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Dynamics of a stage structure prey-predator model with ratio-dependent functional response and anti-predator behavior of adult prey
Solving differential Riccati equations: A nonlinear space-time method using tensor trains
1. | Institute of Mathematics, Technical University of Berlin, 10623 Berlin, Germany |
2. | University of Bath, Department of Mathematical Sciences, BA2 7AY Bath, United Kingdom |
3. | Technische Universität Chemnitz, Department of Mathematics, Scientific Computing Group, 09107 Chemnitz, Germany |
Differential Riccati equations are at the heart of many applications in control theory. They are time-dependent, matrix-valued, and in particular nonlinear equations that require special methods for their solution. Low-rank methods have been used heavily for computing a low-rank solution at every step of a time-discretization. We propose the use of an all-at-once space-time solution leading to a large nonlinear space-time problem for which we propose the use of a Newton–Kleinman iteration. Approximating the space-time problem in a higher-dimensional low-rank tensor form requires fewer degrees of freedom in the solution and in the operator, and gives a faster numerical method. Numerical experiments demonstrate a storage reduction of up to a factor of 100.
References:
[1] |
H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati Equations, Systems Control: Foundations Applications, Birkhäuser Verlag, Basel, 2003, URL https://doi.org/10.1007/978-3-0348-8081-7, In Control and Systems Theory.
doi: 10.1007/978-3-0348-8081-7. |
[2] |
N. Ahmed, G. Matthies, L. Tobiska and H. Xie,
Discontinuous Galerkin time stepping with local projection stabilization for transient convection–diffusion-reaction problems, Computer Methods in Applied Mechanics and Engineering, 200 (2011), 1747-1756.
doi: 10.1016/j.cma.2011.02.003. |
[3] |
J. Ballani and L. Grasedyck,
A projection method to solve linear systems in tensor format, Numerical Linear Algebra with Applications, 20 (2013), 27-43.
doi: 10.1002/nla.1818. |
[4] |
R. Bellman, Dynamic programming, Courier Corporation, 2013. |
[5] |
P. Benner, Z. Bujanović, P. Kürschner and J. Saak,
RADI: A low-rank ADI-type algorithm for large scale algebraic Riccati equations, Numerische Mathematik, 138 (2018), 301-330.
doi: 10.1007/s00211-017-0907-5. |
[6] |
P. Benner, Z. Bujanović, P. Kürschner and J. Saak, A numerical comparison of different solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems, SIAM Journal on Scientific Computing, 42 (2020), A957–A996.
doi: 10.1137/18M1220960. |
[7] |
P. Benner, S. Dolgov, A. Onwunta and M. Stoll,
Low-rank solvers for unsteady Stokes-Brinkman optimal control problem with random data, Computer Methods in Applied Mechanics and Engineering, 304 (2016), 26-54.
doi: 10.1016/j.cma.2016.02.004. |
[8] |
P. Benner and H. Mena, BDF methods for large-scale differential Riccati equations, in in Proceedings of Mathematical Theory of Network and Systems, MTNS, 2004. Google Scholar |
[9] |
P. Benner and H. Mena,
Rosenbrock methods for solving Riccati differential equations, IEEE Transactions on Automatic Control, 58 (2013), 2950-2956.
doi: 10.1109/TAC.2013.2258495. |
[10] |
P. Benner, A. Onwunta and M. Stoll,
Low rank solution of unsteady diffusion equations with stochastic coefficients, SIAM Journal on Uncertainty Quantification, 3 (2015), 622-649.
doi: 10.1137/130937251. |
[11] |
P. Benner and J. Saak,
Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey, GAMM-Mitteilungen, 36 (2013), 32-52.
doi: 10.1002/gamm.201310003. |
[12] |
A. Brooks and T. Hughes,
Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 32 (1982), 199-259.
doi: 10.1016/0045-7825(82)90071-8. |
[13] |
S. Dolgov and B. Khoromskij,
Two-level QTT-Tucker format for optimized tensor calculus, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 593-623.
doi: 10.1137/120882597. |
[14] |
S. Dolgov and M. Stoll, Low-rank solutions to an optimization problem constrained by the Navier-Stokes equations, SIAM Journal on Scientific Computing, 39 (2017), A255–A280.
doi: 10.1137/15M1040414. |
[15] |
S. Dolgov,
TT-GMRES: solution to a linear system in the structured tensor format, Russian Journal of Numerical Analysis and Mathematical Modelling, 28 (2013), 149-172.
