Adaptive POD-DEIM correction for Turing pattern approximation in reaction–diffusion PDE systems

References

[1] A. Alla and M. Falcone, A time-adaptive POD method for optimal control problems, IFAC Proc. Volumes 46 (2013), No. 26, 245–250.10.3182/20130925-3-FR-4043.00042Suche in Google Scholar

[2] A. Alla and J. N. Kutz, Randomized model order reduction, Adv. Comput. Math. 45 (2019), 1251–1271.10.1007/s10444-018-09655-9Suche in Google Scholar

[3] D. Amsallem and C. Farhat, Stabilization of projection-based reduced-order models, Int. J. Numer. Methods Eng. 91 (2012), No. 4, 358–377.10.1002/nme.4274Suche in Google Scholar

[4] U. M. Ascher, S. J. Ruuth, and B. T. R.Wetton, Implicit–explicit methods for time dependent PDE’s, SIAM J. Numer. Anal. 32 (1995), No. 3, 797–823.10.1137/0732037Suche in Google Scholar

[5] U. M. Ascher, K. van den Doel, H. Huang, and B. F. Svaiter, Gradient descent and fast artificial time integration, Math. Modelling Numer. Anal. 43 (2009), No. 4, 689–708.10.1051/m2an/2009025Suche in Google Scholar

[6] F. Ballarin, A. Manzoni, A. Quarteroni, and G. Rozza, Supremizer stabilization of POD–Galerkin approximation of parametrized steady incompressible Navier–Stokes equations, Int. J. Numer. Methods Eng. 102 (2015), No. 5, 1136–1161.10.1002/nme.4772Suche in Google Scholar

[7] 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, Comptes Rendus Mathematique 339 (2004), 667–672.10.1016/j.crma.2004.08.006Suche in Google Scholar

[8] R. Barreira, C. M. Elliott, and A. Madzvamuse, The surface finite element method for pattern formation on evolving biological surfaces, J. Math. Biology 63 (2011), No. 6, 1095–1119.10.1007/s00285-011-0401-0Suche in Google Scholar PubMed

[9] P. Benner and L. Feng, Model order reduction for coupled problems (Survey), Applied and Comput. Math. 14 (2015), 3–22.Suche in Google Scholar

[10] P. Benner, S. Gugercin, and K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev. 57 (2015), 483–531.10.1137/130932715Suche in Google Scholar

[11] M. Benosman, J. Borggaard, O. San, and B. Kramer, Learning-based robust stabilization for reduced-order models of 2D and 3D Boussinesq equations, Applied Math. Modelling 49 (2017), 162–181.10.1016/j.apm.2017.04.032Suche in Google Scholar

[12] B. Bozzini, G. Gambino, D. Lacitignola, S. Lupo, M. Sammartino, and I. Sgura, Weakly nonlinear analysis of Turing patterns in a morphochemical model for metal growth, Computers & Mathematics with Applications 70 (2015), No. 8, 1948–1969.10.1016/j.camwa.2015.08.019Suche in Google Scholar

[13] B. Bozzini, D. Lacitignola, and I. Sgura, Spatio-temporal organization in alloy electrodeposition: a morphochemical mathematical model and its experimental validation, J. Solid State Electrochemistry 17 (2013), No. 2, 467–479.10.1007/s10008-012-1945-7Suche in Google Scholar

[14] B. Bozzini, A. Monti, and I. Sgura, Model-reduction techniques for PDE models with Turing type electrochemical phase formation dynamics, Appl. Eng. Sci. 8 (2021), 1–10.10.1016/j.apples.2021.100074Suche in Google Scholar

[15] M. A. J. Chaplain, M. Ganesh, and I. G. Graham, Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth, J. Math. Biology 42 (2001), No. 5, 387–423.10.1007/s002850000067Suche in Google Scholar PubMed

[16] M. A. J. Chaplain and A. Gerisch, Robust numerical methods for taxis–diffusion–reaction systems: applications to biomedical problems, Math. Comp. Modelling 43 (2006), 49–75.10.1016/j.mcm.2004.05.016Suche in Google Scholar

[17] M. A. J. Chaplain and J. A. Sherratt, A new mathematical model for avascular tumour growth, J. Math. Biol. 43 (2001), 291–312.10.1007/s002850100088Suche in Google Scholar PubMed

[18] S. Chatarantabut and D. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput. 32 (2010), 2737–2764.10.1137/090766498Suche in Google Scholar

[19] M. C. D’Autilia, I. Sgura, and V. Simoncini, Matrix-oriented discretization methods for reaction–diffusion PDEs: Comparisons and applications, Comput. Math. Appl. 79 (2020), 2067–2085.10.1016/j.camwa.2019.10.020Suche in Google Scholar

[20] Z. Drmac and S. Gugercin, A new selection operator for the discrete empirical interpolation method –improved a priori error bound and extensions, SIAM J. Sci. Comput. 38 (2016), A631–A648.10.1137/15M1019271Suche in Google Scholar

