‎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.‎‎‎