[Paper Review] Optimal Control of a Parabolic Distributed Parameter System Using a Barycentric Shifted Gegenbauer Pseudospectral Method

Kareem T. Elgindy|arXiv (Cornell University)|Mar 4, 2016

Advanced Numerical Methods in Computational Mathematics2 references3 citations

TL;DR

This paper proposes a novel barycentric shifted Gegenbauer pseudospectral method (BSGPM) for solving optimal control problems governed by parabolic distributed parameter systems. By reformulating the problem into an integral form and using stable barycentric interpolation with shifted Gegenbauer-Gauss quadrature nodes, the method achieves exponential convergence with high accuracy using a minimal number of collocation points, outperforming existing methods in efficiency and accuracy.

ABSTRACT

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.

Motivation & Objective

• To develop a high-order, stable numerical method for solving optimal control problems governed by parabolic partial differential equations.
• To reduce the infinite-dimensional optimal control problem into a well-conditioned finite-dimensional nonlinear programming problem via integral reformulation and spectral discretization.
• To achieve exponential convergence rates with minimal collocation points through stable barycentric interpolation and shifted Gegenbauer quadratures.
• To improve upon existing methods, such as radial basis functions, by offering superior accuracy and computational efficiency.
• To provide a robust framework applicable to a broad class of PDE-constrained optimal control problems.

Proposed method

• The method reformulates the original PDE-constrained optimal control problem into an integral form using successive integration of the state and control variables.
• It employs shifted Gegenbauer-Gauss (SGG) points as collocation nodes, leveraging a stable barycentric representation of Lagrange interpolating polynomials with explicit barycentric weights.
• Integration matrices are constructed using shifted Gegenbauer quadratures to approximate integral operators accurately and efficiently.
• The infinite-dimensional problem is discretized into a finite-dimensional nonlinear programming problem with linear constraints, solvable via standard optimization solvers.
• The method ensures high-order accuracy by exploiting the spectral properties of Gegenbauer polynomials and the well-conditioning of barycentric interpolation.
• A rigorous error analysis is conducted, proving spectral decay of the error with increasing collocation points.

Experimental results

Research questions

• RQ1 Can a barycentric shifted Gegenbauer pseudospectral method achieve exponential convergence for parabolic optimal control problems?
• RQ2 How does the choice of the shape parameter α in shifted Gegenbauer polynomials affect the accuracy and stability of the solution?
• RQ3 Can the proposed method outperform radial basis function methods in terms of accuracy and computational cost for the same problem?
• RQ4 What is the convergence behavior of the method as the number of collocation points increases?
• RQ5 How well does the method preserve constraints such as initial and boundary conditions in the discrete formulation?

Key findings

• The BSGPM achieves exponential convergence rates, with the error decaying spectrally as the number of collocation points increases.
• Using only N=5 collocation points in each spatial and temporal direction, the method achieves accuracy comparable to radial basis function methods using 120 nodes in each direction.
• For small N, discretizations at SGG points with non-positive α-values (e.g., α = -0.2) yield the smallest maximum error in the initial condition (ψ₁), indicating superior accuracy for small-scale problems.
• The feasibility of the solution and the maximum boundary condition violation (ψ₂) remain near machine epsilon across all tested configurations, indicating strong constraint satisfaction.
• The approximate optimal cost functional Jₙ,ₙ* stabilizes around 15 for all tested N and α values, confirming numerical consistency.
• The method produces highly accurate state and control profiles, as visualized in Figures 2 and 3, even at N=12 with α=-0.2, demonstrating high-resolution approximation with minimal grid points.