×

RBF approximation of three dimensional PDEs using tensor Krylov subspace methods. (English) Zbl 1521.65116


MSC:

65N20 Numerical methods for ill-posed problems for boundary value problems involving PDEs
65D12 Numerical radial basis function approximation
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs

References:

[1] Liu, G. R., Mesh free methods: moving beyond the finite element method (2002), CRC Press: CRC Press Boca Raton, FL
[2] Gladwell, G. M.L.; Willms, N. B., On the mode shape of the Helmholtz equation, J Sound Vib, 188, 419-433 (1995) · Zbl 1232.74048
[3] Wood, A. S.; Tupholme, G. E.; Bhatti, M. I.H.; Heggs, P. J., Steady-state heat transfer through extended plane surfaces, Int Commun Heat Mass Transfer, 22, 99-109 (1995)
[4] Hardy, R., Multiquadric equations of topography and other irregular surfaces, J Geophys Res, 76, 1905-1915 (1971)
[5] Fasshauer, G. E., Meshfree approximation methods with matlab (2007), World Scientific · Zbl 1123.65001
[6] Wendland, H., Scattered data approximation (2005), Cambridge University Press · Zbl 1075.65021
[7] Beatson, R. K.; Cherrie, J. B.; Mouat, C. T., Fast fitting of radial basis functions: method based on preconditioned GMRES iteration, Adv Comput Math, 11, 253-270 (1999) · Zbl 0940.65011
[8] Kolda, T. G.; w. Bader, B., Tensor decompositions and applications, SIAM Rev, 3, 455-500 (2009) · Zbl 1173.65029
[9] Braman, K., Third-order tensors as linear operators on a space of matrices, Linear Algebra Appl, 433, 1241-1253 (2010) · Zbl 1198.15017
[10] Brazell, M.; Li. C. Navasca, N.; Tamon, C., Solving multilinear systems via tensor inversion, SIAM J Matrix Anal Appl, 34, 2, 542-570 (2013) · Zbl 1273.15028
[11] Einstein, A., The foundation of the general theory of relativity, (Kox, AJ; Klein, MJ; Schulmann, R., The collected papers of albert einstein, Vol. 6 (2007), Princeton University Press: Princeton University Press Princeton (NJ)), 146-200
[12] Sarra, S.; Kansa, E., Multiquadric radial basis function approximation methods for the numerical solution of partial differential equations (advances in computational mechanics Vol. 2) (2009), Tech Science Press
[13] Hon, Y. C.; Schaback, R., On unsymmetric collocation by radial basis function, Appl Math Comput, 119, 177-186 (2001) · Zbl 1026.65107
[14] Aussal M. The gypsilabtoolbox for matlab version 0.5. openhmx library.In: Centre de mathematiques appliquees, ecole polytechnique, route de saclay, 91128 palaiseau,France.
[15] Liu, Y.; Sid-Lakhdar, W.; Rebrova, E.; Ghysels, P.; Li, X. S.X., A parallel hierarchical blocked adaptive cross approximation algorithm, Int J High Perform Comput Appl, 34, 4, 394-408 (2020)
[16] Tikhonov, A. N., Regularization of incorrectly posed problems, Soviet Math, 4, 1624-1627 (1963) · Zbl 0183.11601
[17] Wahba, G., Pratical approximation solutions to linear operator equations when the data are noisy, SIAM J Numer Anal, 14, 651-667 (1977) · Zbl 0402.65032
[18] Jbilou A, K.; Messaoudi, H., Sadok global FOM and GMRES algorithms for matrix equations, Appl Numer Math, 31, 49-63 (1999) · Zbl 0935.65024
[19] Bentbib, A. H.; El Guide, M.; Jbilou, K.; Reichel, L., Global Golub-Kahan bidiagonalization applied to large discrete ill-posed problems, J Comput Appl Math, 322, 46-56 (2017) · Zbl 1365.65102
[20] Bentbib, A. H.; El Guide, M.; Jbilou, K.; Onunwor, E.; Reichel, L., Solution methods for linear discrete ill-posed problems for color image restoration, BIT Numer Math, 58, 3, 555-576 (2018) · Zbl 1402.65031
[21] El Guide M, El Ichi A, Jbilou K, Beik FPA. Tensor GMRES and Golub-Kahan bidiagonalization methods via the Einstein product with applications to image and video processing, arXiv preprint arXiv:2005.07458.
[22] Huang, B.; Xie, Y.; Ma, C., Krylov subspace methods to solve a class of tensor equations via the Einstein product, Numer Linear Algebra Appl, 26, Article e2254 pp. (2019) · Zbl 1463.65045
[23] Golub, G. H.; Heath, M.; Wahba, G., Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21, 215-223 (1979) · Zbl 0461.62059
[24] Hansen, P. C., Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev, 34, 561-580 (1992) · Zbl 0770.65026
[25] Hansen, P. C., Regularization tools, a MATLAB package for analysis of discrete regularization problems, Numer Algorithms, 6, 1-35 (1994) · Zbl 0789.65029
[26] Calvetti, D.; Golub, G. H.; Reichel, L., Estimation of the L-curve via Lanczos bidiagonalization, BIT, 39, 603-619 (1999) · Zbl 0945.65044
[27] Calvetti, D.; Hansen, P. C.; Reichel, L., L-Curve curvature bounds via Lanczos bidiagonalization, Electron Trans Numer Anal, 14, 134-149 (2002) · Zbl 1029.65041
[28] Geuzaine, C.; Remacle, J. F., Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, Internat J Numer Methods Engrg, 79, 11, 1309-1331 (2009) · Zbl 1176.74181
[29] Vacca, A.; Guidetti, M., Modelling and experimental validation of external spur gear ma es for fluid power applications, Elsevier Simul Model Pract Theory, 19, 2007-2031 (2011)
[30] Tang, C.; Wang, Y. S.; Gao, J. H.; Guo, H., Fluid-sound coupling simulation and experimental validation for noise characteristics of a variable displacement external gear pump, Noise Control Eng J, 62, 3, 123-131 (2014)
[31] Kimler, M. E.; Martin, C. D., Factorization strategies for third-order tensors, Linear Algebra Appl, 435, 641-658 (2011) · Zbl 1228.15009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.