
- Previous Article
- NACO Home
- This Issue
-
Next Article
Fused LASSO penalized least absolute deviation estimator for high dimensional linear regression
$\mathcal{L}_{∞}$-norm computation for large-scale descriptor systems using structured iterative eigensolvers
1. | Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstraße 1, 39106 Magdeburg, Germany |
2. | McGill University, Reasoning and Learning Laboratory, 3480 University Street, H3A 0E9 Montreal, Quebec, Canada |
3. | Technische Universität Berlin, Institut für Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany |
In this article, we discuss a method for computing the $\mathcal{L}_∞$-norm for transfer functions of descriptor systems using structured iterative eigensolvers. In particular, the algorithm computes some desired imaginary eigenvalues of an even matrix pencil and uses them to determine an upper and lower bound to the $\mathcal{L}_∞$-norm. Finally, we compare our method to a previously developed algorithm using pseudopole sets. Numerical examples demonstrate the reliability and accuracy of the new method along with a significant drop in the runtime.
References:
[1] |
N. Aliyev, P. Benner, E. Mengi, P. Schwerdtner and M. Voigt, Large-scale computation of $\mathcal{L}_∞$-norms by a greedy subspace method, SIAM J. Matrix Anal. Appl., Accepted for publication. Google Scholar |
[2] |
P. Benner, R. Byers, P. Losse, V. Mehrmann and H. Xu, Numerical solution of real skew-Hamiltonian/Hamiltonian eigenproblems, 2007, Unpublished report. Google Scholar |
[3] |
P. Benner, R. Byers, V. Mehrmann and H. Xu,
Numerical computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils, SIAM J. Matrix Anal. Appl., 24 (2002), 165-190.
doi: 10.1137/S0895479800367439. |
[4] |
P. Benner and C. Effenberger,
A rational SHIRA method for the Hamiltonian eigenvalue problem, Taiwanese J. Math., 14 (2010), 805-823.
doi: 10.11650/twjm/1500405868. |
[5] |
P. Benner and H. Faßbender,
An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem, Linear Algebra Appl., 263 (1997), 75-111.
doi: 10.1016/S0024-3795(96)00524-1. |
[6] |
P. Benner, H. Faßbender and M. Stoll,
A Hamiltonian Krylov-Schur-type method based on the symplectic Lanczos process, Linear Algebra Appl., 435 (2011), 578-600.
doi: 10.1016/j.laa.2010.04.048. |
[7] |
P. Benner, V. Sima and M. Voigt,
$\mathcal{L}_∞$-norm computation for continuous-time descriptor systems using structured matrix pencils, IEEE Trans. Automat. Control, 57 (2012), 233-238.
doi: 10.1109/TAC.2011.2161833. |
[8] |
P. Benner, V. Sima and M. Voigt, Robust and efficient algorithms for $\mathcal{L}_∞$-norm computation for descriptor systems, in Proc. 7th IFAC Symposium on Robust Control Design, IFAC, Aalborg, Denmark, 2012,195-200.
doi: 10.3182/20120620-3-DK-2025.00114. |
[9] |
P. Benner, V. Sima and M. Voigt, Algorithm 961 -Fortran 77 subroutines for the solution of skew-Hamiltonian/Hamiltonian eigenproblems, ACM Trans. Math. Software, 42, Paper 24.
doi: 10.1145/2818313. |
[10] |
P. Benner and M. Voigt,
A structured pseudospectral method for $\mathcal{H}_∞$-norm computation of large-scale descriptor systems, Math. Control Signals Systems, 26 (2014), 303-338.
doi: 10.1007/s00498-013-0121-7. |
[11] |
S. Boyd and V. Balakrishnan,
A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its $L_∞$-norm, Systems Control Lett., 15 (1990), 1-7.
doi: 10.1016/0167-6911(90)90037-U. |
[12] |
S. Boyd, V. Balakrishnan and P. Kabamba,
A bisection method for computing the $H_{∞}$ norm of a transfer matrix and related problems, Math. Control Signals Systems, 2 (1989), 207-219.
doi: 10.1007/BF02551385. |
[13] |
N. A. Bruinsma and M. Steinbuch,
A fast algorithm to compute the $\mathcal{H}_∞$-norm of a transfer function matrix, Systems Control Lett., 14 (1990), 287-293.
