doi: 10.3934/jcd.2021025
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Multilinear POD-DEIM model reduction for 2D and 3D semilinear systems of differential equations

Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato, 5, I-40127 Bologna, Italy

Received  March 2021 Revised  July 2021 Early access December 2021

Fund Project: The author is a member of Indam-GNCS, which support is gratefully acknowledged

We are interested in the numerical solution of coupled semilinear partial differential equations (PDEs) in two and three dimensions. Under certain assumptions on the domain, we take advantage of the Kronecker structure arising in standard space discretizations of the differential operators and illustrate how the resulting system of ordinary differential equations (ODEs) can be treated directly in matrix or tensor form. Moreover, in the framework of the proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM) we derive a two- and three-sided model order reduction strategy that is applied directly to the ODE system in matrix and tensor form respectively. We discuss how to integrate the reduced order model and, in particular, how to solve the tensor-valued linear system arising at each timestep of a semi-implicit time discretization scheme. We illustrate the efficiency of the proposed method through a comparison to existing techniques on classical benchmark problems such as the two- and three-dimensional Burgers equation.

Citation: Gerhard Kirsten. Multilinear POD-DEIM model reduction for 2D and 3D semilinear systems of differential equations. Journal of Computational Dynamics, doi: 10.3934/jcd.2021025
References:
[1]

A. Antoulas, C. Beattie and S. Gugercin, Interpolatory Methods for Model Reduction, SIAM, Philidelphia, 2020. doi: 10.1137/1.9781611976083.  Google Scholar

[2]

U. M. AscherS. J. Ruuth and B. T. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), 797-823.  doi: 10.1137/0732037.  Google Scholar

[3]

P. AstridS. WeilandK. Willcox and T. Backx, Missing point estimation in models described by proper orthogonal decomposition, IEEE Trans. Autom. Control, 53 (2008), 2237-2251.  doi: 10.1109/TAC.2008.2006102.  Google Scholar

[4]

M. BarraultY. MadayN. 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), 667-672.  doi: 10.1016/j.crma.2004.08.006.  Google Scholar

[5]

P. Benner, V. Mehrmann and D. Sorensen, Dimension Reduction of Large-scale Systems, Lecture Notes in Computational Science and Engineering, 45. Springer, Berlin, 2005. doi: 10.1007/3-540-27909-1.  Google Scholar

[6]

P. BennerS. Gugercin and K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev., 57 (2015), 483-531.  doi: 10.1137/130932715.  Google Scholar

[7]

D. BonomiA. Manzoni and A. Quarteroni, A matrix DEIM technique for model reduction of nonlinear parametrized problems in cardiac mechanics, Comput. Methods Appl. Mech. Eng., 324 (2017), 300-326.  doi: 10.1016/j.cma.2017.06.011.  Google Scholar

[8]

S. Chaturantabut and D. C. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32 (2010), 2737-2764.  doi: 10.1137/090766498.  Google Scholar

[9]

S. Chaturantabut and D. C. Sorensen, Application of POD and DEIM on dimension reduction of non-linear miscible viscous fingering in porous media, Math. Comput. Modell. Dyn. Syst., 17 (2011), 337-353.  doi: 10.1080/13873954.2011.547660.  Google Scholar

[10]

M. Daub, Mathematical Modeling and Numerical Simulations of the Extrinsic Pro-Apoptotic Signaling Pathway, PhD thesis, University of Stuttgart, 2013. Google Scholar

[11]

M. DaubS. WaldherrF. AllgöwerP. Scheurich and G. Schneider, Death wins against life in a spatially extended apoptosis model, Biosystems, 108 (2012), 45-51.   Google Scholar

[12]

M. C. D'AutiliaI. Sgura and V. Simoncini, Matrix-oriented discretization methods for reaction–diffusion PDEs: Comparisons and applications, Comput. Math. Appl., 79 (2020), 2067-2085.  doi: 10.1016/j.camwa.2019.10.020.  Google Scholar

[13]

A. De Wit, Spatial patterns and spatiotemporal dynamics in chemical systems, Advances in Chemical Physics, 109 (1999), 435-513.  doi: 10.1002/9780470141687.ch5.  Google Scholar

[14]

Z. Drmač 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.  doi: 10.1137/15M1019271.  Google Scholar

[15]

C. A. Fletcher, Generating exact solutions of the two-dimensional Burgers' equations, Int. J. Numer. Methods Fluids, 3 (1983), 213-216.   Google Scholar

[16]

G. GambinoM. Lombardo and M. Sammartino, Pattern selection in the 2D FitzHugh–Nagumo model, Ric. Mat., 68 (2019), 535-549.  doi: 10.1007/s11587-018-0424-6.  Google Scholar

