Math @ Duke

Publications [#243791] of Mauro Maggioni
Papers Published
 Maggioni, M; Jr, JCB; Coifman, RR; Szlam, AD, Biorthogonal diffusion wavelets for multiscale representations on manifolds and graphs, edited by Manos Papadakis and Andrew F. Laine and Michael A. Unser,
Smart Structures and Materials 2005: Active Materials: Behavior and Mechanics, vol. 5914 no. 1
(2005),
pp. 113, SPIE, ISSN 0277786X [1], [doi]
(last updated on 2018/08/17)
Abstract: Recent work by some of the authors presented a novel construction of a multiresolution analysis on manifolds and graphs, acted upon by a given symmetric Markov semigroup {T t} t≥o, for which T t has low rank for large t. 1 This includes important classes of diffusionlike operators, in any dimension, on manifolds, graphs, and in nonhomogeneous media. The dyadic powers of an operator are used to induce a multiresolution analysis, analogous to classical LittlewoodPaley 14 and wavelet theory, while associated wavelet packets can also be constructed. 2 This extends multiscale function and operator analysis and signal processing to a large class of spaces, such as manifolds and graphs, with efficient algorithms. Powers and functions of T (notably its Green's function) are efficiently computed, represented and compressed. This construction is related and generalizes certain Fast Multipole Methods, 3 the wavelet representation of CalderónZygmund and pseudodifferential operators, 4 and also relates to algebraic multigrid techniques. 5 The original diffusion wavelet construction yields orthonormal bases for multiresolution spaces {V j}. The orthogonality requirement has some advantages from the numerical perspective, but several drawbacks in terms of the space and frequency localization of the basis functions. Here we show how to relax this requirement in order to construct biorthogonal bases of diffusion scaling functions and wavelets. This yields more compact representations of the powers of the operator, better localized basis functions. This new construction also applies to non selfadjoint semigroups, arising in many applications.


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