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Time-series forecasting using manifold learning, radial basis function interpolation, and geometric harmonics. (English) Zbl 07876555


MSC:

68Txx Artificial intelligence
62Mxx Inference from stochastic processes
62Pxx Applications of statistics
Full Text: DOI

References:

[1] Granger, C. W. J., Investigating causal relations by econometric models and cross-spectral methods, Econometrica, 37, 424, 1969 · Zbl 1366.91115 · doi:10.2307/1912791
[2] De Gooijer, J. G.; Hyndman, R. J., 25 years of time series forecasting, Int. J. Forecast., 22, 443-473, 2006 · doi:10.1016/j.ijforecast.2006.01.001
[3] Lukoševičius, M.; Jaeger, H., Reservoir computing approaches to recurrent neural network training, Comput. Sci. Rev., 3, 127-149, 2009 · Zbl 1302.68235 · doi:10.1016/j.cosrev.2009.03.005
[4] Wyffels, F.; Schrauwen, B., A comparative study of reservoir computing strategies for monthly time series prediction, Neurocomputing, 73, 1958-1964, 2010 · doi:10.1016/j.neucom.2010.01.016
[5] Roberts, S.; Osborne, M.; Ebden, M.; Reece, S.; Gibson, N.; Aigrain, S., Gaussian processes for time-series modelling, Philos. Trans. Royal Soc. A: Math. Phys. Eng. Sci., 371, 20110550, 2013 · Zbl 1353.62103 · doi:10.1098/rsta.2011.0550
[6] Granger, C. W. J.; Newbold, P., Forecasting Economic Time Series, 2014, Academic Press
[7] Brockwell, P. J.; Brockwell, P. J.; Davis, R. A.; Davis, R. A., Introduction to Time Series and Forecasting, 2016, Springer · Zbl 1355.62001
[8] Greff, K.; Srivastava, R. K.; Koutník, J.; Steunebrink, B. R.; Schmidhuber, J., LSTM: A search space odyssey, IEEE Trans. Neural Netw. Learn. Syst., 28, 2222-2232, 2016 · doi:10.1109/TNNLS.2016.2582924
[9] Pathak, J.; Hunt, B.; Girvan, M.; Lu, Z.; Ott, E., Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach, Phys. Rev. Lett., 120, 024102, 2018 · doi:10.1103/PhysRevLett.120.024102
[10] Zimmermann, R. S.; Parlitz, U., Observing spatio-temporal dynamics of excitable media using reservoir computing, Chaos, 28, 043118, 2018 · doi:10.1063/1.5022276
[11] Vlachas, P. R.; Pathak, J.; Hunt, B. R.; Sapsis, T. P.; Girvan, M.; Ott, E.; Koumoutsakos, P., Backpropagation algorithms and reservoir computing in recurrent neural networks for the forecasting of complex spatiotemporal dynamics, Neural Netw., 126, 191-217, 2020 · doi:10.1016/j.neunet.2020.02.016
[12] Bellmann, R., Dynamic Programming, 1957, Princeton University Press · Zbl 0077.13605
[13] Schmid, P. J., Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656, 5-28, 2010 · Zbl 1197.76091 · doi:10.1017/S0022112010001217
[14] Mann, J.; Kutz, J. N., Dynamic mode decomposition for financial trading strategies, Quant. Finance, 16, 1643-1655, 2016 · Zbl 1400.91558 · doi:10.1080/14697688.2016.1170194
[15] Kutz, J. N.; Brunton, S. L.; Brunton, B. W.; Proctor, J. L., Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems, 2016, SIAM · Zbl 1365.65009
[16] Schölkopf, B.; Smola, A.; Müller, K.-R., Nonlinear component analysis as a kernel eigenvalue problem, Neural Comput., 10, 1299-1319, 1998 · doi:10.1162/089976698300017467
