15 research outputs found
Optimization via Chebyshev Polynomials
This paper presents for the first time a robust exact line-search method
based on a full pseudospectral (PS) numerical scheme employing orthogonal
polynomials. The proposed method takes on an adaptive search procedure and
combines the superior accuracy of Chebyshev PS approximations with the
high-order approximations obtained through Chebyshev PS differentiation
matrices (CPSDMs). In addition, the method exhibits quadratic convergence rate
by enforcing an adaptive Newton search iterative scheme. A rigorous error
analysis of the proposed method is presented along with a detailed set of
pseudocodes for the established computational algorithms. Several numerical
experiments are conducted on one- and multi-dimensional optimization test
problems to illustrate the advantages of the proposed strategy.Comment: 26 pages, 6 figures, 2 table
Optimal Control of a Parabolic Distributed Parameter System Using a Barycentric Shifted Gegenbauer Pseudospectral Method
In this paper, we introduce a novel pseudospectral method for the numerical
solution of optimal control problems governed by a parabolic distributed
parameter system. The infinite-dimensional optimal control problem is reduced
into a finite-dimensional nonlinear programming problem through shifted
Gegenbauer quadratures constructed using a stable barycentric representation of
Lagrange interpolating polynomials and explicit barycentric weights for the
shifted Gegenbauer-Gauss (SGG) points. A rigorous error analysis of the method
is presented, and a numerical test example is given to show the accuracy and
efficiency of the proposed pseudospectral method.Comment: 15 pages, 3 figure
High-Order Numerical Solution of Second-Order One-Dimensional Hyperbolic Telegraph Equation Using a Shifted Gegenbauer Pseudospectral Method
We present a high-order shifted Gegenbauer pseudospectral method (SGPM) to
solve numerically the second-order one-dimensional hyperbolic telegraph
equation provided with some initial and Dirichlet boundary conditions. The
framework of the numerical scheme involves the recast of the problem into its
integral formulation followed by its discretization into a system of
well-conditioned linear algebraic equations. The integral operators are
numerically approximated using some novel shifted Gegenbauer operational
matrices of integration. We derive the error formula of the associated
numerical quadratures. We also present a method to optimize the constructed
operational matrix of integration by minimizing the associated quadrature error
in some optimality sense. We study the error bounds and convergence of the
optimal shifted Gegenbauer operational matrix of integration. Moreover, we
construct the relation between the operational matrices of integration of the
shifted Gegenbauer polynomials and standard Gegenbauer polynomials. We derive
the global collocation matrix of the SGPM, and construct an efficient
computational algorithm for the solution of the collocation equations. We
present a study on the computational cost of the developed computational
algorithm, and a rigorous convergence and error analysis of the introduced
method. Four numerical test examples have been carried out in order to verify
the effectiveness, the accuracy, and the exponential convergence of the method.
The SGPM is a robust technique, which can be extended to solve a wide range of
problems arising in numerous applications.Comment: 36 pages, articl
High-Order, Stable, And Efficient Pseudospectral Method Using Barycentric Gegenbauer Quadratures
The work reported in this article presents a high-order, stable, and
efficient Gegenbauer pseudospectral method to solve numerically a wide variety
of mathematical models. The proposed numerical scheme exploits the stability
and the well-conditioning of the numerical integration operators to produce
well-conditioned systems of algebraic equations, which can be solved easily
using standard algebraic system solvers. The core of the work lies in the
derivation of novel and stable Gegenbauer quadratures based on the stable
barycentric representation of Lagrange interpolating polynomials and the
explicit barycentric weights for the Gegenbauer-Gauss (GG) points. A rigorous
error and convergence analysis of the proposed quadratures is presented along
with a detailed set of pseudocodes for the established computational
algorithms. The proposed numerical scheme leads to a reduction in the
computational cost and time complexity required for computing the numerical
quadrature while sharing the same exponential order of accuracy achieved by
Elgindy and Smith-Miles (2013). The bulk of the work includes three numerical
test examples to assess the efficiency and accuracy of the numerical scheme.
