Cvetkovic, Nada
[Author]
;
Conrad, Tim
[Contributor];
Stoltz, Gabriel
[Contributor]
Convergent discretisation schemes for transition path theory for diffusion processes ; Konvergente Diskretisierungsschemen für transition path Theorie für Diffusionsprozesse
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Media type:
Electronic Thesis;
E-Book;
Doctoral Thesis
Title:
Convergent discretisation schemes for transition path theory for diffusion processes ; Konvergente Diskretisierungsschemen für transition path Theorie für Diffusionsprozesse
Contributor:
Cvetkovic, Nada
[Author]
imprint:
Freie Universität Berlin: Refubium (FU Berlin), 2020
Footnote:
Diese Datenquelle enthält auch Bestandsnachweise, die nicht zu einem Volltext führen.
Description:
Many applications involve analysing dynamical systems that undergo rare transitions between two metastable subsets $A$ and $B$ of state space. For example, in clinical disease modelling, one can model a patient's state as a metastable stochastic process that occasionally transits from a 'healthy' subset $A$ to a 'diseased' subset $B$. We develop a numerical method for analysing the statistics of transitions of a diffusion process between the two disjoint subsets of the bounded state space. Transition path theory (TPT) was first formulated to solve this problem for ergodic diffusion processes. The main object of this theory is a committor function, which at any state of the state space is defined as the probability of reaching the set $B$ before reaching the set $A$, conditioned on starting from the given state. The computation of the committor function requires solving a second order partial differential equation involving the generator of the process. Therefore, TPT for diffusion processes requires the knowledge of the stochastic differential equation governing the process. Furthermore, the high dimensional nature of many problems makes solving such partial differential equations difficult in practice. In order to perform computations, a discretisation method is needed. TPT for Markov jump processes has been developed for this reason. However, space discretisation results in a loss of the Markov property. We discretise the state space using Voronoi tessellations and model the underlying diffusion process by a non-Markovian jump process on the associated Delaunay graph. To this process we associate the analogues of the committor function, isocommittor surfaces, the probability current and streamlines. These objects are the key objects in both TPT for diffusion processes and TPT for Markov jump processes. In this thesis, we define these objects for the non-Markovian jump process described above. All of the objects we define can be computed using sampled trajectories, thus our approach is completely data-driven ...