[17]

Q. Gao and M. Zou, An analytical solution for two and three dimensional nonlinear Burgers' equation, Appl. Math. Modell., 45 (2017), 255-270.  doi: 10.1016/j.apm.2016.12.018.  Google Scholar

[18]

U. Z. GeorgeA. Stéphanou and A. Madzvamuse, Mathematical modelling and numerical simulations of actin dynamics in the eukaryotic cell, J. Math. Biol., 66 (2013), 547-593.  doi: 10.1007/s00285-012-0521-1.  Google Scholar

[19] G. H. Golub and C. F. van Loan, Matrix Computations, 4$^{th}$ edition, Johns Hopkins University Press, Baltimore, 2013.   Google Scholar
[20]

C. Gu, QLMOR: A projection-based nonlinear model order reduction approach using quadratic-linear representation of nonlinear systems, IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., 30 (2011), 1307-1320.   Google Scholar

[21]

N. HalkoP.-G. Martinsson and J. A. Tropp, Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53 (2011), 217-288.  doi: 10.1137/090771806.  Google Scholar

[22]

M. Hinze and S. Volkwein, Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: Error estimates and suboptimal control, Dimension Reduction of Large-Scale Systems, 45 (2005), 261-306.  doi: 10.1007/3-540-27909-1_10.  Google Scholar

[23]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of physiology, 117 (1952), 500-544.   Google Scholar

[24]

W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer Series in Computational Mathematics, 33. Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-09017-6.  Google Scholar

[25]

B. KarasözenM. Uzunca and T. Küçükseyhan, Model order reduction for pattern formation in Fitzhugh-Nagumo equations, Numerical Mathematics and Advanced Applications ENUMATH 2015, 112 (2016), 23-31.  doi: 10.1007/978-3-319-39929-4_3.  Google Scholar

[26]

B. KarasözenM. Uzunca and T. Küçükseyhan, Reduced order optimal control of the convective Fitzhugh–Nagumo equations, Comput. Math. Appl., 79 (2020), 982-995.  doi: 10.1016/j.camwa.2019.08.009.  Google Scholar

[27]

B. KarasözenS. Yıldız and M. Uzunca, Structure preserving model order reduction of shallow water equations, Math. Methods Appl. Sci., 44 (2021), 476-492.  doi: 10.1002/mma.6751.  Google Scholar

[28]

G. Kirsten and V. Simoncini, A matrix-oriented POD-DEIM algorithm applied to nonlinear differential matrix equations, preprint, arXiv: 2006.13289. Google Scholar

[29]

T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), 455-500.  doi: 10.1137/07070111X.  Google Scholar

[30]

B. Kramer, Model Reduction of the Coupled Burgers Equation in Conservation Form, PhD thesis, Virginia Tech, 2011. Google Scholar

[31]

B. Kramer and K. E. Willcox, Nonlinear model order reduction via lifting transformations and proper orthogonal decomposition, AIAA Journal, 57 (2019), 2297-2307.  doi: 10.2514/1.J057791.  Google Scholar

[32]

K. Kunisch and S. Volkwein, Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl., 102 (1999), 345-371.  doi: 10.1023/A:1021732508059.  Google Scholar

[33]

P. K. Maini and H. G. 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. Google Scholar

[34]

H. Malchow, S. Petrovskii and E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulations, Chapman & Hall, CRC, Boca Raton, FL, 2008.  Google Scholar

[35]

The MathWorks, MATLAB 7, r2013b edition, 2013. Google Scholar

[36]

R. MinsterA. K. Saibaba and M. E. Kilmer, Randomized algorithms for low-rank tensor decompositions in the Tucker format, SIAM J. Math. Data Sci., 2 (2020), 189-215.  doi: 10.1137/19M1261043.  Google Scholar

[37]

J. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, 3$^{rd}$ edition, Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.  Google Scholar

[38]

F. NegriA. Manzoni and D. Amsallem, Efficient model reduction of parametrized systems by matrix discrete empirical interpolation, J. Comput. Phys., 303 (2015), 431-454.  doi: 10.1016/j.jcp.2015.09.046.  Google Scholar

[39]

N.-C. NguyenA. T. Patera and J. Peraire, A 'best points' interpolation method for efficient approximation of parametrized functions, Internat. J. Numer. Methods Engrg., 73 (2008), 521-543.  doi: 10.1002/nme.2086.  Google Scholar

[40]

D. Palitta and V. Simoncini, Matrix-equation-based strategies for convection–diffusion equations, BIT, 56 (2016), 751-776.  doi: 10.1007/s10543-015-0575-8.  Google Scholar

