Abstract
We propose a compressive spectral collocation method for the numerical approximation of Partial Differential Equations (PDEs). The approach is based on a spectral Sturm-Liouville approximation of the solution and on the collocation of the PDE in strong form at randomized points, by taking advantage of the compressive sensing principle. The proposed approach makes use of a number of collocation points substantially less than the number of basis functions when the solution to recover is sparse or compressible. Focusing on the case of the diffusion equation, we prove that, under suitable assumptions on the diffusion coefficient, the matrix associated with the compressive spectral collocation approach satisfies the restricted isometry property of compressive sensing with high probability. Moreover, we demonstrate the ability of the proposed method to reduce the computational cost associated with the corresponding full spectral collocation approach while preserving good accuracy through numerical illustrations.
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Notes
- 1.
The substantial independence of the OMP recovery cost with respect to s for the full approach depends on two factors: the particular implementation of OMP in the package OMP-Box and the normalization step \(\widetilde {A} = A M^{-1}\) in Algorithm 1. In fact, in order to speed up the OMP iteration, the function omp of OMP-Box used to produce these results takes \(\widetilde {A}^T\widetilde {A}\) as input. When A is N × N, the cost of computing the matrices \(\widetilde {A}\) and \(\widetilde {A}^T\widetilde {A}\) is independent of s and it turns out to be consistently larger than the cost of OMP itself. As a result, the effect of s on the overall computational cost is negligible. The same remark holds for Fig. 4.
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Acknowledgements
The author acknowledges the support of the Natural Sciences and Engineering Research Council of Canada through grant number 611675 and the Pacific Institute for the Mathematical Sciences (PIMS) through the program “PIMS Postdoctoral Training Centre in Stochastics”. Moreover, the author gratefully acknowledge Ben Adcock and the anonymous reviewer for providing helpful comments on the first version of this manuscript.
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Brugiapaglia, S. (2020). A Compressive Spectral Collocation Method for the Diffusion Equation Under the Restricted Isometry Property. In: D'Elia, M., Gunzburger, M., Rozza, G. (eds) Quantification of Uncertainty: Improving Efficiency and Technology. Lecture Notes in Computational Science and Engineering, vol 137 . Springer, Cham. https://doi.org/10.1007/978-3-030-48721-8_2
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