[21] G. Gambino, M. C. Lombardo, G. Rubino, and M. Sammartino, Pattern selection in the 2D FitzHugh–Nagumo model, Ricerche di Matematica 68 (2019), 535–549.10.1007/s11587-018-0424-6Suche in Google Scholar

[22] D. A. Garzon-Alvarado, A. M. R. Martinez, and D. L. L. Segrera, A model of cerebral cortex formation during fetal development using reaction–diffusion–convection equations with Turing space parameters, Computer Methods and Programs in Biomedicine 104 (2011), No. 3, 489–497.10.1016/j.cmpb.2011.07.001Suche in Google Scholar PubMed

[23] E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak, and E. Meron, Ecosystem engineers: From pattern formation to habitat creation, Phys. Rev. Lett. 93 (2004), 098105.10.1103/PhysRevLett.93.098105Suche in Google Scholar PubMed

[24] G. H. Golub and C. F. Van Loan, Matrix Computations, The Johns Hopkins University Press, 1996.Suche in Google Scholar

[25] C. Gräßle, M. Hinze, and S. Volkwein, Snapshot-Based Methods and Algorithms, Volume 2 of Model Order Reduction, De Gruyter, 2020.Suche in Google Scholar

[26] P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge Monographs on Mechanics, Cambridge University Press, 1996.10.1017/CBO9780511622700Suche in Google Scholar

[27] W. Hundsdorfer and J. Verwer, Numerical solution of time-dependent advection–diffusion–reaction equations, Springer Series in Computational Mathematics 33 (2003).10.1007/978-3-662-09017-6Suche in Google Scholar

[28] B. Karasözen, M. Uzunca, and T. Küçükseyhan, Model order reduction for pattern formation in FitzHugh–Nagumo equations. In: Numerical Mathematics and Advanced Applications ENUMATH 2015 (Eds. B. Karasözen, M. Manguoğlu, M. Tezer-Sezgin, S. Göktepe, and Ö. Uğur), Springer, Cham, 2016, pp. 369–377.10.1007/978-3-319-39929-4_35Suche in Google Scholar

[29] B. Karasözen, G. Mülayim, M. Uzunca, and S. Yıldız, Reduced order modelling of nonlinear cross-diffusion systems, Applied Mathematics and Computation 401 (2021), 126058.10.1016/j.amc.2021.126058Suche in Google Scholar

[30] G. Kirsten, Multilinear POD-DEIM model reduction for 2D and 3D semilinear systems of differential equations, J. Comput. Dynamics (2021), 1–25.10.3934/jcd.2021025Suche in Google Scholar

[31] G. Kirsten and V. Simoncini, A matrix-oriented POD-DEIM algorithm applied to nonlinear differential matrix equations, arXiv:2006.13289, 2021.Suche in Google Scholar

[32] K. Kunisch and S. Volkwein, Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optimization Theory and Applications 102 (1999), 345–371.10.1023/A:1021732508059Suche in Google Scholar

[33] J. N. Kutz, S. L. Brunton, B.W. Brunton, and J. L. Proctor, Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems, SIAM, 2016.10.1137/1.9781611974508Suche in Google Scholar

[34] D. Lacitignola, B. Bozzini, M. Frittelli, and I. Sgura, Turing pattern formation on the sphere for a morphochemical reaction–diffusion model for electrodeposition, Communications in Nonlinear Science and Numerical Simulation 48 (2017), 484–508.10.1016/j.cnsns.2017.01.008Suche in Google Scholar

[35] D. Lacitignola, B. Bozzini, and I. Sgura, Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay, Europ. J. Appl. Math. 26 (2015), No. 2, 143–173.10.1017/S0956792514000370Suche in Google Scholar

[36] J. Lefèvre and J.-F. Mangin, A reaction–diffusion model of human brain development, PLoS Comput. Biology 6 (2010), No. 4, e1000749.10.1371/journal.pcbi.1000749Suche in Google Scholar PubMed PubMed Central

[37] A. Madzvamuse, Time-stepping schemes for moving grid finite elements applied to reaction–diffusion systems on fixed and growing domains, J. Comput. Phys. 214 (2006), No. 2, 239–263.10.1016/j.jcp.2005.09.012Suche in Google Scholar

[38] A. Madzvamuse, A. J.Wathen, and P. K. Maini, A moving grid finite element method applied to a model biological pattern generator, J. Comput. Physics 190 (2003), No. 2, 478–500.10.1016/S0021-9991(03)00294-8Suche in Google Scholar

[39] P. Maini and H. Othmer, Mathematical Models for Biological Pattern Formation. The IMA Volumes in Mathematics and its Applications - Frontiers in application of Mathematics, Springer-Verlag, New York, 2001.10.1007/978-1-4613-0133-2Suche in Google Scholar

[40] H. Malchow, S. Petrowski, and E. Venturino, Spatio Temporal Patterns in Ecology and Epidemiology, Chapman & Hall, UK, 2008.10.1201/9781482286137Suche in Google Scholar

