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Discontinuity Detection by Null Rules for Adaptive Surface Reconstruction

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Abstract

We present a discontinuity detection method based on the so-called null rules, computed as a vector in the null space of certain collocation matrices. These rules are used as weights in a linear combination of function evaluations to indicate the local behavior of the function itself. By analyzing the asymptotic properties of the rules, we introduce two indicators (one for discontinuities of the function and one for discontinuities of its gradient) by locally computing just one rule. This leads to an efficient and reliable scheme, which allows us to effectively detect and classify points close to discontinuities. We then show how this information can be suitably combined with adaptive approximation methods based on hierarchical spline spaces in the reconstruction process of surfaces with discontinuities. The considered adaptive methods exploit the ability of the hierarchical spaces to be locally refined, and fault detection is a natural way to guide the refinement with low computational cost. A selection of test cases is presented to show the effectiveness of our approach.

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Data Availability

The paper contains all the information to simply generate the datasets supporting the conclusions of this article, except for the geological dataset, courtesy of Giacomo Corti (Istituto di Geoscienze e Georisorse CNR, Florence - Italy).

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Acknowledgements

Cesare Bracco, Francesco Calabrò and Carlotta Giannelli are members of INdAM-GNCS and are partially supported by INdAM-GNCS Project (CUP E53C22001930001). Cesare Bracco and Carlotta Giannelli are also partially supported by the National Recovery and Resilience Plan, Mission 4 Component 2 - Investment 1.4 - NATIONAL CENTER FOR HPC, BIG DATA AND QUANTUM COMPUTING - funded by the European Union - NextGenerationEU - (CUP B83C22002830001).

Funding

This work was partially supported by the Istituto Nazionale di Alta Matematica - Gruppo Nazionale per il Calcolo Scientifico (INdAM-GNCS), Italy. Open Access funding enabled and organized by Italy Transformative Agreement.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica “Ulisse Dini”, Università degli Studi di Firenze, Florence, Italy

    Cesare Bracco & Carlotta Giannelli

  2. Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli “Federico II”, Naples, Italy

    Francesco Calabrò

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  1. Cesare Bracco
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  2. Francesco Calabrò
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Correspondence to Cesare Bracco.

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Bracco, C., Calabrò, F. & Giannelli, C. Discontinuity Detection by Null Rules for Adaptive Surface Reconstruction. J Sci Comput 97, 37 (2023). https://doi.org/10.1007/s10915-023-02348-6

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