Numerical simulations of time-dependent problems : applied to epidemiology, artificial intelligence and finance

Résumé :
Numerical simulations encountered in epidemiology, artificial intelligence, and finance alike involve quantities that change over time, be it physical time or a pseudo-time encountered in numerical operations; these are called "time-dependent" problems, the subject of this book. After a brief introductory chapter, ordinary differential equations (ODEs) are addressed, with applications in epidemiology including the now famous SIR model; Runge-Kutta, multi-step and other numerical schemes are introduced and their stability and convergence are analyzed. Next follows the field of optimization with applications in machine learning and artificial intelligence through the notion of automatic differentiation; important objects such as backward computation and adjoint state are described. The final domain of applications concerns situations where uncertainty is present such as the log-normal model in mathematical finance; the numerical schemes of Euler-Maruyama and Milstein for stochastic differential equations (SDE) are introduced and their properties discussed. The book is complemented by exercises and computer implementations in Python as well as their solutions. This Master's level book builds from a course that the author has been teaching since 2005 and is aimed at students, researchers, and professionals working with numerical simulations where time is a variable or where operations unfold sequentially. Former student of ENS Ulm and subsequently research fellow at INRIA, Gabriel Turinici has been a Professor of applied mathematics at the Universit\'e Paris Dauphine - PSL since 2005. His research areas include quantum control, epidemiology, machine learning and finance.
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