| Titre : | Stochastic Koopman Operator for Random Dynamical Systems |
| Auteurs : | Fatima Zohra Mokeddem, Auteur ; YAHIAOUI Lahcene, Auteur |
| Type de document : | texte manuscrit |
| Editeur : | [S.l.] : Université Saïda – Dr. Tahar Moulay – Faculté des Mathématiques, de l’Informatique et de Télécommunications, 2024/2025 |
| Format : | 69 ص |
| Accompagnement : | CD |
| Langues: | Anglais |
| Index. décimale : | BUC-M 003874 |
| Catégories : |
Master Mathématiques Specialization: Stochastic Statistical Analysis of Processes and Applications |
| Mots-clés: | stochastic Koopman operator: random dynamical systems:spectral analysis: semi- group theory: Hankel-DMD: dynamic mode decomposition: numerical approximation. |
| Résumé : |
Abstract
The stochastic Koopman operator characterizes the average evolution of observables in dynamical systems subject to uncertainty or noise. This study focuses on random dynamical systems (RDS) in both discrete and continuous time. We analyze the spectrum and eigenfunctions of several linear RDS. In particular, we examine cases where the family of operators forms a semigroup, which en- ables the definition of an associated generator. A stochastic Hankel-DMD algorithm is introduced to numerically approximate the Koopman eigenvalues and eigenfunctions, along with a proof of convergence. The method is applied to various examples, demonstrating spectral decomposition and model reduction. This approach extends deterministic Koopman theory to systems influenced by randomness. Moreover, we highlight the advantages of refined DMD algorithms in improving the accuracy of the approximated spectra. Conclusion This thesis thoroughly explores the framework of the stochastic Koopman operator, extending the classical Koopman operator theory to random dynamical systems. Given the intrinsic complexity of such systems, often shaped by inherent uncertainties, the objective was to provide theoretical and numerical tools for their linear analysis and the prediction of their observables. The theoretical foundations of the stochastic Koopman operator were first established. This included the formalization of its action on observables across various classes of random systems, ranging from deterministic systems perturbed by noise to stochastic differential equations (SDEs). This groundwork facilitated the characterization of the operator’s fundamental properties, such as linearity and its infinite-dimensional nature, thereby enabling subsequent applications and approx- imations. Building on this theoretical basis, numerical approximation methods for the stochastic Koop- man operator were developed and assessed. Particular emphasis was placed on adapting Dynamic Mode Decomposition (DMD) algorithms originally designed for deterministic systems [34, 35] to enhance robustness under stochastic influences. This adaptation allowed for the extraction of stochastic Koopman modes and eigenvalues, yielding critical insights into the underlying dynam- ics of random systems. The effectiveness and relevance of these numerical approaches were rig- orously validated through a series of examples, including both discrete cases (e.g., noisy rotation on the circle) and continuous ones (e.g., linear and nonlinear SDEs, noisy Van der Pol oscillator). These examples demonstrated the capacity of the methods to capture observable dynamics in the presence of noise. Finally, these tools were applied to practical problems to illustrate their usefulness in tasks such as long-term prediction of observables and the identification of coherent structures in com- plex random systems. These applications underscored the added value of the stochastic Koopman framework over traditional methods for analyzing random systems, particularly in terms of simpli- fying complex dynamics and enhancing predictive capabilities |
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Documents numériques (1)
BUC-M 003874 Adobe Acrobat PDF |
