Refining simplex points for scalable estimation of the Lebesgue constant

To estimate the Lebesgue constant, we propose a point refinement method on the d-dimensional simplex. The proposed method features a smooth gradation of the point resolution, neighbor queries based on neighbor-aware coordinates, and a point refinement that algebraically scales as (d + 1) d. Remarkably, by using neighbor-aware coordinates, the point refinement method is ready to automatically stop using a Lipschitz criterion. For different polynomial degrees and point distributions, we show that our automatic method efficiently reproduces the literature estimations for the triangle and the tetrahedron. Moreover, we efficiently estimate the Lebesgue constant in higher dimensions. Accordingly, up to six dimensions, we conclude that the point refinement method is well-suited to efficiently estimate the Lebesgue constant on simplices. In perspective, for a given polynomial degree, the proposed point refinement method might be relevant to optimize a set of simplex points that guarantees a small interpolation error.
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 715546. This work has also received funding from the Generalitat de Catalunya under grant number 2017 SGR 1731. The work of the second author has been partially supported by Grant IJC2020-045140-I from MCIN/AEI/10.13039/501100011 033 and “European Union NextGenerationEU/PRTR”. The work of the third author has been partially supported by the Spanish Ministerio de Economía y Competitividad under the personal grant agreement RYC-2015-01633.
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