Meshless Galerkin methods using radial basis functions
Mathematics of Computation
On unsymmetric collocation by radial basis functions
Applied Mathematics and Computation
A meshless Galerkin method for Dirichlet problems using radial basis functions
Journal of Computational and Applied Mathematics
A note on the meshless method using radial basis functions
Computers & Mathematics with Applications
Meshless Collocation: Error Estimates with Application to Dynamical Systems
SIAM Journal on Numerical Analysis
Radial basis function meshless method for the steady incompressible Navier-Stokes equations
International Journal of Computer Mathematics
Computers & Mathematics with Applications
Meshfree Approximation Methods with MATLAB
Meshfree Approximation Methods with MATLAB
Optimal Solvers for PDE-Constrained Optimization
SIAM Journal on Scientific Computing
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In this article, we apply the theory of meshfree methods to the problem of PDE-constrained optimization. We derive new collocation-type methods to solve the distributed control problem with Dirichlet boundary conditions and also discuss the Neumann boundary control problem, both involving Poisson's equation. We prove results concerning invertibility of the matrix systems we generate, and discuss a modification to guarantee invertibility. We implement these methods using Matlab, and produce numerical results to demonstrate the methods' capability. We also comment on the methods' effectiveness in comparison to the widely-used finite element formulation of the problem, and make some recommendations as to how this work may be extended.