79 research outputs found
Fast Mesh Refinement in Pseudospectral Optimal Control
Mesh refinement in pseudospectral (PS) optimal control is embarrassingly easy
--- simply increase the order of the Lagrange interpolating polynomial and
the mathematics of convergence automates the distribution of the grid points.
Unfortunately, as increases, the condition number of the resulting linear
algebra increases as ; hence, spectral efficiency and accuracy are lost in
practice. In this paper, we advance Birkhoff interpolation concepts over an
arbitrary grid to generate well-conditioned PS optimal control discretizations.
We show that the condition number increases only as in general, but
is independent of for the special case of one of the boundary points being
fixed. Hence, spectral accuracy and efficiency are maintained as increases.
The effectiveness of the resulting fast mesh refinement strategy is
demonstrated by using \underline{polynomials of over a thousandth order} to
solve a low-thrust, long-duration orbit transfer problem.Comment: 27 pages, 12 figures, JGCD April 201
The Breadth-one -invariant Polynomial Subspace
We demonstrate the equivalence of two classes of -invariant polynomial
subspaces introduced in [8] and [9], i.e., these two classes of subspaces are
different representations of the breadth-one -invariant subspace. Moreover,
we solve the discrete approximation problem in ideal interpolation for the
breadth-one -invariant subspace. Namely, we find the points, such that the
limiting space of the evaluation functionals at these points is the functional
space induced by the given -invariant subspace, as the evaluation points all
coalesce at one point
A Universal Birkhoff Theory for Fast Trajectory Optimization
Over the last two decades, pseudospectral methods based on Lagrange
interpolants have flourished in solving trajectory optimization problems and
their flight implementations. In a seemingly unjustified departure from these
highly successful methods, a new starting point for trajectory optimization is
proposed. This starting point is based on the recently-developed concept of
universal Birkhoff interpolants. The new approach offers a substantial
computational upgrade to the Lagrange theory in completely flattening the rapid
growth of the condition numbers from O(N2) to O(1), where N is the number of
grid points. In addition, the Birkhoff-specific primal-dual computations are
isolated to a well-conditioned linear system even for nonlinear, nonconvex
problems. This is part I of a two-part paper. In part I, a new theory is
developed on the basis of two hypotheses. Other than these hypotheses, the
theoretical development makes no assumptions on the choices of basis functions
or the selection of grid points. Several covector mapping theorems are proved
to establish the mathematical equivalence between direct and indirect Birkhoff
methods. In part II of this paper (with Proulx), it is shown that a select
family of Gegenbauer grids satisfy the two hypotheses required for the theory
to hold. Numerical examples in part II illustrate the power and utility of the
new theory
Implementations of the Universal Birkhoff Theory for Fast Trajectory Optimization
This is part II of a two-part paper. Part I presented a universal Birkhoff
theory for fast and accurate trajectory optimization. The theory rested on two
main hypotheses. In this paper, it is shown that if the computational grid is
selected from any one of the Legendre and Chebyshev family of node points, be
it Lobatto, Radau or Gauss, then, the resulting collection of trajectory
optimization methods satisfy the hypotheses required for the universal Birkhoff
theory to hold. All of these grid points can be generated at an
computational speed. Furthermore, all Birkhoff-generated
solutions can be tested for optimality by a joint application of Pontryagin's-
and Covector-Mapping Principles, where the latter was developed in Part~I. More
importantly, the optimality checks can be performed without resorting to an
indirect method or even explicitly producing the full differential-algebraic
boundary value problem that results from an application of Pontryagin's
Principle. Numerical problems are solved to illustrate all these ideas. The
examples are chosen to particularly highlight three practically useful features
of Birkhoff methods: (1) bang-bang optimal controls can be produced without
suffering any Gibbs phenomenon, (2) discontinuous and even Dirac delta covector
trajectories can be well approximated, and (3) extremal solutions over dense
grids can be computed in a stable and efficient manner
The linear algebra of interpolation with finite applications giving computational methods for multivariate polynomials
Thesis (Ph.D.) University of Alaska Fairbanks, 1988Linear representation and the duality of the biorthonormality relationship express the linear algebra of interpolation by way of the evaluation mapping. In the finite case the standard bases relate the maps to Gramian matrices. Five equivalent conditions on these objects are found which characterize the solution of the interpolation problem. This algebra succinctly describes the solution space of ordinary linear initial value problems. Multivariate polynomial spaces and multidimensional node sets are described by multi-index sets. Geometric considerations of normalization and dimensionality lead to cardinal bases for Lagrange interpolation on regular node sets. More general Hermite functional sets can also be solved by generalized Newton methods using geometry and multi-indices. Extended to countably infinite spaces, the method calls upon theorems of modern analysis
Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection
Matrix functions are a central topic of linear algebra, and problems of their numerical approximation appear increasingly often in scientific computing. We review various rational Krylov methods for the computation of large-scale matrix functions. Emphasis is put on the rational Arnoldi method and variants thereof, namely, the extended Krylov subspace method and the shift-and-invert Arnoldi method, but we also discuss the nonorthogonal generalized Leja point (or PAIN) method. The issue of optimal pole selection for rational Krylov methods applied for approximating the resolvent and exponential function, and functions of Markov type, is treated in some detail
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