[41]

A. T. Patera and G. Rozza, Reduced Basis Approximation and A posteriori Error Estimation for Parametrized Partial Differential Equations, MIT Cambridge, MA, USA, 2007. Google Scholar

[42]

A. Quarteroni, Numerical Models for Differential Problems, vol. 8 of MS & A - Modeling, Simulation and Applications, Springer-Verlag, Milan, 2014. doi: 10.1007/978-88-470-5522-3.  Google Scholar

[43]

S. J. Ruuth, Implicit-explicit methods for reaction-diffusion problems in pattern formation, J. Math. Biol., 34 (1995), 148-176.  doi: 10.1007/BF00178771.  Google Scholar

[44]

S. Sahyoun and S. M. Djouadi, Nonlinear model reduction using space vectors clustering POD with application to the Burgers' equation, 2014 American Control Conference, IEEE, (2014), 1661–1666. doi: 10.1109/ACC.2014.6859104.  Google Scholar

[45]

J. A. Sherratt and M. A. Chaplain, A new mathematical model for avascular tumour growth, J. Math. Biol., 43 (2001), 291-312.  doi: 10.1007/s002850100088.  Google Scholar

[46]

V. Simoncini, Numerical solution of a class of third order tensor linear equations, BUMI, 13 (2020), 429-439.  doi: 10.1007/s40574-020-00247-4.  Google Scholar

[47]

V. Simoncini, Computational methods for linear matrix equations, SIAM Rev., 58 (2016), 377-441.  doi: 10.1137/130912839.  Google Scholar

[48]

R. ŞtefănescuA. Sandu and I. M. Navon, Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations, Internat. J. Numer. Methods Fluids, 76 (2014), 497-521.  doi: 10.1002/fld.3946.  Google Scholar

[49]

J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, SIAM, 2004. doi: 10.1137/1.9780898717938.  Google Scholar

[50]

A. Tveito, H. P. Langtangen, B. F. Nielsen and X. Cai, Elements of Scientific Computing, Texts in Computational Science and Engineering, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11299-7.  Google Scholar

[51]

V. K. Vanag, Waves and patterns in reaction–diffusion systems. Belousov–Zhabotinsky reaction in water-in-oil microemulsions, Phys. Usp., 47 (2004), 923.   Google Scholar

[52]

N. VannieuwenhovenR. Vandebril and K. Meerbergen, A new truncation strategy for the higher-order singular value decomposition, SIAM J. Sci. Comput., 34 (2012), A1027-A1052.  doi: 10.1137/110836067.  Google Scholar

[53]

Y. WangI. M. NavonX. Wang and Y. Cheng, 2D Burgers equation with large Reynolds number using POD/DEIM and calibration, Internat. J. Numer. Methods Fluids, 82 (2016), 909-931.  doi: 10.1002/fld.4249.  Google Scholar

show all references

References:
[1]

A. Antoulas, C. Beattie and S. Gugercin, Interpolatory Methods for Model Reduction, SIAM, Philidelphia, 2020. doi: 10.1137/1.9781611976083.  Google Scholar

[2]

U. M. AscherS. J. Ruuth and B. T. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32 (1995), 797-823.  doi: 10.1137/0732037.  Google Scholar

[3]

P. AstridS. WeilandK. Willcox and T. Backx, Missing point estimation in models described by proper orthogonal decomposition, IEEE Trans. Autom. Control, 53 (2008), 2237-2251.  doi: 10.1109/TAC.2008.2006102.  Google Scholar

[4]

M. BarraultY. MadayN. 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), 667-672.  doi: 10.1016/j.crma.2004.08.006.  Google Scholar

[5]

P. Benner, V. Mehrmann and D. Sorensen, Dimension Reduction of Large-scale Systems, Lecture Notes in Computational Science and Engineering, 45. Springer, Berlin, 2005. doi: 10.1007/3-540-27909-1.  Google Scholar

[6]

P. BennerS. Gugercin and K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev., 57 (2015), 483-531.  doi: 10.1137/130932715.  Google Scholar

[7]

D. BonomiA. Manzoni and A. Quarteroni, A matrix DEIM technique for model reduction of nonlinear parametrized problems in cardiac mechanics, Comput. Methods Appl. Mech. Eng., 324 (2017), 300-326.  doi: 10.1016/j.cma.2017.06.011.  Google Scholar

[8]

S. Chaturantabut and D. C. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32 (2010), 2737-2764.  doi: 10.1137/090766498.  Google Scholar

[9]

S. Chaturantabut and D. C. Sorensen, Application of POD and DEIM on dimension reduction of non-linear miscible viscous fingering in porous media, Math. Comput. Modell. Dyn. Syst., 17 (2011), 337-353.  doi: 10.1080/13873954.2011.547660.  Google Scholar

