Araújo, A.; Patrício, F.; Santos, José L. Optimal solution of a reaction–diffusion system with a control discrete source term. (English) Zbl 1210.65150 Int. J. Numer. Methods Biomed. Eng. 27, No. 2, 186-197 (2011). Summary: We study the numerical behavior of a reaction-diffusion system with a control source point. The main goal consists in estimating the position of the source point that maximizes a given objective function. To reduce the number of variables involved in the optimization algorithm, we first consider the problem with a fixed source point and then, according to the numerical results obtained, we estimate an approximation to the objective function, adjusting, by least squares, a special class of functions that depend on a few number of parameters. With this procedure we obtain an effective way to define the position of the source term. MSC: 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35K57 Reaction-diffusion equations 65K10 Numerical optimization and variational techniques 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 49J20 Existence theories for optimal control problems involving partial differential equations Keywords:finite differences; convergence; optimal control; reaction-diffusion system; optimization algorithm; numerical results; least squares PDFBibTeX XMLCite \textit{A. Araújo} et al., Int. J. Numer. Methods Biomed. Eng. 27, No. 2, 186--197 (2011; Zbl 1210.65150) Full Text: DOI References: [1] Araújo, Optimal solution of a diffusion equation with a discrete source term, Communications in Numerical Methods in Engineering [2] Gobbert, Long-time simulations on high resolution meshes to model calcium waves in a heart cell, SIAM Journal on Scientific Computing 30 (6) pp 2922– (2008) · Zbl 1178.92009 [3] Ashyraliyev M Blom JG Verwer JG On the numerical solution of diffusion-reaction equations with singular source terms 2005 [4] Mehne, A numerical method for solving optimal control problems using state parametrization, Numerical Algorithms 42 pp 165– (2006) · Zbl 1101.65071 [5] Bonnans JF Launay G Large scale direct optimal control applied to the re-entry problem 1994 [6] Dilão R The reaction-diffusion approach to morphogenesis 2005 325 364 [7] Beyer, Analysis of a one-dimensional model for the immersed boundary method, SIAM Journal on Numerical Analysis 29 pp 332– (1992) · Zbl 0762.65052 [8] Kandilarov, The immersed interface method for a nonlinear chemical diffusion equation with local sites of reactions, Numerical Algorithms 36 pp 285– (2004) · Zbl 1074.65102 [9] Kouachi, Existence of global solutions to reaction-diffusion systems with nonhomogeneous boundary conditins via Lyapunov functional, Electronic Journal of Differential Equations 88 pp 1– (2002) · Zbl 0988.35078 [10] Verwer, Convergence of method of lines approximations to partial differential equations, Computing 33 (3-4) pp 297– (1984) · Zbl 0546.65064 [11] Nash, Linear and Nonlinear Programming (1996) [12] Hardin, A new approach to the construction of optimal designs, Journal of Statistical Planning and Inference 37 pp 339– (1993) · Zbl 0799.62082 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.