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doi: 10.3934/dcdsb.2022080
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Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases

1. 

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454003, China

2. 

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 102488, China

3. 

MIIT Key Laboratory of MTCIS, Beijing Institute of Technology, Beijing 102488, China

* Corresponding author: Bing Sun

Received  November 2021 Revised  March 2022 Early access April 2022

Fund Project: The second author is supported in part by the National Natural Science Foundation of China under Grant No. 11471036

In this paper, we are concerned with the space-time spectral discretization of an optimal control problem governed by a fourth-order parabolic partial differential equations (PDEs) in three control constraint cases. The dual Petrov-Galerkin spectral method in time and the spectral method in space are adopted to discrete the continuous system. By means of the obtained optimality condition for the continuous system and that of its spectral discrete system, we establish a priori error estimate for the spectral approximation in details. Four numerical examples are, subsequently, executed to confirm the theoretical results. The experiment results show the high efficiency and a good precision of the space-time spectral method for this kind of problems.

Citation: Zhen-Zhen Tao, Bing Sun. Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022080
References:
[1] R. A. Adams and J. J. Fournier, Sobolev Spaces, 2$^{end}$ edition, Elsevier/Academic Press, Amsterdam, 2003. 
[2]

C. Bernardi, M. Dauge and Y. Maday, Spectral Methods for Axisymmetric Domains, North-Holland, Amsterdam, 1999.

[3]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988. doi: 10.1007/978-3-642-84108-8.

[4]

J. Cao, J. Zhang and X. Yang, Fully-discrete spectral-Galerkin scheme with second-order time-accuracy and unconditionally energy stability for the volume-conserved phase-field lipid vesicle model, J. Comput. Appl. Math., 406 (2022), 113988, 18 pp. doi: 10.1016/j.cam.2021.113988.

[5]

Y. Chen and F. Huang, Galerkin spectral approximation of elliptic optimal control problems with $H^1$-norm state constraint, J. Sci. Comput, 67 (2016), 65-83.  doi: 10.1007/s10915-015-0071-y.

[6]

Y. Chen and F. Huang, Spectral method approximation of flow optimal control problems with $H^1$-norm state constraint, Numer. Math. Theory Methods Appl., 10 (2017), 614-638.  doi: 10.4208/nmtma.2017.m1419.

[7]

Y. ChenY. Huang and N. Yi, A posteriori error estimates of spectral method for optimal control problems governed by parabolic equations, Sci. China Ser. A, 51 (2008), 1376-1390.  doi: 10.1007/s11425-008-0097-9.

[8]

Y. ChenN. Yi and W. Liu, A Legendre-Galerkin spectral method for optimal control problems governed by elliptic equations, SIAM J. Numer Anal, 46 (2008), 2254-2275.  doi: 10.1137/070679703.

[9]

C. M. Elliott and Z. Songmu, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357.  doi: 10.1007/BF00251803.

[10]

W. Gong and M. Hinze, Error estimates for parabolic optimal control problems with control and state constraints, Comput. Optim. Appl., 56 (2013), 131-151.  doi: 10.1007/s10589-013-9541-z.

[11]

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia, PA, 1977.

[12]

B.-Y. Guo, Spectral Methods and Their Applications, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. doi: 10.1142/3662.

[13]

Y. Han, A class of fourth-order parabolic equation with arbitrary initial energy, Nonlinear Anal. Real World Appl., 43 (2018), 451-466.  doi: 10.1016/j.nonrwa.2018.03.009.

[14]

F. Huang and Y. Chen, Error estimates for spectral approximation of elliptic control problems with integral state and control constraints, Comput. Math. Appl., 68 (2014), 789-803.  doi: 10.1016/j.camwa.2014.07.002.

[15]

F. HuangZ. Zheng and Y. Peng, Error estimates of the space-time spectral method for parabolic control problems, Comput. Math. Appl., 75 (2018), 335-348.  doi: 10.1016/j.camwa.2017.09.018.

