of interest, respectively. Computational methods are developed for solving the optimality equations in a matrix free sense which reduces the memory requirements of the algorithms. The result of this is that these methods become more tractable to large systems. Development of time parallel algorithms and preconditioning methods enabled the solution to be distributed in a parallel manner enabling distribution across multiple compute nodes further increasing tractability to large systems. One of the main findings of this thesis was that OCS finds a tangent solution very similar in structure to that of Multiple Shooting Shadowing (MSS), which is a leading Shadowing method. The NV methods investigated, on the other hand, did not produce any similarity with MSS suggesting the Shadowing direction was not found. Further, it was found that OCS predominantly targets the wavenumbers where production of tangent/adjoint energy dominated dissipation. This suggests that OCS applies control on the unstable sub-space and is minimal in the neutral and stable sub-space. The NV methods investigated here did not have this behaviour and were applying large amounts of control across all sub-spaces. The ramification of this is that for an NV method to be optimal, control should be applied mainly in the unstable sub-space. Further, the adjoint formulation applied significantly lower amounts of control than the tangent form. This was due to the terminal boundary condition being identical to that of MSS, and closer to the stabilised solution which required smaller amounts of control. The significance of this is that if an initial condition that is closer to the Shadowing direction for the tangent solution can be found than a smaller amount of control could be applied. A second major finding was that the Shadowing solution was primarily composed of near neutral covariant Lyapunov vectors (CLVs), ones whose Lyapunov exponent was close to zero. This suggests that these near neutral CLVs play an important role in the sensitivity of the system. This thesis also found that the solution generated by OCS is also mainly composed of the near neutral CLVs. This implies, again, that OCS and MSS are finding similar solutions and behave in a comparable manner. NV, on the other hand, did not show as large dominance of the near neutral CLVs suggesting that these methods are applying control in a less than optimal manner and are not targeting the unstable sub-space as well as OCS. A further result discovered in this thesis was that the expected gain, the ratio of the output to input, of the tangent/adjoint solution is proportional to the inverse of the Lyapunov exponent. The significance of this is that CLV modes whose exponents are close to zero will dominate the solution and as discovered previously, these relate to the near neutral CLVs. This suggests that selecting a control term in the NV method that has these features could produce a more accurate solution with minimal tuning.

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Identifiers

ORCID for Davide Lasagna: orcid.org/0000-0002-6501-6041

ORCID for Neil Sandham: orcid.org/0000-0002-5107-0944

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