doi: 10.1016/0167-6911(90)90049-Z. |
[14] |
J. V. Burke, D. Henrion, A. S. Lewis and M. L. Overton, HIFOO-A MATLAB package for fixed-order controller design and $H_∞$ optimization, in Proc. 5th IFAC Syposium on Robust Control Design, Toulouse, France, 2006.
doi: 10.3182/20060705-3-FR-2907.00059. |
[15] |
Y. Chahlaoui and P. Van Dooren, A collection of benchmark examples for model reduction of linear time invariant dynamical systems, Technical Report 2002-2,2002, Available from http://www.slicot.org/index.php?site=benchmodred. Google Scholar |
[16] |
M. A. Freitag, A. Spence and P. V. Dooren,
Calculating the $H_∞$-norm using the implicit determinant method, SIAM J. Matrix Anal. Appl., 35 (2014), 619-634.
doi: 10.1137/130933228. |
[17] |
F. Freitas, J. Rommes and N. Martins,
Gramian-based reduction method applied to large sparse power system descriptor models, IEEE Trans. Power Syst., 23 (2008), 1258-1270.
doi: 10.1109/TPWRS.2008.926693. |
[18] |
N. Guglielmi, M. Gürbüzbalaban and M. L. Overton,
Fast approximation of the $H_∞$ norm via optimization over spectral value sets, SIAM J. Matrix Anal. Appl., 34 (2013), 709-737.
doi: 10.1137/120875752. |
[19] |
P. Losse, V. Mehrmann, L. Poppe and T. Reis,
The modified optimal $\mathcal H_∞$ control problem for descriptor systems, SIAM J. Control Optim., 47 (2008), 2795-2811.
doi: 10.1137/070710093. |
[20] |
N. Martins, P. C. Pellanda and J. Rommes,
Computation of transfer function dominant zeros with applications to oscillation damping control of large power systems, IEEE Trans. Power Syst., 22 (2007), 1657-1664.
doi: 10.1109/TPWRS.2007.907526. |
[21] |
V. Mehrmann, C. Schröder and V. Simoncini,
An implicitly-restarted Krylov subspace method for real symmetric/skew-symmetric eigenproblems, Linear Algebra Appl., 436 (2012), 4070-4087.
doi: 10.1016/j.laa.2009.11.009. |
[22] |
V. Mehrmann and T. Stykel, Balanced truncation model reduction for large-scale systems in descriptor form, in Dimension Reduction of Large-Scale Systems (eds. P. Benner, V. Mehrmann and D. Sorensen), vol. 45 of Lecture Notes Comput. Sci. Eng., Springer-Verlag, Berlin, Heidelberg, New York, 2005, chapter 3, 89-116.
doi: 10.1007/3-540-27909-1_3. |
[23] |
T. Mitchell and M. L. Overton, Fixed low-order controller design and $ H_∞$ optimization for large-scale dynamical systems, in Proc. 8th IFAC Symposium on Robust Control Design, Bratislava, Slovakia, 2015, 25-30.
doi: 10.1016/j.ifacol.2015.09.428. |
[24] |
T. Mitchell and M. L. Overton,
Hybrid expansion-contraction: a robust scaleable method for approxiating the $H_∞$ norm, IMA J. Numer. Anal., 36 (2016), 985-1014.
doi: 10.1093/imanum/drv046. |
[25] |
J. Rommes,
Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problems $Ax = λ Bx$ with singular $B$, Math. Comp., 77 (2008), 995-1015.
doi: 10.1090/S0025-5718-07-02040-6. |
[26] |
J. Rommes and N. Martins,
Efficient computation of multivariable transfer function dominant poles using subspace acceleration, IEEE Trans. Power Syst., 21 (2006), 1471-1487.
doi: 10.1109/TPWRS.2006.881154. |
[27] |
J. Rommes and G. L. G. Sleijpen,
Convergence of the dominant pole algorithm and Rayleigh quotient iteration, SIAM J. Matrix Anal. Appl., 30 (2008), 346-363.
doi: 10.1137/060671401. |
[28] |
A. Ruhe, Rational Krylov algorithms for nonsymmetric eigenvalue problems, in Recent Advances in Iterative Methods (eds. G. Golub, A. Greenbaum and M. Luskin), vol. 60 of IMA Vol. Math. Appl., Springer-Verlag, New York, 1994,149-164.