[16]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971.

[17]

C. Liu, A fourth order parabolic equation with nonlinear principal part, Nonlinear Anal., 68 (2008), 393-401.  doi: 10.1016/j.na.2006.11.005.

[18] W. Liu and N. Yan, Adaptive Finite Element Methods for Optimal Control Governed by PDEs, Science Press, Beijing, 2008. 
[19]

Z. LuF. CaiR. XuC. HouX. Wu and Y. Yang, A posteriori error estimates of $hp$ spectral element method for parabolic optimal control problems, AIMS Math., 7 (2022), 5220-5240.  doi: 10.3934/math.2022291.

[20]

D. MeidnerR. Rannacher and B. Vexler, A priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time, SIAM J. Control Optim., 49 (2011), 1961-1997.  doi: 10.1137/100793888.

[21]

D. MeidnerR. Rannacher and B. Vexler, A priori error estimates for space-time finite element discretization of parabolic optimal control problems. I. Problems without control constraints, SIAM J. Control Optim., 47 (2008), 1150-1177.  doi: 10.1137/070694016.

[22]

I. NeitzelJ. Pfefferer and A. Rösch, Finite element discretization of state-constrained elliptic optimal control problems with semilinear state equation, SIAM J. Control Optim., 53 (2015), 874-904.  doi: 10.1137/140960645.

[23]

I. Neitzel and B. Vexler, A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems, Numer. Math., 120 (2012), 345-386.  doi: 10.1007/s00211-011-0409-9.

[24]

J. Shen, Efficient spectral-Galerkin method I: Direct solvers for second and fourth order equations by using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), 1489-1505.  doi: 10.1137/0915089.

[25] J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006. 
[26]

J. Shen, T. Tang and L.-L. Wang, Spectral Methods. Algorithms, Analysis and Applications, Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.

[27]

J. Shen and L.-L. Wang, Fourierization of the Legendre-Galerkin method and a new space-time spectral method, Appl. Numer. Math., 57 (2007), 710-720.  doi: 10.1016/j.apnum.2006.07.012.

[28]

I. Silberman, Planetary waves in the atmosphere, J. Atmospheric Sciences, 11 (1954), 27-34.  doi: 10.1175/1520-0469(1954)011<0027:PWITA>2.0.CO;2.

[29]

B. Sun, Z.-Z. Tao and Y.-Y. Wang, Dynamic programming viscosity solution approach and its applications to optimal control problems, In Mathematics Applied to Engineering, Modelling, and Social Issues, Springer, Cham, 200 (2019), 363–420.

[30]

Z.-Z. Tao and B. Sun, Galerkin spectral method for a fourth-order optimal control problem with $H^1$-norm state constraint, Comput. Math. Appl., 97 (2021), 1-17.  doi: 10.1016/j.camwa.2021.05.023.

[31]

T. P. Witelski, Similarity solutions of the lubrication equation, Appl. Math. Lett., 10 (1997), 107-113.  doi: 10.1016/S0893-9659(97)00092-X.

[32]

H. Zhang, F. Liu, X. Jiang and I. Turner, Spectral method for the two-dimensional time distributed-order diffusion-wave equation on a semi-infinite domain, J. Comput. Appl. Math., 399 (2022), Paper No. 113712, 15 pp. doi: 10.1016/j.cam.2021.113712.

[33]

J. Zhou and D. Yang, Spectral mixed Galerkin method for state constrained optimal control problem governed by the first bi-harmonic equation, Int. J. Comput. Math., 88 (2011), 2988-3011.  doi: 10.1080/00207160.2011.563845.

[34]

J. ZhouJ. Zhang and X. Xing, Galerkin spectral approximations for optimal control problems governed by the fourth order equation with an integral constraint on state, Comput Math Appl., 72 (2016), 2549-2561.  doi: 10.1016/j.camwa.2016.08.009.

show all references

References:
[1] R. A. Adams and J. J. Fournier, Sobolev Spaces, 2$^{end}$ edition, Elsevier/Academic Press, Amsterdam, 2003. 
[2]

C. Bernardi, M. Dauge and Y. Maday, Spectral Methods for Axisymmetric Domains, North-Holland, Amsterdam, 1999.

[3]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988. doi: 10.1007/978-3-642-84108-8.

