Error Analyses for Nyström Methods for Solving Fredholm Integral and Integro-Differential Equations

This thesis concerns the development and implementation of novel error analyses for ubiquitous Nyström-type methods used in approximating the solution in 1-D of both Fredholm integral- and integro-differential equations of the second-kind, (FIEs) and (FIDEs). The distinctive contribution of the present work is that it offers a new systematic procedure for predicting, to spectral accuracy, error bounds in the numerical solution of FIEs and FIDEs when the solution is, as in most practical applications, a priori unknown. The classic Legendre-based Nyström method is extended through Lagrange interpolation to admit solution of FIEs by collocation on any nodal distribution, in particular, those that are optimal for not only integration but also differentiation. This offers a coupled extension of optimal-error methods for FIEs into those for FIDEs. The so-called FIDE-Nyström method developed herein motivates yet another approach in which (demonstrably ill-conditioned) numerical differentiation is bypassed by reformulating FIDEs as hybrid Volterra-Fredholm integral equations (VFIEs). A novel approach is used to solve the resulting VFIEs that utilises Lagrange interpolation and Gaussian quadrature for the Volterra and Fredholm components respectively. All error bounds implemented for the above numerical methods are obtained from novel, often complex extensions of an established but hitherto-unimplemented theoretical Nyström-error framework. The bounds are computed using only the available computed numerical solution, making the methods of practical value in, e.g., engineering applications. For each method presented, the errors in the numerical solution converge (sometimes exponentially) to zero with N, the number of discrete collocation nodes; this rate of convergence is additionally confirmed via large-N asymptotic estimates. In many cases these bounds are spectrally accurate approximations of the true computed errors; in those cases that the bounds are not, the non-applicability of the theory can be predicted either a priori from the kernel or a posteriori from the numerical solution

A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces

In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in $\mathbb{R}^d$. For two-dimensional surfaces embedded in $\mathbb{R}^3$, these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions (RBFs) and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at "scattered" locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surface-based metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented

Sparse approximation of multilinear problems with applications to kernel-based methods in UQ

We provide a framework for the sparse approximation of multilinear problems and show that several problems in uncertainty quantification fit within this framework. In these problems, the value of a multilinear map has to be approximated using approximations of different accuracy and computational work of the arguments of this map. We propose and analyze a generalized version of Smolyak's algorithm, which provides sparse approximation formulas with convergence rates that mitigate the curse of dimension that appears in multilinear approximation problems with a large number of arguments. We apply the general framework to response surface approximation and optimization under uncertainty for parametric partial differential equations using kernel-based approximation. The theoretical results are supplemented by numerical experiments