doi: 10.1007/978-1-4613-9353-5_10. |
[29] |
C. Schröder, Private communication, 2013. Google Scholar |
[30] |
M. Voigt, On Linear-Quadratic Optimal Control and Robustness of Differential-Algebraic Systems, Logos-Verlag, Berlin, 2015, Also as Dissertation, Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, 2015. Google Scholar |
[31] |
K. Zhou and J. C. Doyle, Essentials of Robust Control, Hemel Hempstead: Prentice Hall, 1997. Google Scholar |
show all references
References:
[1] |
N. Aliyev, P. Benner, E. Mengi, P. Schwerdtner and M. Voigt, Large-scale computation of $\mathcal{L}_∞$-norms by a greedy subspace method, SIAM J. Matrix Anal. Appl., Accepted for publication. Google Scholar |
[2] |
P. Benner, R. Byers, P. Losse, V. Mehrmann and H. Xu, Numerical solution of real skew-Hamiltonian/Hamiltonian eigenproblems, 2007, Unpublished report. Google Scholar |
[3] |
P. Benner, R. Byers, V. Mehrmann and H. Xu,
Numerical computation of deflating subspaces of skew-Hamiltonian/Hamiltonian pencils, SIAM J. Matrix Anal. Appl., 24 (2002), 165-190.
doi: 10.1137/S0895479800367439. |
[4] |
P. Benner and C. Effenberger,
A rational SHIRA method for the Hamiltonian eigenvalue problem, Taiwanese J. Math., 14 (2010), 805-823.
doi: 10.11650/twjm/1500405868. |
[5] |
P. Benner and H. Faßbender,
An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem, Linear Algebra Appl., 263 (1997), 75-111.
doi: 10.1016/S0024-3795(96)00524-1. |
[6] |
P. Benner, H. Faßbender and M. Stoll,
A Hamiltonian Krylov-Schur-type method based on the symplectic Lanczos process, Linear Algebra Appl., 435 (2011), 578-600.
doi: 10.1016/j.laa.2010.04.048. |
[7] |
P. Benner, V. Sima and M. Voigt,
$\mathcal{L}_∞$-norm computation for continuous-time descriptor systems using structured matrix pencils, IEEE Trans. Automat. Control, 57 (2012), 233-238.
doi: 10.1109/TAC.2011.2161833. |
[8] |
P. Benner, V. Sima and M. Voigt, Robust and efficient algorithms for $\mathcal{L}_∞$-norm computation for descriptor systems, in Proc. 7th IFAC Symposium on Robust Control Design, IFAC, Aalborg, Denmark, 2012,195-200.
doi: 10.3182/20120620-3-DK-2025.00114. |
[9] |
P. Benner, V. Sima and M. Voigt, Algorithm 961 -Fortran 77 subroutines for the solution of skew-Hamiltonian/Hamiltonian eigenproblems, ACM Trans. Math. Software, 42, Paper 24.
doi: 10.1145/2818313. |
[10] |
P. Benner and M. Voigt,
A structured pseudospectral method for $\mathcal{H}_∞$-norm computation of large-scale descriptor systems, Math. Control Signals Systems, 26 (2014), 303-338.
doi: 10.1007/s00498-013-0121-7. |
[11] |
S. Boyd and V. Balakrishnan,
A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its $L_∞$-norm, Systems Control Lett., 15 (1990), 1-7.
doi: 10.1016/0167-6911(90)90037-U. |
[12] |
S. Boyd, V. Balakrishnan and P. Kabamba,
A bisection method for computing the $H_{∞}$ norm of a transfer matrix and related problems, Math. Control Signals Systems, 2 (1989), 207-219.
doi: 10.1007/BF02551385. |
[13] |
N. A. Bruinsma and M. Steinbuch,
A fast algorithm to compute the $\mathcal{H}_∞$-norm of a transfer function matrix, Systems Control Lett., 14 (1990), 287-293.
doi: 10.1016/0167-6911(90)90049-Z. |
[14] |
J. V. Burke, D. Henrion, A. S. Lewis and M. L. Overton, HIFOO-A MATLAB package for fixed-order controller design and $H_∞$ optimization, in Proc. 5th IFAC Syposium on Robust Control Design, Toulouse, France, 2006.
