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Title: Model-order reductions for stochastic differential equations
Author: Pangerl, Christian Josef
ISNI:       0000 0004 7963 8209
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2019
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The thesis at hand focuses on approaches towards model-order reduction for stochastic differential equations (SDEs). In particular we consider manifold-based reduction strategies, which rely on a splitting of phase space into subspaces in which the (linearly) stable and (linearly) unstable variables, respectively, assume their values. One then proposes a suitable manifold, which serves as parameterisation of the stable variables in terms of the unstable variables. This manifold can be used to obtain a lower-dimensional system of SDEs for the unstable variables only. It is of interest to compare the qualitative properties of the reduced system to those of the original system. In particular one can ask the following questions: Which features of the original system are preserved in the obtained reduced system and what determines the quality of the reduction? In order to provide an answer to these questions we firstly investigate a so-called ABC system. This SDE is three dimensional having two (linearly) unstable variables denoted by A and B and one (linearly) stable variable denoted by C. We apply the theory of Taylor approximations to random invariant manifolds, which provides us with candidates for suitable parameterisations of the stable variable C in terms of A and B. Based on recent results obtained by Chekroun et al., we introduce the notion of local parameterising manifolds in expectation, which relate the energy unexplained by the parameterisation to the overall energy in the stable variable. This notion enables us to quantify the suitability of a given manifold for reducing the ABC system. The first set of results then establish conditions under which certain examples of random and deterministic manifolds discussed in the literature can be shown to also constitute local parameterising manifolds in expectation. We then consider a three dimensional slow-fast system and take advantage of the associated slow manifold to arrive at a lower-dimensional reduced model. We then seek to derive an upper bound on the discrepancy between appropriate marginals of the invariant measures for the original and reduced system, respectively. More specifically, for the employed metric we show an upper bounded in terms of an additive constant and a quantity capturing the expected deviation between the fast variable and its prediction by means of the slow manifold. Similar results are obtained for stochastic generalisations of the slow manifold. We also prove analogous bounds for a complementary class of SDEs using a simplified approach. In summary we find evidence that quantities related to the parameterisation defect are connected to long-term approximation properties of the reduced relative to the original system on the level of invariant measures.
Supervisor: Rasmussen, Martin ; Lamb, Jeroen S. W. Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral