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Published October 15, 2017 | Submitted
Journal Article Open

Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficients


Implicit schemes are popular methods for the integration of time dependent PDEs such as hyperbolic and parabolic PDEs. However the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application to PDEs with rough coefficients. We present a generalization of gamblets introduced in [62] enabling the resolution of these implicit systems in near-linear complexity and provide rigorous a-priori error bounds on the resulting numerical approximations of hyperbolic and parabolic PDEs. These generalized gamblets induce a multiresolution decomposition of the solution space that is adapted to both the underlying (hyperbolic and parabolic) PDE (and the system of ODEs resulting from space discretization) and to the time-steps of the numerical scheme.

Additional Information

© 2017 Elsevier Inc. Received 24 June 2016, Revised 16 June 2017, Accepted 24 June 2017, Available online 28 June 2017. H. Owhadi gratefully acknowledges the support of the Air Force Office of Scientific Research and the DARPA EQUiPS Program under award number FA9550-16-1-0054 (Computational Information Games). L. Zhang gratefully acknowledges the support of the National Natural Science Foundation of China grant 11471214 and the One Thousand Plan of China for young scientists. The authors also thank two anonymous referees for comments and suggestions.

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