[10]

M. Daub, Mathematical Modeling and Numerical Simulations of the Extrinsic Pro-Apoptotic Signaling Pathway, PhD thesis, University of Stuttgart, 2013. Google Scholar

[11]

M. DaubS. WaldherrF. AllgöwerP. Scheurich and G. Schneider, Death wins against life in a spatially extended apoptosis model, Biosystems, 108 (2012), 45-51.   Google Scholar

[12]

M. C. D'AutiliaI. Sgura and V. Simoncini, Matrix-oriented discretization methods for reaction–diffusion PDEs: Comparisons and applications, Comput. Math. Appl., 79 (2020), 2067-2085.  doi: 10.1016/j.camwa.2019.10.020.  Google Scholar

[13]

A. De Wit, Spatial patterns and spatiotemporal dynamics in chemical systems, Advances in Chemical Physics, 109 (1999), 435-513.  doi: 10.1002/9780470141687.ch5.  Google Scholar

[14]

Z. Drmač 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.  doi: 10.1137/15M1019271.  Google Scholar

[15]

C. A. Fletcher, Generating exact solutions of the two-dimensional Burgers' equations, Int. J. Numer. Methods Fluids, 3 (1983), 213-216.   Google Scholar

[16]

G. GambinoM. Lombardo and M. Sammartino, Pattern selection in the 2D FitzHugh–Nagumo model, Ric. Mat., 68 (2019), 535-549.  doi: 10.1007/s11587-018-0424-6.  Google Scholar

[17]

Q. Gao and M. Zou, An analytical solution for two and three dimensional nonlinear Burgers' equation, Appl. Math. Modell., 45 (2017), 255-270.  doi: 10.1016/j.apm.2016.12.018.  Google Scholar

[18]

U. Z. GeorgeA. Stéphanou and A. Madzvamuse, Mathematical modelling and numerical simulations of actin dynamics in the eukaryotic cell, J. Math. Biol., 66 (2013), 547-593.  doi: 10.1007/s00285-012-0521-1.  Google Scholar

[19] G. H. Golub and C. F. van Loan, Matrix Computations, 4$^{th}$ edition, Johns Hopkins University Press, Baltimore, 2013.   Google Scholar
[20]

C. Gu, QLMOR: A projection-based nonlinear model order reduction approach using quadratic-linear representation of nonlinear systems, IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., 30 (2011), 1307-1320.   Google Scholar

[21]

N. HalkoP.-G. Martinsson and J. A. Tropp, Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53 (2011), 217-288.  doi: 10.1137/090771806.  Google Scholar

[22]

M. Hinze and S. Volkwein, Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: Error estimates and suboptimal control, Dimension Reduction of Large-Scale Systems, 45 (2005), 261-306.  doi: 10.1007/3-540-27909-1_10.  Google Scholar

[23]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of physiology, 117 (1952), 500-544.   Google Scholar

[24]

W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer Series in Computational Mathematics, 33. Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-662-09017-6.  Google Scholar

[25]

B. KarasözenM. Uzunca and T. Küçükseyhan, Model order reduction for pattern formation in Fitzhugh-Nagumo equations, Numerical Mathematics and Advanced Applications ENUMATH 2015, 112 (2016), 23-31.  doi: 10.1007/978-3-319-39929-4_3.  Google Scholar

[26]

B. KarasözenM. Uzunca and T. Küçükseyhan, Reduced order optimal control of the convective Fitzhugh–Nagumo equations, Comput. Math. Appl., 79 (2020), 982-995.  doi: 10.1016/j.camwa.2019.08.009.  Google Scholar

[27]

B. KarasözenS. Yıldız and M. Uzunca, Structure preserving model order reduction of shallow water equations, Math. Methods Appl. Sci., 44 (2021), 476-492.  doi: 10.1002/mma.6751.  Google Scholar

[28]

G. Kirsten and V. Simoncini, A matrix-oriented POD-DEIM algorithm applied to nonlinear differential matrix equations, preprint, arXiv: 2006.13289. Google Scholar

[29]

T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), 455-500.  doi: 10.1137/07070111X.  Google Scholar

[30]

B. Kramer, Model Reduction of the Coupled Burgers Equation in Conservation Form, PhD thesis, Virginia Tech, 2011. Google Scholar

[31]

B. Kramer and K. E. Willcox, Nonlinear model order reduction via lifting transformations and proper orthogonal decomposition, AIAA Journal, 57 (2019), 2297-2307.  doi: 10.2514/1.J057791.  Google Scholar

[32]