doi: 10.3182/20060705-3-FR-2907.00059. |
[15] |
Y. Chahlaoui and P. Van Dooren, A collection of benchmark examples for model reduction of linear time invariant dynamical systems, Technical Report 2002-2,2002, Available from http://www.slicot.org/index.php?site=benchmodred. Google Scholar |
[16] |
M. A. Freitag, A. Spence and P. V. Dooren,
Calculating the $H_∞$-norm using the implicit determinant method, SIAM J. Matrix Anal. Appl., 35 (2014), 619-634.
doi: 10.1137/130933228. |
[17] |
F. Freitas, J. Rommes and N. Martins,
Gramian-based reduction method applied to large sparse power system descriptor models, IEEE Trans. Power Syst., 23 (2008), 1258-1270.
doi: 10.1109/TPWRS.2008.926693. |
[18] |
N. Guglielmi, M. Gürbüzbalaban and M. L. Overton,
Fast approximation of the $H_∞$ norm via optimization over spectral value sets, SIAM J. Matrix Anal. Appl., 34 (2013), 709-737.
doi: 10.1137/120875752. |
[19] |
P. Losse, V. Mehrmann, L. Poppe and T. Reis,
The modified optimal $\mathcal H_∞$ control problem for descriptor systems, SIAM J. Control Optim., 47 (2008), 2795-2811.
doi: 10.1137/070710093. |
[20] |
N. Martins, P. C. Pellanda and J. Rommes,
Computation of transfer function dominant zeros with applications to oscillation damping control of large power systems, IEEE Trans. Power Syst., 22 (2007), 1657-1664.
doi: 10.1109/TPWRS.2007.907526. |
[21] |
V. Mehrmann, C. Schröder and V. Simoncini,
An implicitly-restarted Krylov subspace method for real symmetric/skew-symmetric eigenproblems, Linear Algebra Appl., 436 (2012), 4070-4087.
doi: 10.1016/j.laa.2009.11.009. |
[22] |
V. Mehrmann and T. Stykel, Balanced truncation model reduction for large-scale systems in descriptor form, in Dimension Reduction of Large-Scale Systems (eds. P. Benner, V. Mehrmann and D. Sorensen), vol. 45 of Lecture Notes Comput. Sci. Eng., Springer-Verlag, Berlin, Heidelberg, New York, 2005, chapter 3, 89-116.
doi: 10.1007/3-540-27909-1_3. |
[23] |
T. Mitchell and M. L. Overton, Fixed low-order controller design and $ H_∞$ optimization for large-scale dynamical systems, in Proc. 8th IFAC Symposium on Robust Control Design, Bratislava, Slovakia, 2015, 25-30.
doi: 10.1016/j.ifacol.2015.09.428. |
[24] |
T. Mitchell and M. L. Overton,
Hybrid expansion-contraction: a robust scaleable method for approxiating the $H_∞$ norm, IMA J. Numer. Anal., 36 (2016), 985-1014.
doi: 10.1093/imanum/drv046. |
[25] |
J. Rommes,
Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problems $Ax = λ Bx$ with singular $B$, Math. Comp., 77 (2008), 995-1015.
doi: 10.1090/S0025-5718-07-02040-6. |
[26] |
J. Rommes and N. Martins,
Efficient computation of multivariable transfer function dominant poles using subspace acceleration, IEEE Trans. Power Syst., 21 (2006), 1471-1487.
doi: 10.1109/TPWRS.2006.881154. |
[27] |
J. Rommes and G. L. G. Sleijpen,
Convergence of the dominant pole algorithm and Rayleigh quotient iteration, SIAM J. Matrix Anal. Appl., 30 (2008), 346-363.
doi: 10.1137/060671401. |
[28] |
A. Ruhe, Rational Krylov algorithms for nonsymmetric eigenvalue problems, in Recent Advances in Iterative Methods (eds. G. Golub, A. Greenbaum and M. Luskin), vol. 60 of IMA Vol. Math. Appl., Springer-Verlag, New York, 1994,149-164.
