Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 7 2 2022 09 01 ‎A Solution for Sparse PDE-Constrained Optimization by the Partition of Unity and RBFs 179 192 327 10.22128/gadm.2022.649.1089 EN Majid Darehmiraki Department of Mathematics, Behbahan Khatam Alanbia University of Technology, Khouzestan, Iran. Arezou Rezazadeh Department of mathematics, University of Qom, Qom, Iran. Journal Article 2023 01 30 ‎In this paper‎, ‎we propose a radial basis function partition of unity (RBF-PU) method to solve sparce optimal control problem governed by the elliptic equation‎.‏ The objective function, in addition to the usual quadratic expressions, also includes an ‎L₁-norm‎‎‎ of the control function to compute its spatio sparsity. ‎Meshless methods based on RBF approximation are widely used for solving PDE problems but PDE-constrained optimization problems have been barely solved by RBF methods‎. RBF methods have the benefits of being versatile in terms of geometry, simple to use in higher dimensions, and also having the ability to give spectral convergence. ‎In spite of these advantages‎, ‎when globally RBF collocation methods are used‎, ‎the interpolation matrix becomes dens and computational costs grow with increasing size of data set‎. ‎Thus‎, ‎for overcome on these problemes RBF-PU method will be proposed‎. ‎RBF‎ -‎PU methods reduce the computational effort‎. ‎The aim of this paper is to solve the first-order optimality conditions related to original problem‎.‎‎‎ https://ansne.du.ac.ir/article_327_576cefc998684120f5bdc64841517942.pdf