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

Abstract

Modified Chebyshev-Picard iteration methods are presented for solving boundary value problems. Chebyshev polynomials are used to approximate the state trajectory in Picard iterations, while the boundary conditions are maintained by constraining the coefficients of the Chebyshev polynomials. Using Picard iteration and Clenshaw-Curtis quadrature, the presented methods iteratively refine an orthogonal function approximation ofthe entire state trajectory, in contrast to step-wise, forward integration approaches, which render the methods well-suited for parallel computation because computation of force functions along each path iteration can be rigorously distributed over many parallel cores with negligible cross communication needed. The presented methods solve optimal control problems through Pontryagin's principle without requiring shooting methods or gradient information. The methods are demonstrated to be computationally efficient and strikingly accurate when compared with Battin's method for a classical Lambert's problem and with a Chebyshev pseudo spectral method for an optimal trajectory design problem. The reported simulation results obtained on a serial machine suggest a strong basis for optimism of 'using the presented methods for solving more challenging boundary value problems, especially when highly parallel architectures are fully exploited.

Multiple Revolution Solutions for the Perturbed Lambert Problem using the Method of Particular Solutions and Picard Iteration

“…A decade later Bai and Junkins revisited this approach and developed improved algorithms for solving IVPs and TPBVPs [3,13]. They established new convergence insights and also developed vector-matrix formulations for solving initial and boundary value problems.…”

“…Two alternative methods for solving the perturbed Lambert's problem are the TPBVP implementation of modified Chebyshev-Picard iteration (MCPI) [3,4], and the MCPI-TPBVP algorithm regularized using the Kustaanheimo-Stiefel time transformation [5,6]. The first method converges over a small fraction of an orbit (about a third of an orbit depending on eccentricity) and the second method converges up to about 90% of an orbit.…”

Computational methods for nonlinear dynamical systems

“…For demonstration, the Chebyshev polynomials of the first kind ( ) n T τ are used as basis functions (Bai, andJunkins, 2011, Woollands, etc., 2015). Because the orthogonality of ( ) n T τ is valid for 1 1 τ − ≤ ≤ , time t is replaced by the rescaled time τ , where…”

“…Take the first order correctional formula for instance, one may simply collocate at the boundary point and add the boundary condition as a part of the correctional formula, thus lead to + = E U U describes the boundary condition of the problem. It should be noted that there could be other approaches to incorporate the boundary conditions, such as artificially adapting the first several coefficients of the basis functions in each iteration step, as is adopted by the MCPI method (Bai and Junkins, 2011, Woollands, etc., 2015, Fukushima, 1997.…”

“…Conversely, the second formula is not plagued by this problem because the boundary conditions are naturally satisfied. Obviously, if the basis functions are selected as the Chebyshev polynomials of the first kind, the second iterative formula stated above will give rise to the MCPI method (Bai and Junkins, 2011). The iteration formula works if only the collocations n U is updated in each step, so it is very simple and direct.…”