Persistent homology, the flagship tool from Topological Data Analysis (TDA) has been successfully utilized in many different domains despite the absence of a differentiation framework. Only recently a differential calculus has been defined on the space of persistence diagrams thus unlocking new possibilities for combining persistence with powerful solvers and optimizers. This work explores harnessing persistence differentiation for topologically driven data assimilation and for optimally navigating the parameter space of dynamical systems. Specifically, in Chapter 1, I give an overview of this thesis and present background on optimization and persistence optimization. In Chapter 2, I introduce a new topological data assimilation framework, and demonstrate the capabilities of this new method. In Chapter 3, I show how persistence-based cost functions can be constructed and used to optimally traverse the parameter space of a dynamical system. The cost functions are designed by specifying criteria that correspond to the structure of a desirable target persistence diagram while penalizing undesirable persistence features. Other applications of persistent homology are also presented in Chapter 4 where a texture analysis pipeline was developed to quantify specific features of a texture using TDA. Finally, in Chapter 5, I present a time delay framework for modeling metabolic oscillations in Yeast cells and numerical methods are used to locate parameters of the system that lead to limit cycles.
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