We describe the mathematical representations, data structure and the implementation of the numerical calculus of functions in the software environment multiresolution analysis environment for scientific simulations, MADNESS. In MADNESS, each smooth function is represented using an adaptive pseudo-spectral expansion using the multiwavelet basis to a arbitrary but finite precision. This is an ext...

A novel approach for constructing symmetric compactly supported biorthogonal multiwavelets with short sequences is proposed in this paper. For some symmetric types of biorthogonal multiwavelet systems, starting from the symmetric properties of scaling and wavelet functions, parameterized symmetric forms of polyphase matrices can be derived. Furthermore, according to the matrix equations of the ...

We use the biorthogonal multiwavelets related by differentiation constructed in previous work to construct compactly supported biorthogonal multiwavelet bases for the space of vector fields on the upper half plane R+ such that the reconstruction wavelets are divergence-free and have vanishing normal components on the boundary of R+. Such wavelets are suitable to study the Navier–Stokes equation...

Adaptive Solution of Partial Differential Equations in Multiwavelet Bases B. Alpert,∗,1 G. Beylkin,†,2 D. Gines,† and L. Vozovoi‡,3,4,5 ∗National Institute of Standards and Technology, Boulder, Colorado 80305-3328; †Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309-0526; and ‡School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel E-mail: al...

| In applications using multiwavelets, there is a necessary step of associating a given discrete signal with a function in the scaling function space V 0. This association is equivalent to including a preelter and a posttlter for the lterbank determined by the underlying multiwavelets. We give a construction for orthogonal (paraunitary) FIR pre-lters that preserve the approximation order of a g...

Unshifted and shifted multiscaling functions are used as mathematical models for curve fitting of irregularly sampled data. This pre-processing procedure combined with multiwavelet neural networks for data-adaptive curve fitting is shown to perform well in the case of high resolution. In the case of low resolution it is more accurate than numerical integration and cheaper than matrix inversion....