doi: 10.1515/rnam-2013-0009. |
[16] |
S. Dolgov and D. Savostyanov, Alternating minimal energy methods for linear systems in higher dimensions, SIAM Journal on Scientific Computing, 36 (2014), A2248–A2271.
doi: 10.1137/140953289. |
[17] |
D. Donoho et al., High-dimensional data analysis: The curses and blessings of dimensionality, AMS Math Challenges Lecture, 1 (2000), 32. Google Scholar |
[18] |
F. Feitzinger, T. Hylla and E. Sachs,
Inexact Kleinman–Newton method for Riccati equations, SIAM Journal on Matrix Analysis and Applications, 31 (2009), 272-288.
doi: 10.1137/070700978. |
[19] |
C. Fu and J. Pfaendtner, Lifting the curse of dimensionality on enhanced sampling of reaction networks with parallel bias metadynamics, Journal of Chemical Theory and Computation, 14 (2018), 2516-2525. Google Scholar |
[20] |
L. Grasedyck and W. Hackbusch,
A multigrid method to solve large scale Sylvester equations, SIAM Journal on Matrix Analysis and Applications, 29 (2007), 870-894.
doi: 10.1137/040618102. |
[21] |
L. Grasedyck, D. Kressner and C. Tobler,
A literature survey of low-rank tensor approximation techniques, GAMM-Mitteilungen, 36 (2013), 53-78.
doi: 10.1002/gamm.201310004. |
[22] |
W. Hackbusch, Tensor Spaces And Numerical Tensor Calculus, Springer–Verlag, Berlin, 2012.
doi: 10.1007/978-3-642-28027-6. |
[23] |
W. Hackbusch, B. N. Khoromskij and E. E. Tyrtyshnikov,
Approximate iterations for structured matrices, Numerische Mathematik, 109 (2008), 365-383.
doi: 10.1007/s00211-008-0143-0. |
[24] |
E. Hansen and T. Stillfjord,
Convergence analysis for splitting of the abstract differential Riccati equation, SIAM Journal on Numerical Analysis, 52 (2014), 3128-3139.
doi: 10.1137/130935501. |
[25] |
R. Herzog and E. Sachs, Preconditioned conjugate gradient method for optimal control problems with control and state constraints, SIAM Journal on Matrix Analysis and Applications, 31 (2010), 2291–2317, URL https://doi.org/10.1137/090779127.
doi: 10.1137/090779127. |
[26] |
M. Hinze, Optimal and Instantaneous Control of the Instationary Navier-Stokes Equations, Habilitation, Technisches Universität Dresden, 2002. Google Scholar |
[27] |
S. Holtz, T. Rohwedder and R. Schneider, The alternating linear scheme for tensor optimization in the tensor train format, SIAM Journal on Scientific Computing, 34 (2012), A683–A713.
doi: 10.1137/100818893. |
[28] |
B. Khoromskij and V. Khoromskaia,
Multigrid accelerated tensor approximation of function related multidimensional arrays, SIAM Journal on Scientific Computing, 31 (2009), 3002-3026.
doi: 10.1137/080730408. |
[29] |
G. Kirsten and V. Simoncini, Order reduction methods for solving large-scale differential matrix Riccati equations, accepted SIAM Journal on Scientific Computing.
doi: 10.1137/19M1264217. |
[30] |
T. Kolda and B. Bader,
Tensor decompositions and applications, SIAM Review, 51 (2009), 455-500.
doi: 10.1137/07070111X. |
[31] |
A. Koskela and H. Mena, Analysis of Krylov subspace approximation to large scale differential Riccati equations, arXiv preprint arXiv: 1705.07507. Google Scholar |
[32] |
D. Kressner and C. Tobler,
Krylov subspace methods for linear systems with tensor product structure, SIAM Journal on Matrix Analysis and Applications, 31 (2010), 1688-1714.
doi: 10.1137/090756843. |
[33] |
N. Lang, Numerical Methods for Large-Scale Linear Time-Varying Control Systems and Related Differential Matrix Equations, PhD thesis, Technische Universität Chemnitz, 2018. Google Scholar |
[34] |
N. Lang, H. Mena and J. Saak,
On the benefits of the $LDL^T$ factorization for large-scale differential matrix equation solvers, Linear Algebra and its Applications, 480 (2015), 44-71.
