‎A Solution for Sparse PDE-Constrained Optimization by the Partition of Unity and RBFs

‎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.‎‎‎