[17] Roweis, S. T.; Saul, L. K., Nonlinear dimensionality reduction by locally linear embedding, Science, 290, 2323-2326, 2000 · doi:10.1126/science.290.5500.2323
[18] Tenenbaum, J. B.; De Silva, V.; Langford, J. C., A global geometric framework for nonlinear dimensionality reduction, Science, 290, 2319-2323, 2000 · doi:10.1126/science.290.5500.2319
[19] Balasubramanian, M.; Schwartz, E. L.; Tenenbaum, J. B.; de Silva, V.; Langford, J. C., The Isomap algorithm and topological stability, Science, 295, 7, 2002 · doi:10.1126/science.295.5552.7a
[20] Belkin, M.; Niyogi, P., Laplacian eigenmaps for dimensionality reduction and data representation, Neural Comput., 15, 1373-1396, 2003 · Zbl 1085.68119 · doi:10.1162/089976603321780317
[21] Coifman, R. R.; Lafon, S.; Lee, A. B.; Maggioni, M.; Nadler, B.; Warner, F.; Zucker, S. W., Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps, Proc. Natl. Acad. Sci. U.S.A., 102, 7426-7431, 2005 · Zbl 1405.42043 · doi:10.1073/pnas.0500334102
[22] Coifman, R. R.; Lafon, S., Diffusion maps, Appl. Comput. Harmon. Anal., 21, 5-30, 2006 · Zbl 1095.68094 · doi:10.1016/j.acha.2006.04.006
[23] Coifman, R. R.; Lafon, S., Geometric harmonics: A novel tool for multiscale out-of-sample extension of empirical functions, Appl. Comput. Harmon. Anal., 21, 31-52, 2006 · Zbl 1095.68095 · doi:10.1016/j.acha.2005.07.005
[24] Nadler, B.; Lafon, S.; Coifman, R. R.; Kevrekidis, I. G., Diffusion maps, spectral clustering and reaction coordinates of dynamical systems, Appl. Comput. Harmon. Anal., 21, 113-127, 2006 · Zbl 1103.60069 · doi:10.1016/j.acha.2005.07.004
[25] Coifman, R. R.; Kevrekidis, I. G.; Lafon, S.; Maggioni, M.; Nadler, B., Diffusion maps, reduction coordinates, and low dimensional representation of stochastic systems, Multiscale Model. Simul., 7, 842-864, 2008 · Zbl 1175.60058 · doi:10.1137/070696325
[26] Mezić, I., Analysis of fluid flows via spectral properties of the Koopman operator, Annu. Rev. Fluid Mech., 45, 357-378, 2013 · Zbl 1359.76271 · doi:10.1146/annurev-fluid-011212-140652
[27] Williams, M. O.; Kevrekidis, I. G.; Rowley, C. W., A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition, J. Nonlinear Sci., 25, 1307-1346, 2015 · Zbl 1329.65310 · doi:10.1007/s00332-015-9258-5
[28] Dietrich, F.; Thiem, T. N.; Kevrekidis, I. G., On the Koopman operator of algorithms, SIAM J. Appl. Dyn. Syst., 19, 860-885, 2020 · Zbl 1437.47048 · doi:10.1137/19M1277059
[29] Bollt, E., Attractor modeling and empirical nonlinear model reduction of dissipative dynamical systems, Int. J. Bifurcation Chaos, 17, 1199-1219, 2007 · Zbl 1185.37162 · doi:10.1142/S021812740701777X
[30] Chiavazzo, E.; Gear, C. W.; Dsilva, C. J.; Rabin, N.; Kevrekidis, I. G., Reduced models in chemical kinetics via nonlinear data-mining, Processes, 2, 112-140, 2014 · doi:10.3390/pr2010112
[31] Liu, P.; Safford, H. R.; Couzin, I. D.; Kevrekidis, I. G., Coarse-grained variables for particle-based models: Diffusion maps and animal swarming simulations, Comput. Part. Mech., 1, 425-440, 2014 · doi:10.1007/s40571-014-0030-7
[32] Dsilva, C. J.; Talmon, R.; Gear, C. W.; Coifman, R. R.; Kevrekidis, I. G.
