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Sparse Data-Driven Quadrature Rules via ℓ p -Quasi-Norm Minimization

  • Mattia Manucci EMAIL logo , Jose Vicente Aguado and Domenico Borzacchiello
Published/Copyright: February 15, 2022
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Abstract

This paper is concerned with the use of the focal underdetermined system solver to recover sparse empirical quadrature rules for parametrized integrals from existing data. This algorithm, originally proposed for image and signal reconstruction, relies on an approximated p -quasi-norm minimization. Compared to 1 -norm minimization, the choice of 0<p<1 provides a natural framework to accommodate usual constraints which quadrature rules must fulfil. We also extend an a priori error estimate available for the 1 -norm formulation by considering the error resulting from data compression. Finally, we present numerical examples to investigate the numerical performance of our method and compare our results to both 1 -norm minimization and nonnegative least squares method. Matlab codes related to the numerical examples and the algorithms described are provided.

References

[1] S. S. An, T. Kim and D. L. James, Optimizing cubature for efficient integration of subspace deformations, ACM Trans. Graph. 27 (2008), no. 165, 1–10. 10.1145/1457515.1409118Search in Google Scholar

[2] E. D. Andersen and K. D. Andersen, Presolving in linear programming, Math. Program. 71 (1995), no. 2(A), 221–245. 10.1007/BF01586000Search in Google Scholar

[3] M. Barrault, Y. Maday, N. C. Nguyen and A. T. Patera, An “empirical interpolation” method: Application to efficient reduced-basis discretization of partial differential equations, C. R. Math. Acad. Sci. Paris 339 (2004), no. 9, 667–672. 10.1016/j.crma.2004.08.006Search in Google Scholar

[4] T. Chapman, P. Avery, P. Collins and C. Farhat, Accelerated mesh sampling for the hyper reduction of nonlinear computational models, Internat. J. Numer. Methods Engrg. 109 (2017), no. 12, 1623–1654. 10.1002/nme.5332Search in Google Scholar

[5] S. Chaturantabut and D. C. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput. 32 (2010), no. 5, 2737–2764. 10.1137/090766498Search in Google Scholar

[6] G. B. Dantzig, A. Orden and P. Wolfe, The generalized simplex method for minimizing a linear form under linear inequality restraints, Pacific J. Math. 5 (1955), 183–195. 10.2140/pjm.1955.5.183Search in Google Scholar

[7] M. E. Davies and R. Gribonval, Restricted isometry constants where p sparse recovery can fail for 0&lt;p1 , IEEE Trans. Inform. Theory 55 (2009), no. 5, 2203–2214. 10.1109/TIT.2009.2016030Search in Google Scholar

[8] R. DeVore, S. Foucart, G. Petrova and P. Wojtaszczyk, Computing a quantity of interest from observational data, Constr. Approx. 49 (2019), no. 3, 461–508. 10.1007/s00365-018-9433-7Search in Google Scholar

[9] C. Farhat, T. Chapman and P. Avery, Structure-preserving, stability, and accuracy properties of the energy-conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models, Internat. J. Numer. Methods Engrg. 102 (2015), no. 5, 1077–1110. 10.1002/nme.4820Search in Google Scholar

[10] S. Foucart and M.-J. Lai, Sparsest solutions of underdetermined linear systems via lq -minimization for 0<q1 , Appl. Comput. Harmon. Anal. 26 (2009), no. 3, 395–407. 10.1016/j.acha.2008.09.001Search in Google Scholar

[11] S. Grimberg, C. Farhat, R. Tezaur and C. Bou-Mosleh, Mesh sampling and weighting for the hyperreduction of nonlinear Petrov–Galerkin reduced-order models with local reduced-order bases, Internat. J. Numer. Methods Engrg. 122 (2021), no. 7, 1846–1874. 10.1002/nme.6603Search in Google Scholar

[12] I. F. Gorodnitsky and B. D. Rao, Sparse signal reconstruction from limited data using FOCUSS: A re-weighted minimum norm algorithm, IEEE Trans. Signal Process. 45 (1997), no. 3, 600–616. 10.1109/78.558475Search in Google Scholar

[13] N. Halko, P. G. Martinsson and J. A. Tropp, Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev. 53 (2011), no. 2, 217–288. 10.1137/090771806Search in Google Scholar

[14] J. A. Hernández, M. A. Caicedo and A. Ferrer, Dimensional hyper-reduction of nonlinear finite element models via empirical cubature, Comput. Methods Appl. Mech. Engrg. 313 (2017), 687–722. 10.1016/j.cma.2016.10.022Search in Google Scholar

[15] V. Hernández, J. E. Román and A. Tomás, A robust and efficient parallel SVD solver based on restarted Lanczos bidiagonalization, Electron. Trans. Numer. Anal. 31 (2008), 68–85. Search in Google Scholar

[16] J. S. Hesthaven, G. Rozza and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Springer Briefs Math., Springer, Cham, 2016. 10.1007/978-3-319-22470-1Search in Google Scholar

[17] E. J. Kontoghiorghes, Handbook of Parallel Computing and Statistics, Chapman & Hall/CRC, Boca Raton, 2005. 10.1201/9781420028683Search in Google Scholar

