Computation of Tight Enclosures for Laplacian Eigenvalues
Abstract
Recently, there has been interest in high precision approximations of the first eigenvalue of the Laplace--Beltrami operator on spherical triangles for combinatorial purposes. We compute improved and certified enclosures to these eigenvalues. This is achieved by applying the method of particular solutions in high precision, the enclosure being obtained by a combination of interval arithmetic and Taylor models. The index of the eigenvalue is certified by exploiting the monotonicity of the eigenvalue with respect to the domain. The classically troublesome case of singular corners is handled by combining expansions at all corners and an expansion from an interior point. In particular, this allows us to compute 100 digits of the fundamental eigenvalue for the three-dimensional Kreweras model that has been the object of previous efforts.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
M. S. Ashbaugh and H. A. Levine, Inequalities for the Dirichlet and Neumann eigenvalues of the Laplacian for domains on spheres, in Journées “Équations aux Dérivées Partielles,” Saint-Jean-de-Monts, 1997, École Polytech., Palaiseau, France, 1997, pp. I-1--I-15.
2.
A. Bacher, M. Kauers, and R. Yatchak, Continued classification of 3d lattice models in the positive octant, in FPSAC 2016, Vol. BC Discrete Math. Theoret. Comput. Sci. Proc., 2016, pp. 95--106.
3.
B. S. Balakrishna, On multi-particle Brownian Survivals and the Spherical Laplacian, Technical report 44459, Munich Personal RePEc Archive, 2013, https://mpra.ub.uni-muenchen.de/44459/.
4.
A. H. Barnett and A. Hassell, Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues, SIAM J. Numer. Anal., 49 (2011), pp. 1046--1063, https://doi.org/10.1137/100796637.
5.
H. Behnke and F. Goerisch, Inclusions for eigenvalues of selfadjoint problems, in Topics in Validated Computations, Stud. Comput. Math. 5, North-Holland, Amsterdam, 1994, pp. 277--322.
6.
P. Bérard and G. Besson, Spectres et groupes cristallographiques. II. Domaines sphériques, Ann. Inst. Fourier (Grenoble), 30 (1980), pp. 237--248, http://www.numdam.org/item?id=AIF_1980__30_3_237_0.
7.
P. H. Bérard, Remarques sur la conjecture de Weyl, Compos. Math., 48 (1983), pp. 35--53, http://www.numdam.org/item?id=CM_1983__48_1_35_0.
8.
T. Betcke, The generalized singular value decomposition and the method of particular solutions, SIAM J. Sci. Comput., 30 (2008), pp. 1278--1295, https://doi.org/10.1137/060651057.
9.
T. Betcke and L. N. Trefethen, Reviving the method of particular solutions, SIAM Rev., 47 (2005), pp. 469--491, https://doi.org/10.1137/S0036144503437336.
10.
J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, Julia: A fresh approach to numerical computing, SIAM Rev., 59 (2017), pp. 65--98, https://doi.org/10.1137/141000671.
11.
D. Boffi, Finite element approximation of eigenvalue problems, Acta Numer., 19 (2010), pp. 1--120, https://doi.org/10.1017/S0962492910000012.
12.
B. Bogosel, V. Perrollaz, K. Raschel, and A. Trotignon, 3D Positive lattice walks and spherical triangles, J. Combin. Theory Ser. A, 172 (2020), 105189, https://doi.org/10.1016/j.jcta.2019.105189.
13.
A. Bostan, Computer Algebra for Lattice Path Combinatorics, Technical report, Inria, 2019.
14.
A. Bostan, K. Raschel, and B. Salvy, Non-D-finite excursions in the quarter plane, J. Combin. Theory Ser. A, 121 (2014), pp. 45--63, https://doi.org/10.1016/j.jcta.2013.09.005.
15.
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
16.
E. Cancès, G. Dusson, Y. Maday, B. Stamm, and M. Vohralík, Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: Conforming approximations, SIAM J. Numer. Anal., 55 (2017), pp. 2228--2254, https://doi.org/10.1137/15M1038633.
17.
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1, Wiley, New York, 1962.
18.
D. Denisov and V. Wachtel, Random walks in cones, Ann. Probab., 43 (2015), pp. 992--1044, https://doi.org/10.1214/13-AOP867.
19.
J. Descloux and M. Tolley, An accurate algorithm for computing the eigenvalues of a polygonal membrane, Comput. Methods Appl. Mech. Engrg., 39 (1983), pp. 37--53, https://doi.org/10.1016/0045-7825(83)90072-5.
20.
C. Fieker, W. Hart, T. Hofmann, and F. Johansson, Nemo/Hecke: Computer algebra and number theory packages for the Julia programming language, in Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation, ISSAC '17, New York, NY, 2017, ACM, New York, pp. 157--164, https://doi.org/10.1145/3087604.3087611.
21.
L. Fousse, G. Hanrot, V. Lefèvre, P. Pélissier, and P. Zimmermann, MPFR: A multiple-precision binary floating-point library with correct rounding, ACM Trans. Math. Software, 33 (2007), 13, https://doi.org/10.1145/1236463.1236468.
22.
L. Fox, P. Henrici, and C. Moler, Approximations and bounds for eigenvalues of elliptic operators, SIAM J. Numer. Anal., 4 (1967), pp. 89--102, https://doi.org/10.1137/0704008.
23.
J. Gómez-Serrano and G. Orriols, Any Three Eigenvalues Do Not Determine a Triangle, preprint, https://arxiv.org/abs/1911.06758, 2019.