[33] Brunton, S. L.; Proctor, J. L.; Kutz, J. N., Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci. U.S.A., 113, 3932-3937, 2016 · Zbl 1355.94013 · doi:10.1073/pnas.1517384113
[34] Bhattacharjee, S.; Matouš, K., A nonlinear manifold-based reduced order model for multiscale analysis of heterogeneous hyperelastic materials, J. Comput. Phys., 313, 635-653, 2016 · Zbl 1349.65611 · doi:10.1016/j.jcp.2016.01.040
[35] Wan, Z. Y.; Sapsis, T. P., Reduced-space Gaussian process regression for data-driven probabilistic forecast of chaotic dynamical systems, Phys. D, 345, 40-55, 2017 · Zbl 1380.37111 · doi:10.1016/j.physd.2016.12.005
[36] Nathan Kutz, J.; Proctor, J. L.; Brunton, S. L., Applied Koopman theory for partial differential equations and data-driven modeling of spatio-temporal systems, Complexity, 2018, 6010634 · Zbl 1409.37029 · doi:10.1155/2018/6010634
[37] Chen, W.; Ferguson, A. L., Molecular enhanced sampling with autoencoders: On-the-fly collective variable discovery and accelerated free energy landscape exploration, J. Comput. Chem., 39, 2079-2102, 2018 · doi:10.1002/jcc.25520
[38] Kramer, M. A., Nonlinear principal component analysis using autoassociative neural networks, AIChE J., 37, 233-243, 1991 · doi:10.1002/aic.690370209
[39] Vlachas, P. R.; Byeon, W.; Wan, Z. Y.; Sapsis, T. P.; Koumoutsakos, P., Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks, Proc. R. Soc. A, 474, 20170844, 2018 · Zbl 1402.92030 · doi:10.1098/rspa.2017.0844
[40] Takens, F., “Detecting strange attractors in turbulence,” in Dynamical Systems and Turbulence, Warwick 1980 (Springer, 1981), pp. 366-381. · Zbl 0513.58032
[41] Herzog, S.; Wörgötter, F.; Parlitz, U., Convolutional autoencoder and conditional random fields hybrid for predicting spatial-temporal chaos, Chaos, 29, 123116, 2019 · Zbl 1429.37047 · doi:10.1063/1.5124926
[42] Lee, S.; Kooshkbaghi, M.; Spiliotis, K.; Siettos, C. I.; Kevrekidis, I. G., Coarse-scale PDEs from fine-scale observations via machine learning, Chaos, 30, 013141, 2020 · doi:10.1063/1.5126869
[43] Koronaki, E.; Nikas, A.; Boudouvis, A., A data-driven reduced-order model of nonlinear processes based on diffusion maps and artificial neural networks, Chem. Eng. J., 397, 125475, 2020 · doi:10.1016/j.cej.2020.125475
[44] Isensee, J.; Datseris, G.; Parlitz, U., Predicting spatio-temporal time series using dimension reduced local states, J. Nonlinear Sci., 30, 713-735, 2020 · Zbl 1441.37089 · doi:10.1007/s00332-019-09588-7
[45] Lin, K. K.; Lu, F., Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism, J. Comput. Phys., 424, 109864, 2021 · Zbl 07508469 · doi:10.1016/j.jcp.2020.109864
[46] Gajamannage, K.; Paffenroth, R.; Bollt, E. M., A nonlinear dimensionality reduction framework using smooth geodesics, Pattern Recognit., 87, 226-236, 2019 · doi:10.1016/j.patcog.2018.10.020
[47] Lee, J. M., “Smooth manifolds,” in Introduction to Smooth Manifolds (Springer, 2013), pp. 1-31. · Zbl 1258.53002
[48] Berger, M.; Gostiaux, B., Differential Geometry: Manifolds, Curves, and Surfaces, 2012, Springer Science & Business Media
[49] Kühnel, W., Differential Geometry, 2015, American Mathematical Society
[50] Wang, J., Geometric Structure of High-Dimensional Data and Dimensionality Reduction, 2012, Springer · Zbl 1250.68010
[51] Saul, L. K.; Roweis, S. T., Think globally, fit locally: Unsupervised learning of low dimensional manifolds, J. Mach. Learn. Res., 4, 119-155, 2003 · Zbl 1093.68089
[52] Eckart, C.; Young, G., The approximation of one matrix by another of lower rank, Psychometrika, 1, 211-218, 1936 · JFM 62.1075.02 · doi:10.1007/BF02288367
[53] Mirsky, L., Symmetric gauge functions and unitarily invariant norms, Q. J. Math., 11, 50-59, 1960 · Zbl 0105.01101 · doi:10.1093/qmath/11.1.50
[54] Jones, P. W.; Maggioni, M.; Schul, R., Manifold parametrizations by eigenfunctions of the Laplacian and heat kernels, Proc. Natl. Acad. Sci. U.S.A., 105, 1803-1808, 2008 · Zbl 1215.58012 · doi:10.1073/pnas.0710175104
[55] Marcellino, M.; Stock, J. H.; Watson, M. W., A comparison of direct and iterated multistep AR methods for forecasting macroeconomic time series, J. Econom., 135, 499-526, 2006 · Zbl 1418.62513 · doi:10.1016/j.jeconom.2005.07.020
[56] Rasmussen, C. E.; Williams, C. K. I., Gaussian Processes for Machine Learning, 2018, MIT Press
[57] Corani, G.; Benavoli, A.; Zaffalon, M.