[18] M. Manucci, Accompanying codes published at GitHub, https://github.com/MattiaManucci/Sparse-data-driven-quadrature-rules-via-FOCUSS.git, 2021. Search in Google Scholar

[19] B. K. Natarajan, Sparse approximate solutions to linear systems, SIAM J. Comput. 24 (1995), no. 2, 227–234. 10.1137/S0097539792240406Search in Google Scholar

[20] A. T. Patera and M. Yano, An LP empirical quadrature procedure for parametrized functions, C. R. Math. Acad. Sci. Paris 355 (2017), no. 11, 1161–1167. 10.1016/j.crma.2017.10.020Search in Google Scholar

[21] A. Quarteroni, G. Rozza and A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations and applications, J. Math. Ind. 1 (2011), Article ID 3. 10.1186/2190-5983-1-3Search in Google Scholar

[22] D. Ryckelynck, Hyper-reduction of mechanical models involving internal variables, Internat. J. Numer. Methods Engrg. 77 (2009), no. 1, 75–89. 10.1002/nme.2406Search in Google Scholar

[23] E. K. Ryu and S. P. Boyd, Extensions of Gauss quadrature via linear programming, Found. Comput. Math. 15 (2015), no. 4, 953–971. 10.1007/s10208-014-9197-9Search in Google Scholar

[24] M. K. Sleeman and M. Yano, Goal-oriented model reduction for parametrized time-dependent nonlinear partial differential equations, Comput. Methods Appl. Mech. Engrg. 388 (2022), Paper No. 114206. 10.1016/j.cma.2021.114206Search in Google Scholar

[25] T. Taddei, An offline/online procedure for dual norm calculations of parameterized functionals: Empirical quadrature and empirical test spaces, Adv. Comput. Math. 45 (2019), no. 5–6, 2429–2462. 10.1007/s10444-019-09721-wSearch in Google Scholar

[26] T. Taddei and L. Zhang, A discretize-then-map approach for the treatment of parameterized geometries in model order reduction, Comput. Methods Appl. Mech. Engrg. 384 (2021), Paper No. 113956. 10.1016/j.cma.2021.113956Search in Google Scholar

[27] M. Yano and A. T. Patera, An LP empirical quadrature procedure for reduced basis treatment of parametrized nonlinear PDEs, Comput. Methods Appl. Mech. Engrg. 344 (2019), 1104–1123. 10.1016/j.cma.2018.02.028Search in Google Scholar

Received: 2021-07-15
Revised: 2021-11-16
Accepted: 2022-01-13
Published Online: 2022-02-15
Published in Print: 2022-04-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Abstract

This paper is concerned with the use of the focal underdetermined system solver to recover sparse empirical quadrature rules for parametrized integrals from existing data. This algorithm, originally proposed for image and signal reconstruction, relies on an approximated p -quasi-norm minimization. Compared to 1 -norm minimization, the choice of 0<p<1 provides a natural framework to accommodate usual constraints which quadrature rules must fulfil. We also extend an a priori error estimate available for the 1 -norm formulation by considering the error resulting from data compression. Finally, we present numerical examples to investigate the numerical performance of our method and compare our results to both 1 -norm minimization and nonnegative least squares method. Matlab codes related to the numerical examples and the algorithms described are provided.

References

[1] S. S. An, T. Kim and D. L. James, Optimizing cubature for efficient integration of subspace deformations, ACM Trans. Graph. 27 (2008), no. 165, 1–10. 10.1145/1457515.1409118Search in Google Scholar

[2] E. D. Andersen and K. D. Andersen, Presolving in linear programming, Math. Program. 71 (1995), no. 2(A), 221–245. 10.1007/BF01586000Search in Google Scholar

[3] M. Barrault, Y. Maday, N. C. Nguyen and A. T. Patera, An “empirical interpolation” method: Application to efficient reduced-basis discretization of partial differential equations, C. R. Math. Acad. Sci. Paris 339 (2004), no. 9, 667–672. 10.1016/j.crma.2004.08.006Search in Google Scholar

[4] T. Chapman, P. Avery, P. Collins and C. Farhat, Accelerated mesh sampling for the hyper reduction of nonlinear computational models, Internat. J. Numer. Methods Engrg. 109 (2017), no. 12, 1623–1654. 10.1002/nme.5332Search in Google Scholar

[5] S. Chaturantabut and D. C. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput. 32 (2010), no. 5, 2737–2764. 10.1137/090766498Search in Google Scholar

[6] G. B. Dantzig, A. Orden and P. Wolfe, The generalized simplex method for minimizing a linear form under linear inequality restraints, Pacific J. Math. 5 (1955), 183–195. 10.2140/pjm.1955.5.183Search in Google Scholar

[7] M. E. Davies and R. Gribonval, Restricted isometry constants where p sparse recovery can fail for 0&lt;p1 , IEEE Trans. Inform. Theory 55 (2009), no. 5, 2203–2214. 10.1109/TIT.2009.2016030Search in Google Scholar

