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Published July 1, 2019 | public
Journal Article

Operator-adapted wavelets for finite-element differential forms


We introduce in this paper an operator-adapted multiresolution analysis for finite-element differential forms. From a given continuous, linear, bijective, and self-adjoint positive-definite operator L, a hierarchy of basis functions and associated wavelets for discrete differential forms is constructed in a fine-to-coarse fashion and in quasilinear time. The resulting wavelets are L-orthogonal across all scales, and can be used to derive a Galerkin discretization of the operator such that its stiffness matrix becomes block-diagonal, with uniformly well-conditioned and sparse blocks. Because our approach applies to arbitrary differential p-forms, we can derive both scalar-valued and vector-valued wavelets block-diagonalizing a prescribed operator. We also discuss the generality of the construction by pointing out that it applies to various types of computational grids, offers arbitrary smoothness orders of basis functions and wavelets, and can accommodate linear differential constraints such as divergence-freeness. Finally, we demonstrate the benefits of the corresponding operator-adapted multiresolution decomposition for coarse-graining and model reduction of linear and non-linear partial differential equations.

Additional Information

© 2019 Elsevier Inc. Received 10 September 2018, Revised 26 January 2019, Accepted 3 February 2019, Available online 15 March 2019. H. Owhadi gratefully acknowledges support by the Air Force Office of Scientific Research and the DARPA EQUiPS Program under award number FA9550-16-1-0054 (Computational Information Games), as well as the Air Force Office of Scientific Research under award number FA9550-18-1-0271 (Games for Computation and Learning). M. Desbrun gratefully acknowledges partial support from Pixar Animation Studios, and thanks Jiong Chen and Yiying Tong for various discussions on the topic.

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