Modified Chebyshev-Picard Iteration Methods for Solution of Initial Value and Boundary Value Problems

Abstract

The solution of initial value problems (IVPs) provides the evolution of dynamic\nsystem state history for given initial conditions. Solving boundary value problems\n(BVPs) requires finding the system behavior where elements of the states are defined\nat different times. This dissertation presents a unified framework that applies modified\nChebyshev-Picard iteration (MCPI) methods for solving both IVPs and BVPs.\nExisting methods for solving IVPs and BVPs have not been very successful in\nexploiting parallel computation architectures. One important reason is that most\nof the integration methods implemented on parallel machines are only modified versions\nof forward integration approaches, which are typically poorly suited for parallel\ncomputation.\nThe proposed MCPI methods are inherently parallel algorithms. Using Chebyshev\npolynomials, it is straightforward to distribute the computation of force functions\nand polynomial coefficients to different processors. Combining Chebyshev polynomials\nwith Picard iteration, MCPI methods iteratively refine estimates of the solutions\nuntil the iteration converges. The developed vector-matrix form makes MCPI methods\ncomputationally efficient.\nThe power of MCPI methods for solving IVPs is illustrated through a small perturbation\nfrom the sinusoid motion problem and satellite motion propagation problems.\nCompared with a Runge-Kutta 4-5 forward integration method implemented in MATLAB, MCPI methods generate solutions with better accuracy as well as orders\nof magnitude speedups, prior to parallel implementation. Modifying the algorithm\nto do double integration for second order systems, and using orthogonal polynomials\nto approximate position states lead to additional speedups. Finally, introducing\nperturbation motions relative to a reference motion results in further speedups.\nThe advantages of using MCPI methods to solve BVPs are demonstrated by\naddressing the classical Lambert’s problem and an optimal trajectory design problem.\nMCPI methods generate solutions that satisfy both dynamic equation constraints and\nboundary conditions with high accuracy. Although the convergence of MCPI methods\nin solving BVPs is not guaranteed, using the proposed nonlinear transformations,\nlinearization approach, or correction control methods enlarge the convergence domain.\nParallel realization of MCPI methods is implemented using a graphics card that\nprovides a parallel computation architecture. The benefit from the parallel implementation\nis demonstrated using several example problems. Larger speedups are achieved\nwhen either force functions become more complicated or higher order polynomials are\nused to approximate the solutions.