[58] Nyström, E. J., Über die Praktische Auflösung von Linearen Integralgleichungen mit Anwendungen auf Randwertaufgaben der Potentialtheorie, 1929, Akademische Buchhandlung · JFM 55.0819.02
[59] Monnig, N. D.; Fornberg, B.; Meyer, F. G., Inverting nonlinear dimensionality reduction with scale-free radial basis function interpolation, Appl. Comput. Harmon. Anal., 37, 162-170, 2014 · Zbl 1297.65015 · doi:10.1016/j.acha.2013.10.004
[60] Fornberg, B.; Zuev, J., The runge phenomenon and spatially variable shape parameters in RBF interpolation, Comput. Math. Appl., 54, 379-398, 2007 · Zbl 1128.41001 · doi:10.1016/j.camwa.2007.01.028
[61] Amorim, E.; Brazil, E. V.; Mena-Chalco, J.; Velho, L.; Nonato, L. G.; Samavati, F.; Sousa, M. C., Facing the high-dimensions: Inverse projection with radial basis functions, Comput. Graph., 48, 35-47, 2015 · doi:10.1016/j.cag.2015.02.009
[62] Delves, L. M.; Mohamed, J. L., Computational Methods for Integral Equations, 1988, CUP Archive · Zbl 0662.65111
[63] Press, W. H.; Teukolsky, S. A., Fredholm and Volterra integral equations of the second kind, Comput. Phys., 4, 554-557, 1990 · doi:10.1063/1.4822946
[64] Thiem, T. N.; Kooshkbaghi, M.; Bertalan, T.; Laing, C. R.; Kevrekidis, I. G., Emergent spaces for coupled oscillators, Front. Comput. Neurosci., 14, 36, 2020 · doi:10.3389/fncom.2020.00036
[65] Rabin, N.; Bregman, Y.; Lindenbaum, O.; Ben-Horin, Y.; Averbuch, A., Earthquake-explosion discrimination using diffusion maps, Geophys. J. Int., 207, 1484-1492, 2016 · doi:10.1093/gji/ggw348
[66] Prigogine, I.; Lefever, R., Symmetry breaking instabilities in dissipative systems. II, J. Chem. Phys., 48, 1695-1700, 1968 · doi:10.1063/1.1668896
[67] Baccalá, L. A.; Sameshima, K., Partial directed coherence: A new concept in neural structure determination, Biol. Cybern., 84, 463-474, 2001 · Zbl 1160.92306 · doi:10.1007/PL00007990
[68] Nicolaou, N.; Constandinou, T. G., A nonlinear causality estimator based on non-parametric multiplicative regression, Front. Neuroinform., 10, 19, 2016 · doi:10.3389/fninf.2016.00019
[69] Kevrekidis, P. G.; Siettos, C. I.; Kevrekidis, Y. G., To infinity and some glimpses of beyond, Nat. Commun., 8, 1562, 2017 · doi:10.1038/s41467-017-01502-7
[70] Shampine, L. F.; Reichelt, M. W., The MATLAB ODE suite, SIAM J. Sci. Comput., 18, 1-22, 1997 · Zbl 0868.65040 · doi:10.1137/S1064827594276424
[71] del Canto, A. B., “investpy—Financial Data Extraction from Investing.com with Python,” https://investpy.readthedocs.io/ (2020).
[72] Menkhoff, L.; Sarno, L.; Schmeling, M.; Schrimpf, A., Carry trades and global foreign exchange volatility, J. Finance, 67, 681-718, 2012 · doi:10.1111/j.1540-6261.2012.01728.x
[73] McKinney, W., “Data structures for statistical computing in Python,” in Proceedings of the 9th Python in Science Conference, Vol. 445 (Austin, TX, 2010), pp. 51-56.