The present method provides a strong addition to the arsenal of numerical
pseudospectral methods, and can be extended to solve a wide range of problems
arising in numerous applications.Comment: 30 pages, 10 figures, 1 tabl
Fourier-Gegenbauer Pseudospectral Method for Solving Time-Dependent One-Dimensional Fractional Partial Differential Equations with Variable Coefficients and Periodic Solutions
In this paper, we present a novel pseudospectral (PS) method for solving a
new class of initial-value problems (IVPs) of time-dependent one-dimensional
fractional partial differential equations (FPDEs) with variable coefficients
and periodic solutions. A main ingredient of our work is the use of the
recently developed periodic RL/Caputo fractional derivative (FD) operators with
sliding positive fixed memory length of Bourafa et al. [1] or their reduced
forms obtained by Elgindy [2] as the natural FD operators to accurately model
FPDEs with periodic solutions. The proposed method converts the IVP into a
well-conditioned linear system of equations using the PS method based on
Fourier collocations and Gegenbauer quadratures. The reduced linear system has
a simple special structure and can be solved accurately and rapidly by using
standard linear system solvers. A rigorous study of the error and convergence
of the proposed method is presented. The idea and results presented in this
paper are expected to be useful in the future to address more general problems
involving FPDEs with periodic solutions.Comment: 13 pages, 3 figures. arXiv admin note: text overlap with
arXiv:2304.0445
New Optimal Periodic Control Policy for the Optimal Periodic Performance of a Chemostat Using a Fourier-Gegenbauer-Based Predictor-Corrector Method
In its simplest form, the chemostat consists of microorganisms or cells which
grow continually in a specific phase of growth while competing for a single
limiting nutrient. Under certain conditions on the cells' growth rate,
substrate concentration, and dilution rate, the theory predicts and numerical
experiments confirm that a periodically operated chemostat exhibits an
"over-yielding" state in which the performance becomes higher than that at the
steady-state operation. In this paper we show that an optimal control policy
for maximizing the chemostat performance can be accurately and efficiently
derived numerically using a novel class of integral-pseudospectral methods and
adaptive h-integral-pseudospectral methods composed through a
predictor-corrector algorithm. Some new formulas for the construction of
Fourier pseudospectral integration matrices and barycentric shifted Gegenbauer
quadratures are derived. A rigorous study of the errors and convergence rates
of shifted Gegenbauer quadratures as well as the truncated Fourier series,
interpolation operators, and integration operators for nonsmooth and generally
T-periodic functions is presented. We introduce also a novel adaptive scheme
for detecting jump discontinuities and reconstructing a discontinuous function
from the pseudospectral data. An extensive set of numerical simulations is
presented to support the derived theoretical foundations.Comment: 35 pages, 20 figure
Fourier-Gegenbauer Pseudospectral Method for Solving Periodic Fractional Optimal Control Problems
This paper introduces a new accurate model for periodic fractional optimal
control problems (PFOCPs) using Riemann-Liouville (RL) and Caputo fractional
derivatives (FDs) with sliding fixed memory lengths. The paper also provides a
novel numerical method for solving PFOCPs using Fourier and Gegenbauer
pseudospectral methods. By employing Fourier collocation at equally spaced
nodes and Fourier and Gegenbauer quadratures, the method transforms the PFOCP
into a simple constrained nonlinear programming problem (NLP) that can be
treated easily using standard NLP solvers. We propose a new transformation that
largely simplifies the problem of calculating the periodic FDs of periodic
functions to the problem of evaluating the integral of the first derivatives of
their trigonometric Lagrange interpolating polynomials, which can be treated
accurately and efficiently using Gegenbauer quadratures. We introduce the
notion of the {\alpha}th-order fractional integration matrix with index L based
on Fourier and Gegenbauer pseudospectral approximations, which proves to be
very effective in computing periodic FDs. We also provide a rigorous priori
error analysis to predict the quality of the Fourier-Gegenbauer-based
approximations to FDs. The numerical results of the benchmark PFOCP demonstrate
the performance of the proposed pseudospectral method.Comment: 10 pages, 11 figure
Direct Integral Pseudospectral and Integral Spectral Methods for Solving a Class of Infinite Horizon Optimal Output Feedback Control Problems Using Rational and Exponential Gegenbauer Polynomials