24.
A. Gopal and L. N. Trefethen, Solving Laplace problems with corner singularities via rational functions, SIAM J. Numer. Anal., 57 (2019), pp. 2074--2094, https://doi.org/10.1137/19M125947X.
25.
D. S. Grebenkov and B.-T. Nguyen, Geometrical structure of Laplacian eigenfunctions, SIAM Rev., 55 (2013), pp. 601--667, https://doi.org/10.1137/120880173.
26.
F. Johansson, Arb: Efficient arbitrary-precision midpoint-radius interval arithmetic, IEEE Trans. Comput., 66 (2017), pp. 1281--1292, https://doi.org/10.1109/TC.2017.2690633.
27.
F. Johansson and M. Mezzarobba, Fast and rigorous arbitrary-precision computation of Gauss--Legendre quadrature nodes and weights, SIAM J. Sci. Comput., 40 (2018), pp. C726--C747, https://doi.org/10.1137/18M1170133.
28.
R. S. Jones, Computing ultra-precise eigenvalues of the Laplacian within polygons, Adv. Comput. Math., 43 (2017), pp. 1325--1354, https://doi.org/10.1007/s10444-017-9527-y.
29.
C. Krattenthaler, Lattice path enumeration, in Handbook of Enumerative Combinatorics, Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, FL, 2015, pp. 589--678.
30.
G. Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du B.U.R.O., 6 (1965), pp. 5--105.
31.
J. R. Kuttler and V. G. Sigillito, Eigenvalues of the Laplacian in two dimensions, SIAM Rev., 26 (1984), pp. 163--193, https://doi.org/10.1137/1026033.
32.
X. Liu, A framework of verified eigenvalue bounds for self-adjoint differential operators, Appl. Math. Comput., 267 (2015), pp. 341--355, https://doi.org/10.1016/j.amc.2015.03.048.
33.
X. Liu and S. Oishi, Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape, SIAM J. Numer. Anal., 51 (2013), pp. 1634--1654, https://doi.org/10.1137/120878446.
34.
K. Makino and M. Berz, Taylor models and other validated functional inclusion methods, Internat. J. Pure Appl. Math., 4 (2003), pp. 379--456, http://bt.pa.msu.edu/pub/papers/TMIJPAM03/TMIJPAM03.pdf.
35.
P. K. Mogensen and A. N. Riseth, Optim: A mathematical optimization package for Julia, J. Open Source Software, 3 (2018), 615, https://doi.org/10.21105/joss.00615.
36.
C. B. Moler and L. E. Payne, Bounds for eigenvalues and eigenvectors of symmetric operators, SIAM J. Numer. Anal., 5 (1968), pp. 64--70, https://doi.org/10.1137/0705004.
37.
M. T. Nakao, M. Plum, and Y. Watanabe, Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations, Springer Ser. Comput. Math. 53, Springer, Singapore, 2019, https://doi.org/10.1007/978-981-13-7669-6.
38.
M. T. Nakao, N. Yamamoto, and K. Nagatou, Numerical verifications for eigenvalues of second-order elliptic operators, Jpn. J. Ind. Appl. Math., 16 (1999), pp. 307--320, https://doi.org/10.1007/BF03167360.
39.
F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010.
40.
R. B. Platte and T. A. Driscoll, Computing eigenmodes of elliptic operators using radial basis functions, Comput. Math. Appl., 48 (2004), pp. 561--576, https://doi.org/10.1016/j.camwa.2003.08.007.
41.
M. Plum, Bounds for eigenvalues of second-order elliptic differential operators, Z. Angew. Math. Phys., 42 (1991), pp. 848--863, https://doi.org/10.1007/BF00944567.
42.
J. Ratzkin and A. Treibergs, A capture problem in Brownian motion and eigenvalues of spherical domains, Trans. Amer. Math. Soc., 361 (2009), pp. 391--405, https://doi.org/10.1090/S0002-9947-08-04505-4.
43.
B. Salvy and P. Zimmermann, GFUN: A Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software, 20 (1994), pp. 163--177, https://doi.org/10.1145/178365.178368.
44.
G. Still, Computable bounds for eigenvalues and eigenfunctions of elliptic differential operators, Numer. Math., 54 (1988), pp. 201--223, https://doi.org/10.1007/BF01396975.
45.
A. Strohmaier, Computation of eigenvalues, spectral zeta functions and zeta-determinants on hyperbolic surfaces, in Geometric and Computational Spectral Theory, Contemp. Math. 700, American Mathematical Society, Providence, RI, 2017, pp. 177--205, https://doi.org/10.1090/conm/700/14187.
46.
W. Tucker, Validated numerics, Princeton University Press, Princeton, NJ, 2011.
Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: A3210 - A3232
DOI: 10.1137/20M1326520
ISSN (online): 1095-7197
Copyright
© 2020, Society for Industrial and Applied Mathematics.
History
Submitted: 20 March 2020
Accepted: 27 July 2020
Published online: 19 October 2020
Keywords
MSC codes
Authors
Funding Information
Agence Nationale de la Recherche https://doi.org/10.13039/501100001665 : ANR-19-CE40-0018
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Scale-invariant random geometry from mating of trees: A numerical studyPhysical Review D, Vol. 107, No. 2 | 12 January 2023
- A counterexample to Payne’s nodal line conjecture with few holesCommunications in Nonlinear Science and Numerical Simulation, Vol. 103 | 1 Dec 2021