Wavelet Formulation Research Articles

A Wavelet Optimized Adaptive Multi-domain Method

The formulation and implementation of wavelet based methods for the solution of multi-dimensional partial differential equations in complex geometries is discussed. Utilizing the close connection between Daubechies wavelets and finite difference methods on arbitrary grids, we formulate a wavelet based collocation method, well suited for dealing with general boundary conditions and nonlinearities. To circumvent problems associated with completely arbitrary grids and complex geometries we propose to use a multi-domain formulation in which to solve the partial differential equation, with the ability to adapt the grid as well as the order of the scheme within each subdomain. Besides supplying the required geometric flexibility, the multi-domain formulation also provides a very natural load-balanced data-decomposition, suitable for parallel environments. The performance of the overall scheme is illustrated by solving two dimensional hyperbolic problems.

Analysis and synthesis of feedforward neural networks using discrete affine wavelet transformations

A representation of a class of feedforward neural networks in terms of discrete affine wavelet transforms is developed. It is shown that by appropriate grouping of terms, feedforward neural networks with sigmoidal activation functions can be viewed as architectures which implement affine wavelet decompositions of mappings. It is shown that the wavelet transform formalism provides a mathematical framework within which it is possible to perform both analysis and synthesis of feedforward networks. For the purpose of analysis, the wavelet formulation characterizes a class of mappings which can be implemented by feedforward networks as well as reveals an exact implementation of a given mapping in this class. Spatio-spectral localization properties of wavelets can be exploited in synthesizing a feedforward network to perform a given approximation task. Two synthesis procedures based on spatio-spectral localization that reduce the training problem to one of convex optimization are outlined.