doi: 10.1016/j.laa.2015.04.006. |
[35] |
Y. Lin and V. Simoncini,
A new subspace iteration method for the algebraic Riccati equation, Numerical Linear Algebra with Applications, 22 (2015), 26-47.
doi: 10.1002/nla.1936. |
[36] |
H. Mena, A. Ostermann, L.-M. Pfurtscheller and C. Piazzola,
Numerical low-rank approximation of matrix differential equations, Journal of Computational and Applied Mathematics, 340 (2018), 602-614.
doi: 10.1016/j.cam.2018.01.035. |
[37] |
M. R. Opmeer, Decay of singular values of the Gramians of infinite-dimensional systems, in 2015 European Control Conference (ECC), 2015, 1183–1188. Google Scholar |
[38] |
I. Oseledets,
DMRG approach to fast linear algebra in the TT-format, Computational Methods in Applied Mathematics, 11 (2011), 382-393.
doi: 10.2478/cmam-2011-0021. |
[39] |
I. Oseledets,
Tensor-train decomposition, SIAM Journal on Scientific Computing, 33 (2011), 2295-2317.
doi: 10.1137/090752286. |
[40] |
I. Oseledets, S. Dolgov, V. Kazeev, D. Savostyanov, O. Lebedeva, P. Zhlobich, T. Mach and L. Song, TT-Toolbox, URL https://github.com/oseledets/TT-Toolbox, https://github.com/oseledets/TT-Toolbox. Google Scholar |
[41] |
I. Oseledets and E. Tyrtyshnikov,
Breaking the curse of dimensionality, or how to use SVD in many dimensions, SIAM Journal on Scientific Computing, 31 (2009), 3744-3759.
doi: 10.1137/090748330. |
[42] |
J. Pearson, M. Stoll and A. Wathen, Preconditioners for state-constrained optimal control problems with Moreau-Yosida penalty function, Numerical Linear Algebra with Applications, 21 (2014), 81–97, URL https://doi.org/10.1002/nla.1863.
doi: 10.1002/nla.1863. |
[43] |
J. Saak, M. Köhler and P. Benner, M-M.E.S.S.-2.0 – The matrix equations sparse solvers library, See also: www.mpi-magdeburg.mpg.de/projects/mess.
doi: 10.5281/zenodo.3368844. |
[44] |
C. Schillings and C. Schwab, Sparse, adaptive Smolyak quadratures for Bayesian inverse problems, Inverse Problems, 29 (2013), 065011.
doi: 10.1088/0266-5611/29/6/065011. |
[45] |
C. Silvestre and A. Pascoal, Depth control of the INFANTE AUV using gain-scheduled reduced order output feedback, Control Engineering Practice, 15 (2007), 883-895. Google Scholar |
[46] |
V. Simoncini,
Analysis of the rational Krylov subspace projection method for large-scale algebraic Riccati equations, SIAM Journal on Matrix Analysis and Applications, 37 (2016), 1655-1674.
doi: 10.1137/16M1059382. |
[47] |
T. Stillfjord,
Singular value decay of operator-valued differential Lyapunov and Riccati equations, SIAM Journal on Control and Optimization, 56 (2018), 3598-3618.
doi: 10.1137/18M1178815. |
[48] |
M. Stoll and T. Breiten,
A low-rank in time approach to PDE-constrained optimization, SIAM Journal on Scientific Computing, 37 (2015), 1-19.
doi: 10.1137/130926365. |
[49] |
C. Tobler, Low-rank Tensor Methods for Linear Systems and Eigenvalue Problems, PhD thesis, Diss., Eidgenössische Technische Hochschule ETH Zürich, Nr. 20320, 2012. Google Scholar |
[50] |
F. Tröltzsch, Optimale Steuerung Partieller Differentialgleichungen, Vieweg+Teubner Verlag, 2005. Google Scholar |
[51] |
B. Vandereycken and S. Vandewalle, Local Fourier analysis for tensor-product multigrid, in AIP Conference Proceedings, AIP, 1168 (2009), 354–356. Google Scholar |
[52] |
C. d. Villemagne and R. E. Skelton,
Model reductions using a projection formulation, International Journal of Control, 46 (1987), 2141-2169.
doi: 10.1080/00207178708934040. |
[53] |
Z. Zhang, X. Yang, I. Oseledets, G. Karniadakis and L. Daniel, Enabling high-dimensional hierarchical uncertainty quantification by ANOVA and tensor-train decomposition, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 34 (2015), 63-76. Google Scholar |
show all references
References:
[1] |
H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, Matrix Riccati Equations, Systems Control: Foundations Applications, Birkhäuser Verlag, Basel, 2003, URL https://doi.org/10.1007/978-3-0348-8081-7, In Control and Systems Theory.