[74] Braga, M. D., Risk parity versus other \(\mu \)-free strategies: A comparison in a triple view, Invest. Manag. Financ. Innov., 12, 277-289, 2015
[75] Lehmberg, D.; Dietrich, F.; Köster, G.; Bungartz, H.-J., Datafold: Data-driven models for point clouds and time series on manifolds, J. Open Source Softw., 5, 2283, 2020 · doi:10.21105/joss.02283
[76] Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V.; Vanderplas, J.; Passos, A.; Cournapeau, D.; Brucher, M.; Perrot, M.; Duchesnay, E., Scikit-learn: Machine learning in Python, J. Mach. Learn. Res., 12, 2825-2830, 2011 · Zbl 1280.68189
[77] Seabold, S. and Perktold, J., “Statsmodels: Econometric and statistical modeling with Python,” in Proceedings of the 9th Python in Science Conference, Vol. 57 (Austin, TX, 2010), p. 61.
[78] Lehoucq, R. B.; Sorensen, D. C.; Yang, C., ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, 1998, SIAM · Zbl 0901.65021
[79] Virtanen, P.; Gommers, R.; Oliphant, T. E., SciPy 1.0: Fundamental algorithms for scientific computing in Python, Nat. Methods, 17, 261, 2020 · doi:10.1038/s41592-019-0686-2
[80] Cuppen, J. J. M., A divide and conquer method for the symmetric tridiagonal eigenproblem, Numer. Math., 36, 177, 1980 · Zbl 0431.65022 · doi:10.1007/BF01396757
[81] Anderson, E.; Bai, Z.; Bischof, C.; Blackford, S.; Demmel, J.; Dongarra, J.; Du Croz, J.; Greenbaum, A.; Hammarling, S.; McKenney, A.; Sorensen, D., LAPACK Users’ Guide, 1999, SIAM: SIAM, Philadelphia, PA · Zbl 0934.65030
[82] Dsilva, C. J.; Talmon, R.; Coifman, R. R.; Kevrekidis, I. G., Parsimonious representation of nonlinear dynamical systems through manifold learning: A chemotaxis case study, Appl. Comput. Harmon. Anal., 44, 759-773, 2018 · Zbl 1390.68523 · doi:10.1016/j.acha.2015.06.008
[83] Singer, A.; Erban, R.; Kevrekidis, I. G.; Coifman, R. R., Detecting intrinsic slow variables in stochastic dynamical systems by anisotropic diffusion maps, Proc. Natl. Acad. Sci. U.S.A., 106, 16090-16095, 2009 · doi:10.1073/pnas.0905547106
[84] Byrd, R. H.; Lu, P.; Nocedal, J.; Zhu, C., A limited memory algorithm for bound constrained optimization, SIAM J. Sci. Comput., 16, 1190, 1995 · Zbl 0836.65080 · doi:10.1137/0916069
[85] Zhu, C.; Byrd, R. H.; Lu, P.; Nocedal, J., Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization, ACM Trans. Math. Softw., 23, 550, 1997 · Zbl 0912.65057 · doi:10.1145/279232.279236
[86] Maneewongvatana, S. and Mount, D. M., “Analysis of approximate nearest neighbor searching with clustered point sets,” in Data Structures, Near Neighbor Searches, and Methodology (American Mathematical Society, 2002), Vol. 59, pp. 105-123. · Zbl 1103.68490
[87] Evangelou, N., Dietrich, F., Chiavazzo, E., Lehmberg, D., Meila, M., and Kevrekidis, I. G., “Double diffusion maps and their latent harmonics for scientific computation in latent space,” arXiv:2204.12536 (2022). · Zbl 07690204
[88] Sharpe, W. F., The sharpe ratio, J. Portf. Manag., 21, 49, 1994 · doi:10.3905/jpm.1994.409501
[89] Papaioannou, P.; Russo, L.; Papaioannou, G.; Siettos, C. I., Can social microblogging be used to forecast intraday exchange rates?, NETNOMICS: Econ. Res. Electron. Netw., 14, 47, 2013 · doi:10.1007/s11066-013-9079-3