K. Kunisch and S. Volkwein, Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl., 102 (1999), 345-371.  doi: 10.1023/A:1021732508059.  Google Scholar

[33]

P. K. Maini and H. G. 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. Google Scholar

[34]

H. Malchow, S. Petrovskii and E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulations, Chapman & Hall, CRC, Boca Raton, FL, 2008.  Google Scholar

[35]

The MathWorks, MATLAB 7, r2013b edition, 2013. Google Scholar

[36]

R. MinsterA. K. Saibaba and M. E. Kilmer, Randomized algorithms for low-rank tensor decompositions in the Tucker format, SIAM J. Math. Data Sci., 2 (2020), 189-215.  doi: 10.1137/19M1261043.  Google Scholar

[37]

J. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, 3$^{rd}$ edition, Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003.  Google Scholar

[38]

F. NegriA. Manzoni and D. Amsallem, Efficient model reduction of parametrized systems by matrix discrete empirical interpolation, J. Comput. Phys., 303 (2015), 431-454.  doi: 10.1016/j.jcp.2015.09.046.  Google Scholar

[39]

N.-C. NguyenA. T. Patera and J. Peraire, A 'best points' interpolation method for efficient approximation of parametrized functions, Internat. J. Numer. Methods Engrg., 73 (2008), 521-543.  doi: 10.1002/nme.2086.  Google Scholar

[40]

D. Palitta and V. Simoncini, Matrix-equation-based strategies for convection–diffusion equations, BIT, 56 (2016), 751-776.  doi: 10.1007/s10543-015-0575-8.  Google Scholar

[41]

A. T. Patera and G. Rozza, Reduced Basis Approximation and A posteriori Error Estimation for Parametrized Partial Differential Equations, MIT Cambridge, MA, USA, 2007. Google Scholar

[42]

A. Quarteroni, Numerical Models for Differential Problems, vol. 8 of MS & A - Modeling, Simulation and Applications, Springer-Verlag, Milan, 2014. doi: 10.1007/978-88-470-5522-3.  Google Scholar

[43]

S. J. Ruuth, Implicit-explicit methods for reaction-diffusion problems in pattern formation, J. Math. Biol., 34 (1995), 148-176.  doi: 10.1007/BF00178771.  Google Scholar

[44]

S. Sahyoun and S. M. Djouadi, Nonlinear model reduction using space vectors clustering POD with application to the Burgers' equation, 2014 American Control Conference, IEEE, (2014), 1661–1666. doi: 10.1109/ACC.2014.6859104.  Google Scholar

[45]

J. A. Sherratt and M. A. Chaplain, A new mathematical model for avascular tumour growth, J. Math. Biol., 43 (2001), 291-312.  doi: 10.1007/s002850100088.  Google Scholar

[46]

V. Simoncini, Numerical solution of a class of third order tensor linear equations, BUMI, 13 (2020), 429-439.  doi: 10.1007/s40574-020-00247-4.  Google Scholar

[47]

V. Simoncini, Computational methods for linear matrix equations, SIAM Rev., 58 (2016), 377-441.  doi: 10.1137/130912839.  Google Scholar

[48]

R. ŞtefănescuA. Sandu and I. M. Navon, Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations, Internat. J. Numer. Methods Fluids, 76 (2014), 497-521.  doi: 10.1002/fld.3946.  Google Scholar

[49]

J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, SIAM, 2004. doi: 10.1137/1.9780898717938.  Google Scholar

[50]

A. Tveito, H. P. Langtangen, B. F. Nielsen and X. Cai, Elements of Scientific Computing, Texts in Computational Science and Engineering, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11299-7.  Google Scholar

[51]

V. K. Vanag, Waves and patterns in reaction–diffusion systems. Belousov–Zhabotinsky reaction in water-in-oil microemulsions, Phys. Usp., 47 (2004), 923.   Google Scholar

[52]

N. VannieuwenhovenR. Vandebril and K. Meerbergen, A new truncation strategy for the higher-order singular value decomposition, SIAM J. Sci. Comput., 34 (2012), A1027-A1052.  doi: 10.1137/110836067.  Google Scholar

[53]

Y. WangI. M. NavonX. Wang and Y. Cheng, 2D Burgers equation with large Reynolds number using POD/DEIM and calibration, Internat. J. Numer. Methods Fluids, 82 (2016), 909-931.  doi: 10.1002/fld.4249.  Google Scholar