[4]

J. Cao, J. Zhang and X. Yang, Fully-discrete spectral-Galerkin scheme with second-order time-accuracy and unconditionally energy stability for the volume-conserved phase-field lipid vesicle model, J. Comput. Appl. Math., 406 (2022), 113988, 18 pp. doi: 10.1016/j.cam.2021.113988.

[5]

Y. Chen and F. Huang, Galerkin spectral approximation of elliptic optimal control problems with $H^1$-norm state constraint, J. Sci. Comput, 67 (2016), 65-83.  doi: 10.1007/s10915-015-0071-y.

[6]

Y. Chen and F. Huang, Spectral method approximation of flow optimal control problems with $H^1$-norm state constraint, Numer. Math. Theory Methods Appl., 10 (2017), 614-638.  doi: 10.4208/nmtma.2017.m1419.

[7]

Y. ChenY. Huang and N. Yi, A posteriori error estimates of spectral method for optimal control problems governed by parabolic equations, Sci. China Ser. A, 51 (2008), 1376-1390.  doi: 10.1007/s11425-008-0097-9.

[8]

Y. ChenN. Yi and W. Liu, A Legendre-Galerkin spectral method for optimal control problems governed by elliptic equations, SIAM J. Numer Anal, 46 (2008), 2254-2275.  doi: 10.1137/070679703.

[9]

C. M. Elliott and Z. Songmu, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357.  doi: 10.1007/BF00251803.

[10]

W. Gong and M. Hinze, Error estimates for parabolic optimal control problems with control and state constraints, Comput. Optim. Appl., 56 (2013), 131-151.  doi: 10.1007/s10589-013-9541-z.

[11]

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia, PA, 1977.

[12]

B.-Y. Guo, Spectral Methods and Their Applications, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. doi: 10.1142/3662.

[13]

Y. Han, A class of fourth-order parabolic equation with arbitrary initial energy, Nonlinear Anal. Real World Appl., 43 (2018), 451-466.  doi: 10.1016/j.nonrwa.2018.03.009.

[14]

F. Huang and Y. Chen, Error estimates for spectral approximation of elliptic control problems with integral state and control constraints, Comput. Math. Appl., 68 (2014), 789-803.  doi: 10.1016/j.camwa.2014.07.002.

[15]

F. HuangZ. Zheng and Y. Peng, Error estimates of the space-time spectral method for parabolic control problems, Comput. Math. Appl., 75 (2018), 335-348.  doi: 10.1016/j.camwa.2017.09.018.

[16]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971.

[17]

C. Liu, A fourth order parabolic equation with nonlinear principal part, Nonlinear Anal., 68 (2008), 393-401.  doi: 10.1016/j.na.2006.11.005.

[18] W. Liu and N. Yan, Adaptive Finite Element Methods for Optimal Control Governed by PDEs, Science Press, Beijing, 2008. 
[19]

Z. LuF. CaiR. XuC. HouX. Wu and Y. Yang, A posteriori error estimates of $hp$ spectral element method for parabolic optimal control problems, AIMS Math., 7 (2022), 5220-5240.  doi: 10.3934/math.2022291.

[20]

D. MeidnerR. Rannacher and B. Vexler, A priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time, SIAM J. Control Optim., 49 (2011), 1961-1997.  doi: 10.1137/100793888.

[21]

D. MeidnerR. Rannacher and B. Vexler, A priori error estimates for space-time finite element discretization of parabolic optimal control problems. I. Problems without control constraints, SIAM J. Control Optim., 47 (2008), 1150-1177.  doi: 10.1137/070694016.