[90] Rasmussen, C. E., “Gaussian processes in machine learning,” in Summer School on Machine Learning (Springer, 2003), pp. 63-71. · Zbl 1120.68436
[91] Cheng, C.; Sa-Ngasoongsong, A.; Beyca, O.; Le, T.; Yang, H.; Kong, Z.; Bukkapatnam, S. T., Time series forecasting for nonlinear and non-stationary processes: A review and comparative study, IIE Trans., 47, 1053-1071, 2015 · doi:10.1080/0740817X.2014.999180
[92] Saad, Y., A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14, 461-469, 1993 · Zbl 0780.65022 · doi:10.1137/0914028
[93] Beatson, R. K.; Cherrie, J. B.; Mouat, C. T., Fast fitting of radial basis functions: Methods based on preconditioned GMRES iteration, Adv. Comput. Math., 11, 253-270, 1999 · Zbl 0940.65011 · doi:10.1023/A:1018932227617
[94] Eldén, L.; Simoncini, V., Solving ill-posed linear systems with GMRES and a singular preconditioner, SIAM J. Matrix Anal. Appl., 33, 1369-1394, 2012 · Zbl 1263.65033 · doi:10.1137/110832793
[95] Schwartz, A.; Talmon, R., Intrinsic isometric manifold learning with application to localization, SIAM J. Imaging Sci., 12, 1347-1391, 2019 · doi:10.1137/18M1198752
[96] Wu, H.-T.; Wu, N., Think globally, fit locally under the manifold setup: Asymptotic analysis of locally linear embedding, Ann. Stat., 46, 3805-3837, 2018 · Zbl 1405.62058
[97] Wu, H.-T.; Wu, N.
[98] Malik, J.; Shen, C.; Wu, H.-T.; Wu, N., Connecting dots: From local covariance to empirical intrinsic geometry and locally linear embedding, Pure Appl. Anal., 1, 515-542, 2019 · Zbl 1433.62142 · doi:10.2140/paa.2019.1.515
[99] Theodoropoulos, C.; Qian, Y.-H.; Kevrekidis, I. G., “Coarse” stability and bifurcation analysis using time-steppers: A reaction-diffusion example, Proc. Natl. Acad. Sci. U.S.A., 97, 9840-9843, 2000 · Zbl 1064.65121 · doi:10.1073/pnas.97.18.9840
[100] Kevrekidis, I. G.; Gear, C. W.; Hyman, J. M.; Kevrekidis, P. G.; Runborg, O.; Theodoropoulos, C., Equation-free, coarse-grained multiscale computation: Enabling microscopic simulators to perform system-level analysis, Commun. Math. Sci., 1, 715-762, 2003 · Zbl 1086.65066 · doi:10.4310/CMS.2003.v1.n4.a5
[101] Siettos, C. I.; Graham, M. D.; Kevrekidis, I. G., Coarse Brownian dynamics for nematic liquid crystals: Bifurcation, projective integration, and control via stochastic simulation, J. Chem. Phys., 118, 10149-10156, 2003 · doi:10.1063/1.1572456
[102] Kevrekidis, I. G.; Gear, C. W.; Hummer, G., Equation-free: The computer-aided analysis of complex multiscale systems, AIChE J., 50, 1346-1355, 2004 · doi:10.1002/aic.10106
[103] Gear, C. W.; Li, J.; Kevrekidis, I. G., The gap-tooth method in particle simulations, Phys. Lett. A, 316, 190-195, 2003 · Zbl 1031.82024 · doi:10.1016/j.physleta.2003.07.004
[104] Samaey, G.; Roose, D.; Kevrekidis, I. G., The gap-tooth scheme for homogenization problems, Multiscale Model. Simul., 4, 278-306, 2005 · Zbl 1092.35009 · doi:10.1137/030602046
[105] Samaey, G.; Kevrekidis, I. G.; Roose, D., Patch dynamics with buffers for homogenization problems, J. Comput. Phys., 213, 264-287, 2006 · Zbl 1092.65074 · doi:10.1016/j.jcp.2005.08.010
[106] Choi, H.; Choi, S., Robust kernel Isomap, Pattern Recognit., 40, 853-862, 2007 · Zbl 1118.68188 · doi:10.1016/j.patcog.2006.04.025
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