[8] R. DeVore, S. Foucart, G. Petrova and P. Wojtaszczyk, Computing a quantity of interest from observational data, Constr. Approx. 49 (2019), no. 3, 461–508. 10.1007/s00365-018-9433-7Search in Google Scholar

[9] C. Farhat, T. Chapman and P. Avery, Structure-preserving, stability, and accuracy properties of the energy-conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models, Internat. J. Numer. Methods Engrg. 102 (2015), no. 5, 1077–1110. 10.1002/nme.4820Search in Google Scholar

[10] S. Foucart and M.-J. Lai, Sparsest solutions of underdetermined linear systems via lq -minimization for 0<q1 , Appl. Comput. Harmon. Anal. 26 (2009), no. 3, 395–407. 10.1016/j.acha.2008.09.001Search in Google Scholar

[11] S. Grimberg, C. Farhat, R. Tezaur and C. Bou-Mosleh, Mesh sampling and weighting for the hyperreduction of nonlinear Petrov–Galerkin reduced-order models with local reduced-order bases, Internat. J. Numer. Methods Engrg. 122 (2021), no. 7, 1846–1874. 10.1002/nme.6603Search in Google Scholar

[12] I. F. Gorodnitsky and B. D. Rao, Sparse signal reconstruction from limited data using FOCUSS: A re-weighted minimum norm algorithm, IEEE Trans. Signal Process. 45 (1997), no. 3, 600–616. 10.1109/78.558475Search in Google Scholar

[13] N. Halko, P. G. Martinsson and J. A. Tropp, Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev. 53 (2011), no. 2, 217–288. 10.1137/090771806Search in Google Scholar

[14] J. A. Hernández, M. A. Caicedo and A. Ferrer, Dimensional hyper-reduction of nonlinear finite element models via empirical cubature, Comput. Methods Appl. Mech. Engrg. 313 (2017), 687–722. 10.1016/j.cma.2016.10.022Search in Google Scholar

[15] V. Hernández, J. E. Román and A. Tomás, A robust and efficient parallel SVD solver based on restarted Lanczos bidiagonalization, Electron. Trans. Numer. Anal. 31 (2008), 68–85. Search in Google Scholar

[16] J. S. Hesthaven, G. Rozza and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Springer Briefs Math., Springer, Cham, 2016. 10.1007/978-3-319-22470-1Search in Google Scholar

[17] E. J. Kontoghiorghes, Handbook of Parallel Computing and Statistics, Chapman & Hall/CRC, Boca Raton, 2005. 10.1201/9781420028683Search in Google Scholar

[18] M. Manucci, Accompanying codes published at GitHub, https://github.com/MattiaManucci/Sparse-data-driven-quadrature-rules-via-FOCUSS.git, 2021. Search in Google Scholar

[19] B. K. Natarajan, Sparse approximate solutions to linear systems, SIAM J. Comput. 24 (1995), no. 2, 227–234. 10.1137/S0097539792240406Search in Google Scholar

[20] A. T. Patera and M. Yano, An LP empirical quadrature procedure for parametrized functions, C. R. Math. Acad. Sci. Paris 355 (2017), no. 11, 1161–1167. 10.1016/j.crma.2017.10.020Search in Google Scholar

[21] A. Quarteroni, G. Rozza and A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations and applications, J. Math. Ind. 1 (2011), Article ID 3. 10.1186/2190-5983-1-3Search in Google Scholar

[22] D. Ryckelynck, Hyper-reduction of mechanical models involving internal variables, Internat. J. Numer. Methods Engrg. 77 (2009), no. 1, 75–89. 10.1002/nme.2406Search in Google Scholar

[23] E. K. Ryu and S. P. Boyd, Extensions of Gauss quadrature via linear programming, Found. Comput. Math. 15 (2015), no. 4, 953–971. 10.1007/s10208-014-9197-9Search in Google Scholar

[24] M. K. Sleeman and M. Yano, Goal-oriented model reduction for parametrized time-dependent nonlinear partial differential equations, Comput. Methods Appl. Mech. Engrg. 388 (2022), Paper No. 114206. 10.1016/j.cma.2021.114206Search in Google Scholar

[25] T. Taddei, An offline/online procedure for dual norm calculations of parameterized functionals: Empirical quadrature and empirical test spaces, Adv. Comput. Math. 45 (2019), no. 5–6, 2429–2462. 10.1007/s10444-019-09721-wSearch in Google Scholar

[26] T. Taddei and L. Zhang, A discretize-then-map approach for the treatment of parameterized geometries in model order reduction, Comput. Methods Appl. Mech. Engrg. 384 (2021), Paper No. 113956. 10.1016/j.cma.2021.113956Search in Google Scholar

[27] M. Yano and A. T. Patera, An LP empirical quadrature procedure for reduced basis treatment of parametrized nonlinear PDEs, Comput. Methods Appl. Mech. Engrg. 344 (2019), 1104–1123. 10.1016/j.cma.2018.02.028Search in Google Scholar

Received: 2021-07-15
Revised: 2021-11-16
Accepted: 2022-01-13
Published Online: 2022-02-15
Published in Print: 2022-04-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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