doi: 10.1007/978-1-4613-9353-5_10. |
[29] |
C. Schröder, Private communication, 2013. Google Scholar |
[30] |
M. Voigt, On Linear-Quadratic Optimal Control and Robustness of Differential-Algebraic Systems, Logos-Verlag, Berlin, 2015, Also as Dissertation, Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, 2015. Google Scholar |
[31] |
K. Zhou and J. C. Doyle, Essentials of Robust Control, Hemel Hempstead: Prentice Hall, 1997. Google Scholar |




parameter | description | default value |
desired relative accuracy of |
1e-06 | |
number of dominant poles computed in initial stage | 20 | |
relative cutoff value for the dominance of the poles to | 0.5 | |
determine the initial eigenvalues | ||
minimum relative distance between two subsequent shifts | 0.01 | |
maximum search space dimension even IRA | 8 | |
maximum number of restarts for even IRA | 30 | |
number of eigenvalues calculated per shift in even IRA | 4 | |
tolerance on the eigenvalue residual for even IRA | 1e-06 | |
relative eigenvalue distance for frequency sweep | 1e-05 | |
relative shift displacement | 5e-04 |
parameter | description | default value |
desired relative accuracy of |
1e-06 | |
number of dominant poles computed in initial stage | 20 | |
relative cutoff value for the dominance of the poles to | 0.5 | |
determine the initial eigenvalues | ||
minimum relative distance between two subsequent shifts | 0.01 | |
maximum search space dimension even IRA | 8 | |
maximum number of restarts for even IRA | 30 | |
number of eigenvalues calculated per shift in even IRA | 4 | |
tolerance on the eigenvalue residual for even IRA | 1e-06 | |
relative eigenvalue distance for frequency sweep | 1e-05 | |
relative shift displacement | 5e-04 |
# | example | n | m | p | computed |
optimal frequency |
time in s | |||
pseudopoles | matrix pencil | pseudopoles | matrix pencil | pseudopoles | matrix pencil | |||||
1 | 48 | 1 | 1 | 5.27633e-03 | 5.27634e-03 | 5.20608e+00 | 5.20584e+00 | 1.54 | 0.54 | |
2 | 84 | 1 | 1 | 1.08358e+01 | 1.08358e+01 | 0.00000e+00 | 0.00000e+00 | 2.08 | 0.61 | |
3 | 120 | 2 | 2 | 2.31982e+06 | 2.31982e+06 | 2.25682e+01 | 2.25682e+01 | 2.70 | 0.54 | |
4 | 270 | 3 | 3 | 1.15887e-01 | 1.15887e-01 | 7.75093e-01 | 7.75089e-01 | 2.69 | 0.61 | |
5 | 348 | 1 | 1 | 4.55487e+03 | 4.55488e+03 | 1.04575e-01 | 1.04573e-01 | 50.22 | 38.15 | |
6 | 528 | 1 | 1 | 3.97454e+00 | 3.97454e+00 | 7.53151e+03 | 7.53152e+03 | 1.78 | 0.79 | |
7 | 1028 | 1 | 1 | 3.44317e+00 | 3.44317e+00 | 7.61831e+03 | 7.61835e+03 | 4.22 | 1.33 | |
8 | 2028 | 1 | 1 | 3.34732e+00 | 3.34732e+00 | 6.95875e+03 | 6.95873e+03 | 4.77 | 2.12 | |
9 | 4028 | 1 | 1 | 3.37016e+00 | 3.37016e+00 | 6.96149e+03 | 6.96147e+03 | 10.50 | 4.80 | |
10 | 528 | 3 | 3 | 4.05662e+00 | 4.05662e+00 | 7.53181e+03 | 7.53182e+03 | 3.08 | 1.29 | |
11 | 1028 | 3 | 3 | 3.87260e+00 | 3.87260e+00 | 5.06412e+03 | 5.06412e+03 | 3.63 | 1.62 | |
12 | 2028 | 3 | 3 | 3.81767e+00 | 3.81767e+00 | 5.07107e+03 | 5.07107e+03 | 5.83 | 2.83 | |
