Chebyshev pseudospectral method
Encyclopedia
The Chebyshev pseudospectral method for optimal control
problems is based on Chebyshev polynomials of the first kind
. Unlike the Legendre pseudospectral method
, the Chebyshev pseudospectral (PS) method does not immediately offer high-accuracy quadrature solutions. Consequently, two different versions of the method have been proposed: one by Elnagar et al, and another by Fahroo and Ross. The two versions differ in their quadrature techniques. The Fahroo–Ross method is more commonly used today due to the ease in implementation of the Clenshaw–Curtis quadrature technique (in contrast to Elnagar-Kazemi's cell-averaging method). In 2008, Trefethen showed that the Clenshaw–Curtis method was nearly as accurate as Gauss quadrature.
This breakthrough result opened the door for a covector mapping theorem for Chebyshev PS methods. A complete mathematical theory for Chebyshev PS methods was finally developed in 2009 by Gong, Ross and Fahroo.
to solve optimal control
problems; however, none of these methods belong to the class of pseudospectral methods
.
Optimal control
Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and his collaborators in the Soviet Union and Richard Bellman in the United States.-General method:Optimal...
problems is based on Chebyshev polynomials of the first kind
Chebyshev polynomials
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...
. Unlike the Legendre pseudospectral method
Legendre pseudospectral method
The Legendre pseudospectral method for optimal control problems is based on Legendre polynomials. It is part of the larger pseudospectral optimal control theory, that was originally proposed by Elnagar and coworkers in 1995. Since then, Ross, Fahroo and co-workers have analysed, extended and...
, the Chebyshev pseudospectral (PS) method does not immediately offer high-accuracy quadrature solutions. Consequently, two different versions of the method have been proposed: one by Elnagar et al, and another by Fahroo and Ross. The two versions differ in their quadrature techniques. The Fahroo–Ross method is more commonly used today due to the ease in implementation of the Clenshaw–Curtis quadrature technique (in contrast to Elnagar-Kazemi's cell-averaging method). In 2008, Trefethen showed that the Clenshaw–Curtis method was nearly as accurate as Gauss quadrature.
This breakthrough result opened the door for a covector mapping theorem for Chebyshev PS methods. A complete mathematical theory for Chebyshev PS methods was finally developed in 2009 by Gong, Ross and Fahroo.
Other Chebyshev nethods
The Chebyshev PS method is frequently confused with other Chebyshev methods. Prior to the advent of PS methods, many authors proposed using Chebyshev polynomialsChebyshev polynomials
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...
to solve optimal control
Optimal control
Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and his collaborators in the Soviet Union and Richard Bellman in the United States.-General method:Optimal...
problems; however, none of these methods belong to the class of pseudospectral methods
Pseudo-spectral method
Pseudo-spectral methods are a class of numerical methods used in applied mathematics and scientific computing for the solution of PDEs, such as the direct simulation of a particle with an arbitrary wavefunction interacting with an arbitrary potential...
.