This study is concerned with the numerical solution of a class of
infinite-horizon linear regulation problems with state equality constraints and
output feedback control. We propose two numerical methods to convert the
optimal control problem into nonlinear programming problems (NLPs) using
collocations in a semi-infinite domain based on rational Gegenbauer (RG) and
exponential Gegenbauer (EG) basis functions. We introduce new properties of
these basis functions and derive their quadratures and associated truncation
errors. A rigorous stability analysis of the RG and EG interpolations is also
presented. The effects of various parameters on the accuracy and efficiency of
the proposed methods are investigated. The performance of the developed
integral spectral method is demonstrated using two benchmark test problems
related to a simple model of a divert control system and the lateral dynamics
of an F-16 aircraft. Comparisons of the results of the current study with
available numerical solutions show that the developed numerical scheme is
efficient and exhibits faster convergence rates and higher accuracy.Comment: 27 pages, 24 figure
A Direct Integral Pseudospectral Method for Solving a Class of Infinite-Horizon Optimal Control Problems Using Gegenbauer Polynomials and Certain Parametric Maps
We present a novel direct integral pseudospectral (PS) method (a direct IPS
method) for solving a class of continuous-time infinite-horizon optimal control
problems (IHOCs). The method transforms the IHOCs into finite-horizon optimal
control problems (FHOCs) in their integral forms by means of certain parametric
mappings, which are then approximated by finite-dimensional nonlinear
programming problems (NLPs) through rational collocations based on Gegenbauer
polynomials and Gegenbauer-Gauss-Radau (GGR) points. The paper also analyzes
the interplay between the parametric maps, barycentric rational collocations
based on Gegenbauer polynomials and GGR points, and the convergence properties
of the collocated solutions for IHOCs. Some novel formulas for the construction
of the rational interpolation weights and the GGR-based integration and
differentiation matrices in barycentric-trigonometric forms are derived. A
rigorous study on the error and convergence of the proposed method is
presented. A stability analysis based on the Lebesgue constant for GGR-based
rational interpolation is investigated. Two easy-to-implement pseudocodes of
computational algorithms for computing the barycentric-trigonometric rational
weights are described. Two illustrative test examples are presented to support
the theoretical results. We show that the proposed collocation method leveraged
with a fast and accurate NLP solver converges exponentially to near-optimal
approximations for a coarse collocation mesh grid size. The paper also shows
that typical direct spectral/PS- and IPS-methods based on classical Jacobi
polynomials and certain parametric maps usually diverge as the number of
collocation points grow large, if the computations are carried out using
floating-point arithmetic and the discretizations use a single mesh grid
whether they are of Gauss/Gauss-Radau (GR) type or equally-spaced.Comment: 33 pages, 19 figure
Distributed optimal control of viscous Burgers' equation via a high-order, linearization, integral, nodal discontinuous Gegenbauer-Galerkin method
We developed a novel direct optimization method to solve distributed optimal control of viscous Burgers' equation over a finite-time horizon by minimizing the distance between the state function and a desired target state profile along with the energy of the control. Through a novel linearization strategy, well-conditioned integral reformulations, optimal Gegenbauer barycentric quadratures, and nodal discontinuous Galerkin discretizations, the method reduces such optimal control problems into finite-dimensional, nonlinear programming problems subject to linear algebraic system of equations and discrete mixed path inequality constraints that can be solved easily using standard optimization software. The proposed method produces "an auxiliary control function" that provides a useful model to explicitly define the optimal controller of the state variable. We present an error analysis of the semidiscretization and full discretization of the weak form of the reduced equality constraint system equations to demonstrate the exponential convergence of the method. The accuracy of the proposed method is examined using two numerical examples for various target state functions in the existence/absence of control bounds. The proposed method is exponentially convergent in both space and time, thus producing highly accurate approximations using a significantly small number of collocation points