[22]

I. NeitzelJ. Pfefferer and A. Rösch, Finite element discretization of state-constrained elliptic optimal control problems with semilinear state equation, SIAM J. Control Optim., 53 (2015), 874-904.  doi: 10.1137/140960645.

[23]

I. Neitzel and B. Vexler, A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems, Numer. Math., 120 (2012), 345-386.  doi: 10.1007/s00211-011-0409-9.

[24]

J. Shen, Efficient spectral-Galerkin method I: Direct solvers for second and fourth order equations by using Legendre polynomials, SIAM J. Sci. Comput., 15 (1994), 1489-1505.  doi: 10.1137/0915089.

[25] J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006. 
[26]

J. Shen, T. Tang and L.-L. Wang, Spectral Methods. Algorithms, Analysis and Applications, Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.

[27]

J. Shen and L.-L. Wang, Fourierization of the Legendre-Galerkin method and a new space-time spectral method, Appl. Numer. Math., 57 (2007), 710-720.  doi: 10.1016/j.apnum.2006.07.012.

[28]

I. Silberman, Planetary waves in the atmosphere, J. Atmospheric Sciences, 11 (1954), 27-34.  doi: 10.1175/1520-0469(1954)011<0027:PWITA>2.0.CO;2.

[29]

B. Sun, Z.-Z. Tao and Y.-Y. Wang, Dynamic programming viscosity solution approach and its applications to optimal control problems, In Mathematics Applied to Engineering, Modelling, and Social Issues, Springer, Cham, 200 (2019), 363–420.

[30]

Z.-Z. Tao and B. Sun, Galerkin spectral method for a fourth-order optimal control problem with $H^1$-norm state constraint, Comput. Math. Appl., 97 (2021), 1-17.  doi: 10.1016/j.camwa.2021.05.023.

[31]

T. P. Witelski, Similarity solutions of the lubrication equation, Appl. Math. Lett., 10 (1997), 107-113.  doi: 10.1016/S0893-9659(97)00092-X.

[32]

H. Zhang, F. Liu, X. Jiang and I. Turner, Spectral method for the two-dimensional time distributed-order diffusion-wave equation on a semi-infinite domain, J. Comput. Appl. Math., 399 (2022), Paper No. 113712, 15 pp. doi: 10.1016/j.cam.2021.113712.

[33]

J. Zhou and D. Yang, Spectral mixed Galerkin method for state constrained optimal control problem governed by the first bi-harmonic equation, Int. J. Comput. Math., 88 (2011), 2988-3011.  doi: 10.1080/00207160.2011.563845.

[34]

J. ZhouJ. Zhang and X. Xing, Galerkin spectral approximations for optimal control problems governed by the fourth order equation with an integral constraint on state, Comput Math Appl., 72 (2016), 2549-2561.  doi: 10.1016/j.camwa.2016.08.009.