13 | 4028 | 3 | 3 | 3.80375e+00 | 3.80376e+00 | 5.07279e+03 | 5.07279e+03 | 10.88 | 5.23 | |
14 | 480 | 1 | 1 | 3.52651e-01 | 3.52541e-01 | 5.46349e+00 | 5.46349e+00 | 20.80 | 9.31 | |
15 | 682 | 1 | 1 | 3.97454e+00 | 3.97454e+00 | 7.53151e+03 | 7.53152e+03 | 2.29 | 1.03 | |
16 | 1182 | 1 | 1 | 3.44317e+00 | 3.44317e+00 | 7.61831e+03 | 7.61835e+03 | 4.67 | 1.53 | |
17 | 2182 | 1 | 1 | 3.34732e+00 | 3.34732e+00 | 6.95875e+03 | 6.95873e+03 | 5.35 | 2.47 | |
18 | 4182 | 1 | 1 | 3.37016e+00 | 3.37016e+00 | 6.96149e+03 | 6.96147e+03 | 10.41 | 4.51 | |
19 | 682 | 3 | 3 | 4.05662e+00 | 4.05662e+00 | 7.53181e+03 | 7.53182e+03 | 3.56 | 1.36 | |
20 | 1182 | 3 | 3 | 3.87260e+00 | 3.87260e+00 | 5.06412e+03 | 5.06412e+03 | 4.03 | 1.73 | |
21 | 2182 | 3 | 3 | 3.81767e+00 | 3.81767e+00 | 5.07107e+03 | 5.07107e+03 | 6.11 | 2.85 | |
22 | 4182 | 3 | 3 | 3.80375e+00 | 3.80376e+00 | 5.07279e+03 | 5.07279e+03 | 11.28 | 5.19 | |
23 | 7135 | 4 | 4 | 2.01956e+02 | 2.01956e+02 | 3.81763e+00 | 3.81763e+00 | 35.43 | 14.43 | |
24 | 9735 | 4 | 4 | 1.60427e+02 | 1.60427e+02 | 4.93005e+00 | 4.93005e+00 | 69.65 | 16.37 | |
25 | 11305 | 4 | 4 | 1.97389e+02 | 1.97389e+02 | 5.64575e+00 | 5.64650e+00 | 61.65 | 17.91 | |
26 | 13275 | 4 | 4 | 2.04168e+02 | 2.04168e+02 | 5.53766e+00 | 5.53767e+00 | 167.10 | 31.98 | |
27 | 15066 | 4 | 4 | 1.97064e+02 | 1.97064e+02 | 6.39968e+00 | 6.39879e+00 | 102.11 | 29.99 | |
28 | 16861 | 4 | 4 | 1.89579e+02 | 1.89579e+02 | 5.88971e+00 | 5.89023e+00 | 146.18 | 31.62 | |
29 | 21128 | 4 | 4 | 2.09445e+02 | 2.09445e+02 | 5.55792e+00 | 5.55839e+00 | 91.05 | 34.73 | |
30 | 13250 | 1 | 1 | 4.05605e+00 | 4.05605e+00 | 1.09165e+00 | 1.09165e+00 | 39.24 | 16.80 | |
31 | 13309 | 8 | 8 | 5.34292e-02 | 5.34293e-02 | 1.03313e+00 | 1.03308e+00 | 78.47 | 23.25 | |
32 | 13251 | 28 | 28 | 1.18618e-01 | 1.18618e-01 | 1.07935e+00 | 1.07935e+00 | 85.36 | 35.45 | |
33 | 13250 | 46 | 46 | 2.05631e+02 | 2.05631e+02 | 1.07908e+00 | 1.07908e+00 | 115.91 | 49.13 |
# | example | n | m | p | computed |
optimal frequency |
time in s | |||
pseudopoles | matrix pencil | pseudopoles | matrix pencil | pseudopoles | matrix pencil | |||||
1 | 48 | 1 | 1 | 5.27633e-03 | 5.27634e-03 | 5.20608e+00 | 5.20584e+00 | 1.54 | 0.54 | |
2 | 84 | 1 | 1 | 1.08358e+01 | 1.08358e+01 | 0.00000e+00 | 0.00000e+00 | 2.08 | 0.61 | |
3 | 120 | 2 | 2 | 2.31982e+06 | 2.31982e+06 | 2.25682e+01 | 2.25682e+01 | 2.70 | 0.54 | |
4 | 270 | 3 | 3 | 1.15887e-01 | 1.15887e-01 | 7.75093e-01 | 7.75089e-01 | 2.69 | 0.61 | |
5 | 348 | 1 | 1 | 4.55487e+03 | 4.55488e+03 | 1.04575e-01 | 1.04573e-01 | 50.22 | 38.15 | |
6 | 528 | 1 | 1 | 3.97454e+00 | 3.97454e+00 | 7.53151e+03 | 7.53152e+03 | 1.78 | 0.79 | |
7 | 1028 | 1 | 1 | 3.44317e+00 | 3.44317e+00 | 7.61831e+03 | 7.61835e+03 | 4.22 | 1.33 | |
8 | 2028 | 1 | 1 | 3.34732e+00 | 3.34732e+00 | 6.95875e+03 | 6.95873e+03 | 4.77 | 2.12 | |
9 | 4028 | 1 | 1 | 3.37016e+00 | 3.37016e+00 | 6.96149e+03 | 6.96147e+03 | 10.50 | 4.80 | |
10 | 528 | 3 | 3 | 4.05662e+00 | 4.05662e+00 | 7.53181e+03 | 7.53182e+03 | 3.08 | 1.29 | |
