64 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
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
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A boundary integral method for modelling vibroacoustic energy distributions in uncertain built up structures
A phase-space boundary integral method is developed for modelling stochastic high-frequency acoustic and vibrational energy transport in both single and multi-domain problems. The numerical implementation is carried out using the collocation method in both the position and momentum phase-space variables. One of the major developments of this work is the systematic convergence study, which demonstrates that the proposed numerical schemes exhibit convergence rates that could be expected from theoretical estimates under the right conditions. For the discretisation with respect to the momentum variable, we employ spectrally convergent basis approximations using both Legendre polynomials and Gaussian radial basis functions. The former have the advantage of being simpler to apply in general without the need for preconditioning techniques. The Gaussian basis is introduced with the aim of achieving more efficient computations in the weak noise case with near-deterministic dynamics. Numerical results for a series of coupled domain problems are presented, and demonstrate the potential for future applications to larger scale problems from industry
A bibliography on parallel and vector numerical algorithms
This is a bibliography of numerical methods. It also includes a number of other references on machine architecture, programming language, and other topics of interest to scientific computing. Certain conference proceedings and anthologies which have been published in book form are listed also
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Modified Fourier expansions: theory, construction and applications
Modified Fourier expansions present an alternative to more standard algorithms for the approximation of nonperiodic functions in bounded domains. This thesis addresses the theory of such expansions, their effective construction and computation, and their application to the numerical solution of partial differential equations.
As the name indicates, modified Fourier expansions are closely related to classical Fourier series. The latter are naturally defined in the d-variate cube, and, in an analogous fashion, we primarily study modified Fourier expansions in this domain. However, whilst Fourier coefficients are commonly computed with the Fast Fourier Transform (FFT), we use modern numerical quadratures instead. In contrast to the FFT, such schemes are adaptive, leading to great potential savings in computational cost.
Standard algorithms for the approximation of nonperiodic functions in -variate cubes exhibit complexities that grow exponentially with dimension. The aforementioned quadratures permit the design of approximations based on modified Fourier expansions that do not possess this feature. Consequently, such schemes are increasingly effective in higher dimensions. When applied to the numerical solution of boundary value problems, such savings in computational cost impart benefits over more commonly used polynomial-based methods. Moreover, regardless of the dimensionality of the problem, modified Fourier methods lead to well-conditioned matrices and corresponding linear systems that can be solved cheaply with standard iterative techniques.
The theoretical component of this thesis furnishes modified Fourier expansions with a convergence analysis in arbitrary dimensions. In particular, we prove uniform convergence of modified Fourier expansions under rather general conditions. Furthermore, it is known that the notion of modified Fourier expansions can be effectively generalised, resulting in a family of approximation bases sharing many of the features of the modified Fourier case. The purpose of such a generalisation is to obtain both faster rates and higher degrees of convergence. Having detailed the approximation-theoretic properties of modified Fourier expansions, we extend this analysis to the general case and thereby verify this improvement.
A central drawback of these expansions is that their convergence rate is both fixed and typically slow. This makes the construction of effective convergence acceleration techniques imperative. In the final part of this thesis, we design and analyse a robust method, applicable in arbitrary numbers of dimensions, for accelerating convergence of modified Fourier expansions. When employed in the approximation of multivariate functions, this culminates in efficient, high-order approximants comprising relatively small numbers of terms
Lectures on Computational Numerical Analysis of Partial Differential Equations
From Chapter 1:
The purpose of these lectures is to present a set of straightforward numerical methods with applicability to essentially any problem associated with a partial differential equation (PDE) or system of PDEs independent of type, spatial dimension or form of nonlinearity.https://uknowledge.uky.edu/me_textbooks/1002/thumbnail.jp
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