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| Compressive sensing enables accurate recovery of approximately sparse vectors from incomplete information. We apply this principle to the numerical solution of parametric partial differential equations (PDEs with random coefficients). In fact, one can show that the solution of certain parametric PDEs is analytic in the parameters which can be exploited to show convergence rates for nonlinear (sparse) approximation. Building on this fact, we show that methods from compressive sensing can be used to compute approximations from samples (snapshots) of the parametric PDEs, which in turn can be computed by standard methods for usual (nonparametric) PDEs. We provide theoretical approximation rates for this scheme. |
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