11 | 1028 | 3 | 3 | 3.87260e+00 | 3.87260e+00 | 5.06412e+03 | 5.06412e+03 | 3.63 | 1.62 | |
12 | 2028 | 3 | 3 | 3.81767e+00 | 3.81767e+00 | 5.07107e+03 | 5.07107e+03 | 5.83 | 2.83 | |
13 | 4028 | 3 | 3 | 3.80375e+00 | 3.80376e+00 | 5.07279e+03 | 5.07279e+03 | 10.88 | 5.23 | |
14 | 480 | 1 | 1 | 3.52651e-01 | 3.52541e-01 | 5.46349e+00 | 5.46349e+00 | 20.80 | 9.31 | |
15 | 682 | 1 | 1 | 3.97454e+00 | 3.97454e+00 | 7.53151e+03 | 7.53152e+03 | 2.29 | 1.03 | |
16 | 1182 | 1 | 1 | 3.44317e+00 | 3.44317e+00 | 7.61831e+03 | 7.61835e+03 | 4.67 | 1.53 | |
17 | 2182 | 1 | 1 | 3.34732e+00 | 3.34732e+00 | 6.95875e+03 | 6.95873e+03 | 5.35 | 2.47 | |
18 | 4182 | 1 | 1 | 3.37016e+00 | 3.37016e+00 | 6.96149e+03 | 6.96147e+03 | 10.41 | 4.51 | |
19 | 682 | 3 | 3 | 4.05662e+00 | 4.05662e+00 | 7.53181e+03 | 7.53182e+03 | 3.56 | 1.36 | |
20 | 1182 | 3 | 3 | 3.87260e+00 | 3.87260e+00 | 5.06412e+03 | 5.06412e+03 | 4.03 | 1.73 | |
21 | 2182 | 3 | 3 | 3.81767e+00 | 3.81767e+00 | 5.07107e+03 | 5.07107e+03 | 6.11 | 2.85 | |
22 | 4182 | 3 | 3 | 3.80375e+00 | 3.80376e+00 | 5.07279e+03 | 5.07279e+03 | 11.28 | 5.19 | |
23 | 7135 | 4 | 4 | 2.01956e+02 | 2.01956e+02 | 3.81763e+00 | 3.81763e+00 | 35.43 | 14.43 | |
24 | 9735 | 4 | 4 | 1.60427e+02 | 1.60427e+02 | 4.93005e+00 | 4.93005e+00 | 69.65 | 16.37 | |
25 | 11305 | 4 | 4 | 1.97389e+02 | 1.97389e+02 | 5.64575e+00 | 5.64650e+00 | 61.65 | 17.91 | |
26 | 13275 | 4 | 4 | 2.04168e+02 | 2.04168e+02 | 5.53766e+00 | 5.53767e+00 | 167.10 | 31.98 | |
27 | 15066 | 4 | 4 | 1.97064e+02 | 1.97064e+02 | 6.39968e+00 | 6.39879e+00 | 102.11 | 29.99 | |
28 | 16861 | 4 | 4 | 1.89579e+02 | 1.89579e+02 | 5.88971e+00 | 5.89023e+00 | 146.18 | 31.62 | |
29 | 21128 | 4 | 4 | 2.09445e+02 | 2.09445e+02 | 5.55792e+00 | 5.55839e+00 | 91.05 | 34.73 | |
30 | 13250 | 1 | 1 | 4.05605e+00 | 4.05605e+00 | 1.09165e+00 | 1.09165e+00 | 39.24 | 16.80 | |
31 | 13309 | 8 | 8 | 5.34292e-02 | 5.34293e-02 | 1.03313e+00 | 1.03308e+00 | 78.47 | 23.25 | |
32 | 13251 | 28 | 28 | 1.18618e-01 | 1.18618e-01 | 1.07935e+00 | 1.07935e+00 | 85.36 | 35.45 | |
33 | 13250 | 46 | 46 | 2.05631e+02 | 2.05631e+02 | 1.07908e+00 | 1.07908e+00 | 115.91 | 49.13 |
[1] |
Monica Motta, Caterina Sartori. On ${\mathcal L}^1$ limit solutions in impulsive control. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1201-1218. doi: 10.3934/dcdss.2018068 |
[2] |
Lidan Li, Hongwei Zhang, Liwei Zhang. Inverse quadratic programming problem with $ l_1 $ norm measure. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-13. doi: 10.3934/jimo.2019061 |
[3] |
Lin Du, Yun Zhang. $\mathcal{H}_∞$ filtering for switched nonlinear systems: A state projection method. Journal of Industrial & Management Optimization, 2018, 14 (1) : 19-33. doi: 10.3934/jimo.2017035 |
[4] |
Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116 |
[5] |
Yeping Li, Jie Liao. Stability and $ L^{p}$ convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1281-1302. doi: 10.3934/cpaa.2019062 |
[6] |