[41] T. T. Marquez-Lago and P. Padilla, A selection criterion for patterns in reaction–diffusion systems, Theor. Biol. Med. Modelling 11 (2014), No. 7, 1093–1121.10.1186/1742-4682-11-7Suche in Google Scholar PubMed PubMed Central

[42] C. Mou, H. Liu, D. R.Wells, and T. Iliescu, Data-driven correction reduced order models for the quasi-geostrophic equations: a numerical investigation, Int. J. Comut. Fluid Dyn. 34 (2020), 147–159.10.1080/10618562.2020.1723556Suche in Google Scholar

[43] J. D. Murray, Mathematical Biology II –Spatial Models and Biomedical Applications. Interdisciplinary Applied Mathematics, Vol. 18, Springer-Verlag, Berlin–Heidelberg, 2003.10.1080/10618562.2020.1723556Suche in Google Scholar

[44] H. Müller and S. Volkwein, Model reduction by proper orthogonal decomposition for lambda–omega systems, In: Proc. of European Conf. on Computational Fluid Dynamics (ECCOMAS CFD), Egmont aan Zee, 2006 (Eds. P.Wesseling, E. Onate, and J. Periaux), 2006.10.1007/b98869Suche in Google Scholar

[45] M. G. Neubert and H. Caswell, Alternatives to resilience for measuring the responses of ecological systems to perturbations, Ecology, The Ecological Society of America 78 (1997), 653–665.10.1890/0012-9658(1997)078[0653:ATRFMT]2.0.CO;2Suche in Google Scholar

[46] M. G. Neubert, H. Caswell, and J. D. Murray, Transient dynamics and pattern formation: reactivity is necessary for Turing instabilities, Math. Biosciences 175 (2002), 1–11.10.1016/S0025-5564(01)00087-6Suche in Google Scholar

[47] K. J. Painter, Mathematical models for chemotaxis and their applications in self-organisation phenomena, J. Theoretical Biology 481 (2019), 162–182.10.1016/j.jtbi.2018.06.019Suche in Google Scholar PubMed

[48] B. Peherstorfer, D. Butnaru, K. Willcox, and H. Bungartz, Localized discrete empirical interpolation method, SIAM J. Sci. Comp. 36 (2014), A168–A192.10.1137/130924408Suche in Google Scholar

[49] T. Reis and T. Stykel, Stability analysis and model order reduction of coupled systems, Mathematical and Computer Modelling of Dynamical Systems 13 (2007), No. 5, 413–436.10.1080/13873950701189071Suche in Google Scholar

[50] T. Reis and T. Stykel, A survey on model reduction of coupled systems, In: Model Order Reduction: Theory, Research Aspects and Applications, Springer, Berlin–Heidelberg, 2008, pp. 133–155.10.1007/978-3-540-78841-6_7Suche in Google Scholar

[51] S. J. Ruuth, Implicit–explicit methods for reaction–diffusion problems in pattern formation, J. Math. Biology 34 (1995), No. 2, 148–176.10.1007/BF00178771Suche in Google Scholar

[52] G. Settanni and I. Sgura, Devising efficient numerical methods for oscillating patterns in reaction–diffusion systems, J. Comput. Appl. Math. 292 (2016), 674–693.10.1016/j.cam.2015.04.044Suche in Google Scholar

[53] I. Sgura, B. Bozzini, and D. Lacitignola, Numerical approximation of Turing patterns in electrodeposition by ADI methods, J. Comput. Appl. Math. 236 (2012), No. 16, 4132–4147.10.1016/j.cam.2012.03.013Suche in Google Scholar

[54] I. Sgura, A. Lawless, and B. Bozzini, Parameter estimation for a morphochemical reaction–diffusion model of electrochemical pattern formation, Inverse Probl. Sci. Eng. 27 (2019), 618–647.10.1080/17415977.2018.1490278Suche in Google Scholar

[55] J. Sherratt, Turing pattern in desert, In: How the World Computes (Eds. S. B. Cooper and A. Dawar), Lecture Notes in Computer Science, Vol. 7318, 2012.10.1007/978-3-642-30870-3_67Suche in Google Scholar

[56] L. Sirovich, Turbulence and the dynamics of coherent structures, Parts I-II, Quarterly of Applied Mathematics (1987), 561–59.10.1090/qam/910462Suche in Google Scholar

[57] D.Wells, Z.Wang, X. Xie, and T. Iliescu, An evolve-then-filter regularized reduced order model for convection-dominated flows, Int. J. Numer. Methods Fluids 84 (2017), No. 10, 598–615.10.1002/fld.4363Suche in Google Scholar

[58] X. Xie, M. Mohebujjaman, L. G. Rebholz, and T. Iliescu, Data-driven filtered reduced order modeling of fluid flows, SIAM J. Sci. Com-put. 40 (2018), B834–B857.10.1137/17M1145136Suche in Google Scholar

[59] X. Xie, C.Webster, and T. Iliescu, Closure learning for nonlinear model reduction using deep residual neural network, Fluids 5 (2020), No. 1, 39.10.3390/fluids5010039Suche in Google Scholar