چكيده به لاتين
This thesis studies on stable methods for evaluating Radial basis functions (RBFs). RBFs are a powerful tool for approximating the solution of high-dimensional problems. They are often referred to as a meshfree method and can be spectrally accurate. The best accuracy can often be achieved when the so-called shape parameter of the basis functions is small, which in turn tends to make the interpolation matrix increasingly ill-conditioned. To overcome such instability in the numerical method, which arises from even the most stable problems, one needs to stabilize the method. In this thesis, we present a new stable method for evaluating Gaussian radial basis function interpolants based on the eigenfunction expansion for Gaussian RBFs. We develop our approach in one and two-dimensional spaces, with the extension to higher dimensions proceeding analogously. In the univariate setting, the orthogonality of the eigenfunctions and our special collocation locations give rise to easily computable cardinal basis functions. High accuracy, simple implementation and low complexity for high-dimensional problems are the advantages of our approach. Also, we investigate the numerical solution of two-dimensional time-dependent convection-diffusion-reaction equations with nonhomogeneous boundary conditions. We first approximate the equation in space by a stable Gaussian RBF method and obtain a matrix system of ODEs. The advantage of our method is that, by avoiding Kronecker products, this system can be solved by using one of the standard methods for ODEs. For the linear case, we show that the matrix system of ODEs becomes a Sylvester-type equation, and for the nonlinear case, we solve it by some predictor-corrector schemes like Adams-Bashforth and implicit explicit (IMEX) methods. The accuracy, robustness and computational efficiency of the method are tested by numerically solving several interpolations and boundary value problems in one and two dimensions like Poisson equation, Helmholtz equation, and time-dependent convection-diffusion-reaction equations.