Tuan Anh Dao, Michael Reissig. $ L^1 $ estimates for oscillating integrals and their applications to semi-linear models with $ \sigma $-evolution like structural damping. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5431-5463. doi: 10.3934/dcds.2019222 |
[7] |
Dajana Conte, Raffaele D'Ambrosio, Beatrice Paternoster. On the stability of $\vartheta$-methods for stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2695-2708. doi: 10.3934/dcdsb.2018087 |
[8] |
Jingwen Wu, Jintao Hu, Hongjiong Tian. Functionally-fitted block $ \theta $-methods for ordinary differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020164 |
[9] |
Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $ L^p$ type critical Besov spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124 |
[10] |
Woocheol Choi, Yong-Cheol Kim. $L^p$ mapping properties for nonlocal Schrödinger operators with certain potentials. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5811-5834. doi: 10.3934/dcds.2018253 |
[11] |
Yongkuan Cheng, Yaotian Shen. Generalized quasilinear Schrödinger equations with concave functions $ l(s^2) $. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1311-1343. doi: 10.3934/dcds.2019056 |
[12] |
Xinghong Pan, Jiang Xu. Global existence and optimal decay estimates of the compressible viscoelastic flows in $ L^p $ critical spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2021-2057. doi: 10.3934/dcds.2019085 |
[13] |
Abdelwahab Bensouilah, Sahbi Keraani. Smoothing property for the $ L^2 $-critical high-order NLS Ⅱ. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2961-2976. doi: 10.3934/dcds.2019123 |
[14] |
Silvia Frassu. Nonlinear Dirichlet problem for the nonlocal anisotropic operator $ L_K $. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1847-1867. doi: 10.3934/cpaa.2019086 |
[15] |
Yupeng Li, Wuchen Li, Guo Cao. Image segmentation via $ L_1 $ Monge-Kantorovich problem. Inverse Problems & Imaging, 2019, 13 (4) : 805-826. doi: 10.3934/ipi.2019037 |
[16] |
Justin Forlano. Almost sure global well posedness for the BBM equation with infinite $ L^{2} $ initial data. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 267-318. doi: 10.3934/dcds.2020011 |
[17] |
Jinrui Huang, Wenjun Wang, Huanyao Wen. On $ L^p $ estimates for a simplified Ericksen-Leslie system. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1485-1507. doi: 10.3934/cpaa.2020075 |
[18] |
Pak Tung Ho. Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry. Communications on Pure & Applied Analysis, 2020, 19 (2) : 715-722. doi: 10.3934/cpaa.2020033 |
[19] |
Yong Ren, Huijin Yang, Wensheng Yin. Weighted exponential stability of stochastic coupled systems on networks with delay driven by $ G $-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3379-3393. doi: 10.3934/dcdsb.2018325 |
[20] |
Edcarlos D. Silva, José Carlos de Albuquerque, Uberlandio Severo. On a class of linearly coupled systems on $ \mathbb{R}^N $ involving asymptotically linear terms. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3089-3101. doi: 10.3934/cpaa.2019138 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]