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Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization from Part V - Applications, Developments, and Open Problems
Published online by Cambridge University Press: 10 October 2019
This chapter reviews the application of gamblets to the opening of the complexity bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficients. In this setting, the spatial component of the PDE is an elliptic operator on a Sobolev space, and since the natural truncationof the gamblet decomposition of the operator is rank-revealing, in that it is adapted to the eigensubspaces of the operator, these gamblet decompositionscan be naturally applied to the simulation of time-dependent operators. These gamblets are not only adapted to the coefficients of the underlying PDE but also to the numerical scheme used forits time discretization. For higher order PDEs these gamblets may be complex-valued.
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