On the continuity of pullback attractors for evolution processes

In this paper we give general results on the continuity of pullback attractors for nonlinear evolution processes. We then revisit results of [D. Li, P.E. Kloeden, Equi-attraction and the continuous dependence of pullback attractors on parameters, Stoch. Dyn. 4 (3) (2004) 373-384] which show that, under certain conditions, continuity is equivalent to uniformity of attraction over a range of parameters (""equi-attraction""): we are able to simplify their proofs and weaken the conditions required for this equivalence to hold. Generalizing a classical autonomous result [A.V. Babin, M.I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992] we give bounds on the rate of convergence of attractors when the family is uniformly exponentially attracting. To apply these results in a more concrete situation we show that a non-autonomous regular perturbation of a gradient-like system produces a family of pullback attractors that are uniformly exponentially attracting: these attractors are therefore continuous, and we can give an explicit bound on the distance between members of this family. (C) 2009 Elsevier Ltd. All rights reserved.

Fiedler trees for multiscale surface analysis

In this work we introduce a new hierarchical surface decomposition method for multiscale analysis of surface meshes. In contrast to other multiresolution methods, our approach relies on spectral properties of the surface to build a binary hierarchical decomposition. Namely, we utilize the first nontrivial eigenfunction of the Laplace-Beltrami operator to recursively decompose the surface. For this reason we coin our surface decomposition the Fiedler tree. Using the Fiedler tree ensures a number of attractive properties, including: mesh-independent decomposition, well-formed and nearly equi-areal surface patches, and noise robustness. We show how the evenly distributed patches can be exploited for generating multiresolution high quality uniform meshes. Additionally, our decomposition permits a natural means for carrying out wavelet methods, resulting in an intuitive method for producing feature-sensitive meshes at multiple scales. Published by Elsevier Ltd.