Distributed approximating functionals, partial differential equations, and signal processing

This work deals with the distributed approximating functional (DAF) approach for solving partial differential equations (PDEs) and problems in signal processing. In the first part, the basic formalism of the DAF theory is presented and three concrete realizations of the DAFs are provided. The DAFs have the ability to approximate a function and its derivatives with a controllable accuracy, which is the principal reason for their success in solving PDEs, and especially for some nonlinear PDEs whose solutions are sensitive to the accuracy of approximating spatial derivatives. The second part concerns the DAF application to several PDEs encountered in science and engineering. We focus on using the DAF approach to discretize the spatial derivatives. It is shown that compared with other methods, for every problem we tested, our DAF-based approach can provide either the most accurate, or at least equally accurate, numerical solutions. Because of lower numerical accuracy, many numerical integrators tend to produce numerically induced phenomena. In contrast, our DAF-based approach can avoid these spurious phenomena for similar numerical parameters. The third part proposes several numerical algorithms to remove noise from one-dimensional signals and two-dimensional images, based on the well-tempered DAFs. In the case that the signal is band-limited in Fourier space and the noise is mainly in the high frequency region, direct convolution of the signal with the DAF kernel can be used to remove the noise. To avoid “aliasing” using this filtering technique, a well-tempered DAF-based padding scheme is employed to impose periodic boundary conditions prior to the convolution. Another application of the well-tempered DAFs is to remove noise from corrupted images. This is done by applying an anistrophic diffusion operator to the digital image data.