176 research outputs found
Sequential quadrature methods for RDO
Abstract
This paper presents a comparative study between a large number of different existing sequential quadrature schemes suitable for Robust Design Optimization (RDO), with the inclusion of two partly original approaches. Efficiency of the different integration strategies is evaluated in terms of accuracy and computational effort: main goal of this paper is the identification of an integration strategy able to provide the integral value with a prescribed accuracy using a limited number of function samples. Identification of the different qualities of the various integration schemes is obtained utilizing both algebraic and practical test cases. Differences in the computational effort needed by the different schemes is evidenced, and the implications on their application to practical RDO problems is highlighted
Improved Laguerre Spectral Methods with Less Round-off Errors and Better Stability
Laguerre polynomials are orthogonal polynomials defined on positive half line
with respect to weight . They have wide applications in scientific and
engineering computations. However, the exponential growth of Laguerre
polynomials of high degree makes it hard to apply them to complicated systems
that need to use large numbers of Laguerre bases. In this paper, we introduce
modified three-term recurrence formula to reduce the round-off error and to
avoid overflow and underflow issues in generating generalized Laguerre
polynomials and Laguerre functions. We apply the improved Laguerre methods to
solve an elliptic equation defined on the half line. More than one thousand
Laguerre bases are used in this application and meanwhile accuracy close to
machine precision is achieved. The optimal scaling factor of Laguerre methods
are studied and found to be independent of number of quadrature points in two
cases that Laguerre methods have better convergence speeds than mapped Jacobi
methods.Comment: 19pages, 8 figure
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Numerical Solution of Linear Ordinary Differential Equations and Differential-Algebraic Equations by Spectral Methods
This thesis involves the implementation of spectral methods, for numerical solution of linear Ordinary Differential Equations (ODEs) and linear Differential-Algebraic Equations (DAEs). First we consider ODEs with some ordinary problems, and then, focus on those problems in which the solution function or some coefficient functions have singularities. Then, by expressing weak and strong aspects of spectral methods to solve these kinds of problems, a modified pseudo-spectral method which is more efficient than other spectral methods is suggested and tested on some examples.
We extend the pseudo-spectral method to solve a system of linear ODEs and linear DAEs and compare this method with other methods such as Backward Difference Formulae (BDF), and implicit Runge-Kutta (RK) methods using some numerical examples. Furthermore, by using appropriate choice of Gauss-Chebyshev-Radau points, we will show that this method can be used to solve a linear DAE whenever some of coefficient functions have singularities by providing some examples. We also used some problems that have already been considered by some authors by finite difference methods, and compare their results with ours.
Finally, we present a short survey of properties and numerical methods for solving DAE problems and then we extend the pseudo-spectral method to solve DAE problems with variable coefficient functions. Our numerical experience shows that spectral and pseudo-spectral methods and their modified versions are very promising for linear ODE and linear DAE problems with solution or coefficient functions having singularities.
In section 3.2, a modified method for solving an ODE is introduced which is new work. Furthermore, an extension of this method for solving a DAE or system of ODEs which has been explained in section 4.6 of chapter four is also a new idea and has not been done by anyone previously.
In all chapters, wherever we talk about ODE or DAE we mean linear
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
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