doi: 10.1007/978-3-0348-8081-7. |
[2] |
N. Ahmed, G. Matthies, L. Tobiska and H. Xie,
Discontinuous Galerkin time stepping with local projection stabilization for transient convection–diffusion-reaction problems, Computer Methods in Applied Mechanics and Engineering, 200 (2011), 1747-1756.
doi: 10.1016/j.cma.2011.02.003. |
[3] |
J. Ballani and L. Grasedyck,
A projection method to solve linear systems in tensor format, Numerical Linear Algebra with Applications, 20 (2013), 27-43.
doi: 10.1002/nla.1818. |
[4] |
R. Bellman, Dynamic programming, Courier Corporation, 2013. |
[5] |
P. Benner, Z. Bujanović, P. Kürschner and J. Saak,
RADI: A low-rank ADI-type algorithm for large scale algebraic Riccati equations, Numerische Mathematik, 138 (2018), 301-330.
doi: 10.1007/s00211-017-0907-5. |
[6] |
P. Benner, Z. Bujanović, P. Kürschner and J. Saak, A numerical comparison of different solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems, SIAM Journal on Scientific Computing, 42 (2020), A957–A996.
doi: 10.1137/18M1220960. |
[7] |
P. Benner, S. Dolgov, A. Onwunta and M. Stoll,
Low-rank solvers for unsteady Stokes-Brinkman optimal control problem with random data, Computer Methods in Applied Mechanics and Engineering, 304 (2016), 26-54.
doi: 10.1016/j.cma.2016.02.004. |
[8] |
P. Benner and H. Mena, BDF methods for large-scale differential Riccati equations, in in Proceedings of Mathematical Theory of Network and Systems, MTNS, 2004. Google Scholar |
[9] |
P. Benner and H. Mena,
Rosenbrock methods for solving Riccati differential equations, IEEE Transactions on Automatic Control, 58 (2013), 2950-2956.
doi: 10.1109/TAC.2013.2258495. |
[10] |
P. Benner, A. Onwunta and M. Stoll,
Low rank solution of unsteady diffusion equations with stochastic coefficients, SIAM Journal on Uncertainty Quantification, 3 (2015), 622-649.
doi: 10.1137/130937251. |
[11] |
P. Benner and J. Saak,
Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey, GAMM-Mitteilungen, 36 (2013), 32-52.
doi: 10.1002/gamm.201310003. |
[12] |
A. Brooks and T. Hughes,
Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 32 (1982), 199-259.
doi: 10.1016/0045-7825(82)90071-8. |
[13] |
S. Dolgov and B. Khoromskij,
Two-level QTT-Tucker format for optimized tensor calculus, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 593-623.
doi: 10.1137/120882597. |
[14] |
S. Dolgov and M. Stoll, Low-rank solutions to an optimization problem constrained by the Navier-Stokes equations, SIAM Journal on Scientific Computing, 39 (2017), A255–A280.
doi: 10.1137/15M1040414. |
[15] |
S. Dolgov,
TT-GMRES: solution to a linear system in the structured tensor format, Russian Journal of Numerical Analysis and Mathematical Modelling, 28 (2013), 149-172.
doi: 10.1515/rnam-2013-0009. |
[16] |
S. Dolgov and D. Savostyanov, Alternating minimal energy methods for linear systems in higher dimensions, SIAM Journal on Scientific Computing, 36 (2014), A2248–A2271.
doi: 10.1137/140953289. |
[17] |
D. Donoho et al., High-dimensional data analysis: The curses and blessings of dimensionality, AMS Math Challenges Lecture, 1 (2000), 32. Google Scholar |
[18] |
F. Feitzinger, T. Hylla and E. Sachs,
Inexact Kleinman–Newton method for Riccati equations, SIAM Journal on Matrix Analysis and Applications, 31 (2009), 272-288.
doi: 10.1137/070700978. |
[19] |
C. Fu and J. Pfaendtner, Lifting the curse of dimensionality on enhanced sampling of reaction networks with parallel bias metadynamics, Journal of Chemical Theory and Computation, 14 (2018), 2516-2525. Google Scholar |
[20] |
L. Grasedyck and W. Hackbusch,
A multigrid method to solve large scale Sylvester equations, SIAM Journal on Matrix Analysis and Applications, 29 (2007), 870-894.
