If computation has emerged alongside theory and experimentation as the third pillar of science, then parallel computing is the strut that supports the ever increasing demands of computational science. My dissertation focuses on the discovery and application of novel parallel algorithms for time-independent (static) Hamilton-Jacobi (HJ) equations. These equations arise in physical, mechanical, and dynamical systems and often present unique computational challenges. The computational methods presented here make possible the solution of problems that were previously intractable. Advancements in algorithms and applications are reciprocal; interesting applications drive computational advances and efficient algorithms facilitate new areas of research. In addition to the parallel algorithms I present the formulation of isochrons of a dynamical system as the solution of a linear HJ equation. The solution of this equation for high dimensional systems was previously impractical.

The Fast Sweeping Method (FSM) was invented to solve the Eikonal equation, a specific case of the general HJ equation. This problem has applications in optimal control, image processing, and the study of seismic waves. The FSM is a clever finite difference method by which the solution to the Eikonal equation is computed in O(N) time, where N is the total number of points in the uniform grid. This method, however, inherently sequential in nature; a parallel computing strategy is not straightforward. Previously existing parallel approaches add additional computational expense and are, therefore, suboptimal. My dissertation presents an optimal parallel algorithm that preserves the exact efficiency of the serial method. Numerical results show high parallel efficiency on shared memory computer architecture.

The modern supercomputer consists of an array of distributed compute nodes each with their own shared memory. This heterogeneous architecture demands a method specifically tailored to its strengths for maximum efficiency. I have developed a hybrid massively parallel fast sweeping method to take advantage of this computing power. In demonstrating this extreme computational power I have applied the method to a wide array of HJ problems including seismic wave propagation, optimal control, and dynamic games. Theoretical estimates give a factor of 2.0-2.5 improvement over previous state-of-the-art methods and numerical results confirm. On hundreds or thousands of processors, this is a significant improvement.

Another strategy for increasing the efficiency and effectiveness of a finite difference methods is to use nonuniform grids like tree-based data structures or triangulated meshes. Fast sweeping methods exist for these types of grids, but similar to the uniform case, they are strictly serial in design. I have developed a graph-theoretic approach for analyzing an arbitrary grid and determining an optimal parallel algorithm. This can be used on any serial FSM on any type of grid and give a parallel ordering. I conduct numerical experiments and again, show high parallel efficiency on shared memory computer architecture.

Lastly, I will present a new mathematical approach for computing isochrons of a dynamical system with a stable periodic orbit. Many useful dynamical systems are in high dimensions and are prohibitively expensive to compute solutions. The parallel algorithms previously mentioned make the computation of solution to these problems feasible. I show numerical results giving the isochrons of the Hodgkin-Huxley model of a squid axon and the Wilson-Callaway model of a dopaminergic neuron. The knowledge of the isochrons of these systems can be used to inform control approaches, for instance the control of neurons in the human brain for the treatment of Parkinson's disease.