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

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. 10.1145/1457515.1409118Suche 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/BF01586000Suche 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.2004.08.006Suche 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.5332Suche 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. 10.1137/090766498Suche 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.1955.5.183Suche in Google Scholar

[7] M. E. Davies and R. Gribonval, Restricted isometry constants where ℓ p sparse recovery can fail for 0 & l ⁢ t ; p ≤ 1 , IEEE Trans. Inform. Theory 55 (2009), no. 5, 2203–2214. 10.1109/TIT.2009.2016030Suche 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/s00365-018-9433-7Suche 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.4820Suche in Google Scholar

[10] S. Foucart and M.-J. Lai, Sparsest solutions of underdetermined linear systems via l q -minimization for 0 < q ≤ 1 , Appl. Comput. Harmon. Anal. 26 (2009), no. 3, 395–407. 10.1016/j.acha.2008.09.001Suche 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.6603Suche 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. 10.1109/78.558475Suche 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. 10.1137/090771806Suche 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.2016.10.022Suche 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. Suche 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. 10.1007/978-3-319-22470-1Suche in Google Scholar

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

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

[19] B. K. Natarajan, Sparse approximate solutions to linear systems, SIAM J. Comput. 24 (1995), no. 2, 227–234. 10.1137/S0097539792240406Suche 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.2017.10.020Suche 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. 10.1186/2190-5983-1-3Suche 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.2406Suche 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/s10208-014-9197-9Suche 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.2021.114206Suche 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-wSuche 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.2021.113956Suche 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.2018.02.028Suche in Google Scholar