doi: 10.1137/040618102. |
[21] |
L. Grasedyck, D. Kressner and C. Tobler,
A literature survey of low-rank tensor approximation techniques, GAMM-Mitteilungen, 36 (2013), 53-78.
doi: 10.1002/gamm.201310004. |
[22] |
W. Hackbusch, Tensor Spaces And Numerical Tensor Calculus, Springer–Verlag, Berlin, 2012.
doi: 10.1007/978-3-642-28027-6. |
[23] |
W. Hackbusch, B. N. Khoromskij and E. E. Tyrtyshnikov,
Approximate iterations for structured matrices, Numerische Mathematik, 109 (2008), 365-383.
doi: 10.1007/s00211-008-0143-0. |
[24] |
E. Hansen and T. Stillfjord,
Convergence analysis for splitting of the abstract differential Riccati equation, SIAM Journal on Numerical Analysis, 52 (2014), 3128-3139.
doi: 10.1137/130935501. |
[25] |
R. Herzog and E. Sachs, Preconditioned conjugate gradient method for optimal control problems with control and state constraints, SIAM Journal on Matrix Analysis and Applications, 31 (2010), 2291–2317, URL https://doi.org/10.1137/090779127.
doi: 10.1137/090779127. |
[26] |
M. Hinze, Optimal and Instantaneous Control of the Instationary Navier-Stokes Equations, Habilitation, Technisches Universität Dresden, 2002. Google Scholar |
[27] |
S. Holtz, T. Rohwedder and R. Schneider, The alternating linear scheme for tensor optimization in the tensor train format, SIAM Journal on Scientific Computing, 34 (2012), A683–A713.
doi: 10.1137/100818893. |
[28] |
B. Khoromskij and V. Khoromskaia,
Multigrid accelerated tensor approximation of function related multidimensional arrays, SIAM Journal on Scientific Computing, 31 (2009), 3002-3026.
doi: 10.1137/080730408. |
[29] |
G. Kirsten and V. Simoncini, Order reduction methods for solving large-scale differential matrix Riccati equations, accepted SIAM Journal on Scientific Computing.
doi: 10.1137/19M1264217. |
[30] |
T. Kolda and B. Bader,
Tensor decompositions and applications, SIAM Review, 51 (2009), 455-500.
doi: 10.1137/07070111X. |
[31] |
A. Koskela and H. Mena, Analysis of Krylov subspace approximation to large scale differential Riccati equations, arXiv preprint arXiv: 1705.07507. Google Scholar |
[32] |
D. Kressner and C. Tobler,
Krylov subspace methods for linear systems with tensor product structure, SIAM Journal on Matrix Analysis and Applications, 31 (2010), 1688-1714.
doi: 10.1137/090756843. |
[33] |
N. Lang, Numerical Methods for Large-Scale Linear Time-Varying Control Systems and Related Differential Matrix Equations, PhD thesis, Technische Universität Chemnitz, 2018. Google Scholar |
[34] |
N. Lang, H. Mena and J. Saak,
On the benefits of the $LDL^T$ factorization for large-scale differential matrix equation solvers, Linear Algebra and its Applications, 480 (2015), 44-71.
doi: 10.1016/j.laa.2015.04.006. |
[35] |
Y. Lin and V. Simoncini,
A new subspace iteration method for the algebraic Riccati equation, Numerical Linear Algebra with Applications, 22 (2015), 26-47.
doi: 10.1002/nla.1936. |
[36] |
H. Mena, A. Ostermann, L.-M. Pfurtscheller and C. Piazzola,
Numerical low-rank approximation of matrix differential equations, Journal of Computational and Applied Mathematics, 340 (2018), 602-614.
doi: 10.1016/j.cam.2018.01.035. |
[37] |
M. R. Opmeer, Decay of singular values of the Gramians of infinite-dimensional systems, in 2015 European Control Conference (ECC), 2015, 1183–1188. Google Scholar |
[38] |
I. Oseledets,
DMRG approach to fast linear algebra in the TT-format, Computational Methods in Applied Mathematics, 11 (2011), 382-393.
doi: 10.2478/cmam-2011-0021. |
[39] |
I. Oseledets,
Tensor-train decomposition, SIAM Journal on Scientific Computing, 33 (2011), 2295-2317.