Figure 1.  Example 1: Average relative error 28 (left) and online computational time (right) of the reduced order model and the full order model for different values of $ \tau $
Figure 2.  Example 2: $ u_1(x,y,0.5) $ discretized with $ n = 200 $. The exact solution (left), the $\mathsf{ho-pod-deim}$ approximation (middle), and the relative error mesh between the two (right)
Figure 3.  Example 2: $ u_2(x,y,0.5) $ discretized with $ n = 200 $. The exact solution (left), the $\mathsf{ho-pod-deim}$ approximation (middle), and the relative error mesh between the two (right)
Figure 4.  Example 2: The average relative error through $ n_{\mathfrak t} = 2n $ timesteps between the $\mathsf{ho-pod-deim}$ approximation $ \widetilde{\bf U}_1(t) $ ($ \widetilde{\bf U}_2(t) $) and the exact solution $ u_1(x,y,t) $ ($ u_2(x,y,t) $)
53] for increasing $ n $">Figure 5.  Example 2: A comparison of the time required offline for basis construction (left) and online for integration (right) between $\mathsf{ho-pod-deim}$ and $\mathsf{pod-deim}$ [53] for increasing $ n $
Figure 6.  Example 3: A comparison of the offline time for increasing dimension $ n $, between $\mathsf{ho-pod}$ and $\mathsf{pod}$ (left) and $\mathsf{ho-deim}$ and $\mathsf{deim}$ (right)
Figure 7.  Example 3: A comparison of the time to solve all linear systems of the form 25, for different values of $ \tau $, between $\mathsf{t3-sylv}$ and Vec-lin. The $ x- $axis displays the maximum dimension of the three vectorized equations for different values of $ \tau $
Table 1.  Example 1. Dim. of $ {\textsf{ho-pod}} $ and $ {\textsf{ho-deim}} $ bases obtained for different $\tau$. The full order model has dimension $n = 1200$
$\tau$ ${\bf U}_i$ left dim. $\mathsf{ho-pod}$ right dim. $\mathsf{ho-pod}$ left dim. $\mathsf{ho-deim}$ right dim. $\mathsf{ho-deim}$
$10^{-2}$ ${\mathbf{U}}_1$ 7 7 11 11
${\mathbf{U}}_2$ 9 10 - -
$10^{-4}$ ${\mathbf{U}}_1$ 18 20 23 23
${\mathbf{U}}_2$ 19 20 - -
$10^{-6}$ ${\mathbf{U}}_1$ 31 33 32 34
${\mathbf{U}}_2$ 29 31 - -
$10^{-8}$ ${\mathbf{U}}_1$ 43 46 44 47
${\mathbf{U}}_2$ 37 40 - -
$\tau$ ${\bf U}_i$ left dim. $\mathsf{ho-pod}$ right dim. $\mathsf{ho-pod}$ left dim. $\mathsf{ho-deim}$ right dim. $\mathsf{ho-deim}$
$10^{-2}$ ${\mathbf{U}}_1$ 7 7 11 11
${\mathbf{U}}_2$ 9 10 - -
$10^{-4}$ ${\mathbf{U}}_1$ 18 20 23 23
${\mathbf{U}}_2$ 19 20 - -
$10^{-6}$ ${\mathbf{U}}_1$ 31 33 32 34
${\mathbf{U}}_2$ 29 31 - -
$10^{-8}$ ${\mathbf{U}}_1$ 43 46 44 47
${\mathbf{U}}_2$ 37 40 - -
Table 2.  A breakdown of the $\mathsf{(ho)-pod}$ and $\mathsf{(ho)-deim}$ basis dimensions and the memory requirements for four different state space dimensions. Note that $ \tau = 1/n^2 $