Figure 1.  Error estimates in Example 4.1
Figure 2.  Error estimates versus $ M $ with $ N = 16 $ in Example 4.2
Figure 3.  Error estimates in Example 4.3
Figure 4.  Error estimates in Example 4.4
Table 1.  The numerical results of error estimates versus $ M $ with $ N = 16 $ in Example 4.1
$ M $ 4 6 8 10 12
$ \left\|u-u_S \right\|_X $ 2.6735e-1 2.0432e-2 8.0550e-4 1.9906e-5 4.0128e-7
$ \left\|y-y_S \right\|_X $ 2.5661e-1 1.9019e-2 7.4403e-4 1.8202e-5 3.6027e-7
$ \left\|p-p_S \right\|_X $ 2.7749e-1 2.0434e-2 8.0541e-4 1.9922e-5 4.0364e-7
$ M $ 4 6 8 10 12
$ \left\|u-u_S \right\|_X $ 2.6735e-1 2.0432e-2 8.0550e-4 1.9906e-5 4.0128e-7
$ \left\|y-y_S \right\|_X $ 2.5661e-1 1.9019e-2 7.4403e-4 1.8202e-5 3.6027e-7
$ \left\|p-p_S \right\|_X $ 2.7749e-1 2.0434e-2 8.0541e-4 1.9922e-5 4.0364e-7
Table 2.  The numerical results of error estimates versus $ N $ with $ M = 16 $ in Example 4.1
$ N $ 4 6 8 10 12
$ \left\|u-u_S \right\|_X $ 7.2398e-1 1.8638e-1 2.4561e-2 2.1601e-3 1.2982e-4
$ \left\|y-y_S \right\|_X $ 6.6338e-1 1.6126e-1 2.1630e-2 1.8708e-3 1.1242e-4
$ \left\|p-p_S \right\|_X $ 7.6287e-1 1.8557e-1 2.4975e-2 2.1601e-3 1.2982e-4
$ N $ 4 6 8 10 12
$ \left\|u-u_S \right\|_X $ 7.2398e-1 1.8638e-1 2.4561e-2 2.1601e-3 1.2982e-4
$ \left\|y-y_S \right\|_X $ 6.6338e-1 1.6126e-1 2.1630e-2 1.8708e-3 1.1242e-4
$ \left\|p-p_S \right\|_X $ 7.6287e-1 1.8557e-1 2.4975e-2 2.1601e-3 1.2982e-4
Table 3.  The numerical results of error estimates versus $ M $ with $ N = 16 $ in Example 4.2
$ M $ 2 4 6 8 10 12
$ \left\|u-u_S \right\|_X $ 1.5461e-0 2.4357e-1 1.8796e-2 6.5755e-4 1.5708e-5 9.6694e-5
$ \left\|y-y_S \right\|_X $ 2.3426e-1 8.7262e-3 2.6265e-3 1.0560e-3 2.4378e-4 1.6250e-4
$ \left\|p-p_S \right\|_X $ 1.2673e-0 2.4358e-1 1.7679e-2 6.5756e-4 1.5549e-5 9.6694e-5
$ M $ 2 4 6 8 10 12
$ \left\|u-u_S \right\|_X $ 1.5461e-0 2.4357e-1 1.8796e-2 6.5755e-4 1.5708e-5 9.6694e-5
$ \left\|y-y_S \right\|_X $ 2.3426e-1 8.7262e-3 2.6265e-3 1.0560e-3 2.4378e-4 1.6250e-4
$ \left\|p-p_S \right\|_X $ 1.2673e-0 2.4358e-1 1.7679e-2 6.5756e-4 1.5549e-5 9.6694e-5
Table 4.  The numerical results of error estimates versus $ N $ with $ M = 16 $ in Example 4.2
$ N $ 4 6 8 10 12
$ \left\|u-u_S \right\|_X $ 6.5948e-1 1.8336e-1 2.5193e-2 2.1668e-3 1.2954e-4
$ \left\|y-y_S \right\|_X $ 1.2300e-1 5.4459e-3 1.5343e-4 4.5379e-5 4.5297e-5
$ \left\|p-p_S \right\|_X $ 6.5948e-1 1.8336e-1 2.5193e-2 2.1667e-3 1.2953e-4
$ N $ 4 6 8 10 12
$ \left\|u-u_S \right\|_X $ 6.5948e-1 1.8336e-1 2.5193e-2 2.1668e-3 1.2954e-4
