Computational methods inspired by chemistry: Multiscale modeling and mechanisms of control

The control of molecular systems inspired the two themes of this thesis: mechanism analysis and multiscale modeling. The goal of mechanism analysis is to understand and quantify the underlying mechanism of interaction governing laser controlled quantum systems. Despite successful optimal control simulations and experiments, little is known about the underlying mechanism. Mechanism is identified as the set of significant quantum pathways connecting the initial state of a system with a target state. The probability amplitudes of mechanistic pathways are determined by a procedure called Coding Hamiltonians to Access Mechanistic Pathways. Several coding schemes are proposed, and numerical simulations reveal high order multiphoton processes. The mechanisms are robust with respect to reasonable levels of noise.; Coding methods are adapted for laboratory use: only the electric field may be modulated and only observable data may be collected. Under ideal conditions, mechanism extraction requires the solution of a system of quadratic equations. Existence, uniqueness, and symmetry in the solutions are addressed. The system is shown to be overdetermined in most reasonable cases. An analytic solution is presented for a simple case, and an algorithm is given for use under the rotating wave approximation.; Shortcomings of the ideal setting are discussed, including noisy observable data, limited ability to modulate the electric field, and numerical challenges. A simulation demonstrates mechanism extraction from observable data.; The second part of the thesis studies multiscale algorithms produced within the Heterogeneous Multiscale Methods (HMM) framework for application to highly oscillatory ODES. A class of numerical ODE schemes that operate in two time scales is developed. The methods are applied to compute the average path of Kapitza's pendulum. A consistency condition for the initialization of microscopic data given the macroscopic state is derived. A proof and numerical simulations show that the proposed methods approximate the analytically averaged equation, and thus correctly compute the average path of the pendulum.; The final chapter extends these ideas to autonomous systems. The consistency condition must be calculated numerically because no explicit forcing exists. Simulations of a mass and spring system compare and evaluate the effectiveness of various HMM style algorithms in this setting.