$ n $ $\mathsf{algorithm}$ $ {\bf U}_i $ $\mathsf{pod dim.}$ $\mathsf{deim dim.}$ ${\textsf{offline}}$$\mathsf{memory}$ $\mathsf{online}$$\mathsf{memory}$
$ 60 $ $\mathsf{ho-pod-deim}$ $ {\bf U}_1 $ 9/9 18/18 $ 98 n $ $ 54n $
$ {\bf U}_2 $ 9/9 18/18 $ 98n $ $ 54n $
$\mathsf{pod-deim}$ [53] $ {\bf U}_1 $ 5 14 $ 400n^2 $ $ 19n^2 $
$ {\bf U}_2 $ 4 14 $ 400n^2 $ $ 18n^2 $
$ 200 $ $\mathsf{ho-pod-deim}$ $ {\bf U}_1 $ 13/13 24/25 $ 153n $ $ 75n $
$ {\bf U}_2 $ 12/12 24/25 $ 153n $ $ 73n $
$\mathsf{pod-deim}$ [53] $ {\bf U}_1 $ 9 23 $ 400n^2 $ $ 32n^2 $
$ {\bf U}_2 $ 8 23 $ 400n^2 $ $ 31n^2 $
$ 600 $ $\mathsf{ho-pod-deim}$ $ {\bf U}_1 $ 16/17 32/32 $ 196n $ $ 97n $
$ {\bf U}_2 $ 16/16 32/32 $ 194n $ $ 96n $
$\mathsf{pod-deim}$ [53] $ {\bf U}_1 $ 15 28 $ 400 n^2 $ $ 43n^2 $
$ {\bf U}_2 $ 14 28 $ 400n^2 $ $ 42n^2 $
$ 1200 $ $\mathsf{ho-pod-deim}$ $ {\bf U}_1 $ 19/19 36/39 $ 219n $ $ 113n $
$ {\bf U}_2 $ 19/19 36/39 $ 215n $ $ 113 n $
$\mathsf{pod-deim}$ [53] $ {\bf U}_1 $ 19 31 $ 400n^2 $ $ 50 n^2 $
$ {\bf U}_2 $ 18 31 $ 400 n^2 $ $ 50 n^2 $
$ n $ $\mathsf{algorithm}$ $ {\bf U}_i $ $\mathsf{pod dim.}$ $\mathsf{deim dim.}$ ${\textsf{offline}}$$\mathsf{memory}$ $\mathsf{online}$$\mathsf{memory}$
$ 60 $ $\mathsf{ho-pod-deim}$ $ {\bf U}_1 $ 9/9 18/18 $ 98 n $ $ 54n $
$ {\bf U}_2 $ 9/9 18/18 $ 98n $ $ 54n $
$\mathsf{pod-deim}$ [53] $ {\bf U}_1 $ 5 14 $ 400n^2 $ $ 19n^2 $
$ {\bf U}_2 $ 4 14 $ 400n^2 $ $ 18n^2 $
$ 200 $ $\mathsf{ho-pod-deim}$ $ {\bf U}_1 $ 13/13 24/25 $ 153n $ $ 75n $
$ {\bf U}_2 $ 12/12 24/25 $ 153n $ $ 73n $
$\mathsf{pod-deim}$ [53] $ {\bf U}_1 $ 9 23 $ 400n^2 $ $ 32n^2 $
$ {\bf U}_2 $ 8 23 $ 400n^2 $ $ 31n^2 $
$ 600 $ $\mathsf{ho-pod-deim}$ $ {\bf U}_1 $ 16/17 32/32 $ 196n $ $ 97n $
$ {\bf U}_2 $ 16/16 32/32 $ 194n $ $ 96n $
$\mathsf{pod-deim}$ [53] $ {\bf U}_1 $ 15 28 $ 400 n^2 $ $ 43n^2 $
$ {\bf U}_2 $ 14 28 $ 400n^2 $ $ 42n^2 $
$ 1200 $ $\mathsf{ho-pod-deim}$ $ {\bf U}_1 $ 19/19 36/39 $ 219n $ $ 113n $
$ {\bf U}_2 $ 19/19 36/39 $ 215n $ $ 113 n $
$\mathsf{pod-deim}$ [53] $ {\bf U}_1 $ 19 31 $ 400n^2 $ $ 50 n^2 $
$ {\bf U}_2 $ 18 31 $ 400 n^2 $ $ 50 n^2 $
Table 3.  Example 3. Dim. of $\mathsf{ho-pod}$ and $\mathsf{ho-deim}$ bases and the average error at 300 timesteps for increasing $ r $. The full order model has dimension $ n = 150 $ and $ \tau = 10^{-4} $
$ r $ $ u $ $ k_1 $ $ k_2 $ $ k_3 $ $ p_1 $ $ p_2 $ $ p_3 $ $\mathsf{error} \bar{\mathcal{E}}(\pmb{\mathcal{ U}}) $
$ 10 $ $ u_1 $ 4 7 10 7 12 16 $ 1\cdot 10^{-4} $
$ u_2 $ 7 7 7 9 12 13 $ 6\cdot 10^{-5} $
$ u_3 $ 8 12 8 9 16 13 $ 1\cdot 10^{-4} $
$ 100 $ $ u_1 $ 6 11 15 10 17 20 $ 3\cdot 10^{-5} $