$ \left\|y-y_S \right\|_X $ 1.2300e-1 5.4459e-3 1.5343e-4 4.5379e-5 4.5297e-5
$ \left\|p-p_S \right\|_X $ 6.5948e-1 1.8336e-1 2.5193e-2 2.1667e-3 1.2953e-4
Table 5.  The numerical results of error estimates versus $ M $ with $ N = 14 $ in Example 4.3
$ M $ 4 6 8 10 12
$ \left\|u-u_S \right\|_X $ 2.8277e-1 2.2323e-2 9.0317e-4 2.4526e-5 9.1509e-6
$ \left\|y-y_S \right\|_X $ 2.2620e-1 1.7289e-2 7.0066e-4 1.8947e-5 6.8776e-6
$ \left\|p-p_S \right\|_X $ 2.9416e-1 2.2327e-2 9.0309e-4 2.4546e-5 9.1127e-6
$ M $ 4 6 8 10 12
$ \left\|u-u_S \right\|_X $ 2.8277e-1 2.2323e-2 9.0317e-4 2.4526e-5 9.1509e-6
$ \left\|y-y_S \right\|_X $ 2.2620e-1 1.7289e-2 7.0066e-4 1.8947e-5 6.8776e-6
$ \left\|p-p_S \right\|_X $ 2.9416e-1 2.2327e-2 9.0309e-4 2.4546e-5 9.1127e-6
Table 6.  The numerical results of error estimates versus $ N $ with $ M = 14 $ in Example 4.3
$ N $ 4 6 8 10 12
$ \left\|u-u_S \right\|_X $ 9.0104e-1 2.4823e-2 3.7304e-2 3.3514e-3 2.0369e-4
$ \left\|y-y_S \right\|_X $ 6.5356e-1 1.9859e-2 2.8635e-2 2.5156e-3 1.5277e-4
$ \left\|p-p_S \right\|_X $ 8.7113e-1 2.6477e-2 3.8175e-2 3.3540e-3 2.0369e-4
$ N $ 4 6 8 10 12
$ \left\|u-u_S \right\|_X $ 9.0104e-1 2.4823e-2 3.7304e-2 3.3514e-3 2.0369e-4
$ \left\|y-y_S \right\|_X $ 6.5356e-1 1.9859e-2 2.8635e-2 2.5156e-3 1.5277e-4
$ \left\|p-p_S \right\|_X $ 8.7113e-1 2.6477e-2 3.8175e-2 3.3540e-3 2.0369e-4
Table 7.  The numerical results of error estimates versus $ M $ with $ N = 14 $ in Example 4.4
$ M $ 4 6 8 10 12
$ \left\|u-u_S \right\|_X $ 9.2474e-1 6.1536e-2 2.4340e-3 5.5967e-5 8.9987e-7
$ \left\|y-y_S \right\|_X $ 4.1858e-2 3.7996e-3 1.6333e-4 3.7134e-6 5.7744e-8
$ \left\|p-p_S \right\|_X $ 8.4912e-1 6.1544e-2 2.3427e-3 5.5097e-5 8.8207e-7
$ M $ 4 6 8 10 12
$ \left\|u-u_S \right\|_X $ 9.2474e-1 6.1536e-2 2.4340e-3 5.5967e-5 8.9987e-7
$ \left\|y-y_S \right\|_X $ 4.1858e-2 3.7996e-3 1.6333e-4 3.7134e-6 5.7744e-8
$ \left\|p-p_S \right\|_X $ 8.4912e-1 6.1544e-2 2.3427e-3 5.5097e-5 8.8207e-7
Table 8.  The numerical results of error estimates versus $ N $ with $ M = 14 $ in Example 4.4
$ N $ 4 6 8 10 12
$ \|u-u_S\|_X $ 2.1487e-1 9.4348e-3 2.7567e-4 4.9387e-6 7.1424e-8
$ \|y-y_S\|_X $ 3.1639e-1 1.3647e-2 3.7899e-4 7.1197e-6 1.0138e-7
$ \|p-p_S\|_X $ 2.1487e-1 9.2748e-3 2.5742e-4 4.8196e-6 6.9612e-8
$ N $ 4 6 8 10 12
$ \|u-u_S\|_X $ 2.1487e-1 9.4348e-3 2.7567e-4 4.9387e-6 7.1424e-8
$ \|y-y_S\|_X $ 3.1639e-1 1.3647e-2 3.7899e-4 7.1197e-6 1.0138e-7
$ \|p-p_S\|_X $ 2.1487e-1 9.2748e-3 2.5742e-4 4.8196e-6 6.9612e-8
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