doi: 10.1137/090752286. |
[40] |
I. Oseledets, S. Dolgov, V. Kazeev, D. Savostyanov, O. Lebedeva, P. Zhlobich, T. Mach and L. Song, TT-Toolbox, URL https://github.com/oseledets/TT-Toolbox, https://github.com/oseledets/TT-Toolbox. Google Scholar |
[41] |
I. Oseledets and E. Tyrtyshnikov,
Breaking the curse of dimensionality, or how to use SVD in many dimensions, SIAM Journal on Scientific Computing, 31 (2009), 3744-3759.
doi: 10.1137/090748330. |
[42] |
J. Pearson, M. Stoll and A. Wathen, Preconditioners for state-constrained optimal control problems with Moreau-Yosida penalty function, Numerical Linear Algebra with Applications, 21 (2014), 81–97, URL https://doi.org/10.1002/nla.1863.
doi: 10.1002/nla.1863. |
[43] |
J. Saak, M. Köhler and P. Benner, M-M.E.S.S.-2.0 – The matrix equations sparse solvers library, See also: www.mpi-magdeburg.mpg.de/projects/mess.
doi: 10.5281/zenodo.3368844. |
[44] |
C. Schillings and C. Schwab, Sparse, adaptive Smolyak quadratures for Bayesian inverse problems, Inverse Problems, 29 (2013), 065011.
doi: 10.1088/0266-5611/29/6/065011. |
[45] |
C. Silvestre and A. Pascoal, Depth control of the INFANTE AUV using gain-scheduled reduced order output feedback, Control Engineering Practice, 15 (2007), 883-895. Google Scholar |
[46] |
V. Simoncini,
Analysis of the rational Krylov subspace projection method for large-scale algebraic Riccati equations, SIAM Journal on Matrix Analysis and Applications, 37 (2016), 1655-1674.
doi: 10.1137/16M1059382. |
[47] |
T. Stillfjord,
Singular value decay of operator-valued differential Lyapunov and Riccati equations, SIAM Journal on Control and Optimization, 56 (2018), 3598-3618.
doi: 10.1137/18M1178815. |
[48] |
M. Stoll and T. Breiten,
A low-rank in time approach to PDE-constrained optimization, SIAM Journal on Scientific Computing, 37 (2015), 1-19.
doi: 10.1137/130926365. |
[49] |
C. Tobler, Low-rank Tensor Methods for Linear Systems and Eigenvalue Problems, PhD thesis, Diss., Eidgenössische Technische Hochschule ETH Zürich, Nr. 20320, 2012. Google Scholar |
[50] |
F. Tröltzsch, Optimale Steuerung Partieller Differentialgleichungen, Vieweg+Teubner Verlag, 2005. Google Scholar |
[51] |
B. Vandereycken and S. Vandewalle, Local Fourier analysis for tensor-product multigrid, in AIP Conference Proceedings, AIP, 1168 (2009), 354–356. Google Scholar |
[52] |
C. d. Villemagne and R. E. Skelton,
Model reductions using a projection formulation, International Journal of Control, 46 (1987), 2141-2169.
doi: 10.1080/00207178708934040. |
[53] |
Z. Zhang, X. Yang, I. Oseledets, G. Karniadakis and L. Daniel, Enabling high-dimensional hierarchical uncertainty quantification by ANOVA and tensor-train decomposition, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 34 (2015), 63-76. Google Scholar |










Time ( |
Memory (MB) | Time ( |
Memory (MB) | |
$\mathtt{ode23}$ | 731 | 2048 | 894 | 20480 |
$\mathtt{tt}$ | 6142 | 14.61 | 11940 | 18.6 |
$\mathtt{MMESS (split)}$ | 6.5 | 74.32 | 86 | 916 |
$\mathtt{MMESS (BDF)}$ | 48 | 115.3 | 172 | 1045 |
$\mathtt{RKSM-DRE}$ | 12 | 29.44 | 153 | 273 |
Time ( |
Memory (MB) | Time ( |
Memory (MB) | |
$\mathtt{ode23}$ | 731 | 2048 | 894 | 20480 |
$\mathtt{tt}$ | 6142 | 14.61 | 11940 | 18.6 |
$\mathtt{MMESS (split)}$ | 6.5 | 74.32 | 86 | 916 |
$\mathtt{MMESS (BDF)}$ | 48 | 115.3 | 172 | 1045 |
$\mathtt{RKSM-DRE}$ | 12 | 29.44 | 153 | 273 |
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