$ u_2 $ 10 11 11 12 17 17 $ 4\cdot 10^{-5} $
$ u_3 $ 10 16 12 12 21 17 $ 4\cdot 10^{-5} $
$ 500 $ $ u_1 $ 9 15 19 13 23 26 $ 2\cdot 10^{-5} $
$ u_2 $ 11 16 17 14 23 23 $ 3\cdot 10^{-5} $
$ u_3 $ 12 19 16 14 25 23 $ 4\cdot 10^{-5} $
$ r $ $ u $ $ k_1 $ $ k_2 $ $ k_3 $ $ p_1 $ $ p_2 $ $ p_3 $ $\mathsf{error} \bar{\mathcal{E}}(\pmb{\mathcal{ U}}) $
$ 10 $ $ u_1 $ 4 7 10 7 12 16 $ 1\cdot 10^{-4} $
$ u_2 $ 7 7 7 9 12 13 $ 6\cdot 10^{-5} $
$ u_3 $ 8 12 8 9 16 13 $ 1\cdot 10^{-4} $
$ 100 $ $ u_1 $ 6 11 15 10 17 20 $ 3\cdot 10^{-5} $
$ u_2 $ 10 11 11 12 17 17 $ 4\cdot 10^{-5} $
$ u_3 $ 10 16 12 12 21 17 $ 4\cdot 10^{-5} $
$ 500 $ $ u_1 $ 9 15 19 13 23 26 $ 2\cdot 10^{-5} $
$ u_2 $ 11 16 17 14 23 23 $ 3\cdot 10^{-5} $
$ u_3 $ 12 19 16 14 25 23 $ 4\cdot 10^{-5} $
Table 4.  Example 3. Memory and CPU time required for basis construction and integration. The full order model has dimension $ n = 150 $ and $ \tau = 10^{-4} $
r Online memory Basis time(s) FOM time(s) ROM time(s)
10 $ 177n $ 20 1641 1.9
100 $ 245n $ 20 1641 2.2
500 $ 318n $ 20 1641 3.3
r Online memory Basis time(s) FOM time(s) ROM time(s)
10 $ 177n $ 20 1641 1.9
100 $ 245n $ 20 1641 2.2
500 $ 318n $ 20 1641 3.3
Table 5.  Example 4. Dim. of ${\mathsf{ho-pod}}$ and ${\mathsf{ho-deim }}$ bases and further computational detalis for $\tau = 10^{-2}$ and $n = 150$
$r_0$ ${\bf U}_i$ $\mathsf{pod dim.}$ ($k_1/k_2/k_3$) $\mathsf{deim dim.}$ ($p_1/p_2/p_3$) $\mathsf{online}$ $\mathsf{memory}$ $\mathsf{online}$ $\mathsf{time (s)}$ $\mathsf{error}$
$0.1$ ${\mathbf{U}}_1$ 2/2/2 5/5/5 $21n$ 1.29 $3\cdot10^{-4} $
${\mathbf{U}}_2$ 2/2/2 3/3/3 $15n$ 1.20 $4\cdot10^{-4}$
${\mathbf{U}}_3$ 18/18/18 - $54n$ 1.50 $3\cdot10^{-4}$
${\mathbf{U}}_4$ 9/9/9 - $27n$ 0.63 $1\cdot10^{-2} $
$0.3$ ${\mathbf{U}}_1$ 8/8/8 9/9/9 $51n$ 1.56 $3\cdot10^{-4}$
${\mathbf{U}}_2$ 8/8/8 9/9/9 $51n$ 1.13 $3\cdot10^{-4}$
${\mathbf{U}}_3$ 43/43/42 - $128n$ 11.25 $3\cdot10^{-3}$
${\mathbf{U}}_4$ 31/31/30 - $92n$ 4.27 $5\cdot10^{-3} $
$r_0$ ${\bf U}_i$ $\mathsf{pod dim.}$ ($k_1/k_2/k_3$) $\mathsf{deim dim.}$ ($p_1/p_2/p_3$) $\mathsf{online}$ $\mathsf{memory}$ $\mathsf{online}$ $\mathsf{time (s)}$ $\mathsf{error}$
$0.1$ ${\mathbf{U}}_1$ 2/2/2 5/5/5 $21n$ 1.29 $3\cdot10^{-4} $
${\mathbf{U}}_2$ 2/2/2 3/3/3 $15n$ 1.20 $4\cdot10^{-4}$
${\mathbf{U}}_3$ 18/18/18 - $54n$ 1.50 $3\cdot10^{-4}$
${\mathbf{U}}_4$ 9/9/9 - $27n$ 0.63 $1\cdot10^{-2} $
$0.3$ ${\mathbf{U}}_1$ 8/8/8 9/9/9 $51n$ 1.56 $3\cdot10^{-4}$
${\mathbf{U}}_2$ 8/8/8 9/9/9 $51n$ 1.13 $3\cdot10^{-4}$
${\mathbf{U}}_3$ 43/43/42 - $128n$ 11.25 $3\cdot10^{-3}$
${\mathbf{U}}_4$ 31/31/30 - $92n$ 4.27 $5\cdot10^{-3} $
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