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November 2017, 22(9): 3439-3458. doi: 10.3934/dcdsb.2017174

Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs

1. 

College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

2. 

School of Mathematics & Institute of Finance, Shandong University, Jinan 250100, China

3. 

LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author

Received  April 2016 Revised  April 2017 Published  July 2017

Fund Project: This work was supported by the National Natural Science Foundations of China under grants 91630312,91630203,11571351,11171189 and 11571206. The last author was also supported by NCMIS

This is the second part of a series papers on multi-step schemes for solving coupled forward backward stochastic differential equations (FBSDEs). We extend the basic idea in our former paper [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751] to solve high-dimensional FBSDEs, by using the spectral sparse grid approximations. The main issue for solving high-dimensional FBSDEs is to build an efficient spatial discretization, and deal with the related high-dimensional conditional expectations and interpolations. In this work, we propose the sparse grid spatial discretization. The sparse grid Gaussian-Hermite quadrature rule is used to approximate the conditional expectations. And for the associated high-dimensional interpolations, we adopt a spectral expansion of functions in polynomial spaces with respect to the spatial variables, and use the sparse grid approximations to recover the expansion coefficients. The FFT algorithm is used to speed up the recovery procedure, and the entire algorithm admits efficient and highly accurate approximations in high dimensions. Several numerical examples are presented to demonstrate the efficiency of the proposed methods.

Citation: Yu Fu, Weidong Zhao, Tao Zhou. Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3439-3458. doi: 10.3934/dcdsb.2017174
References:
[1]

V. BarthelmannE. Novak and K. Ritter, High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math., 12 (2000), 273-288. doi: 10.1023/A:1018977404843.

[2]

C. Bender and J. Zhang, Time discretization and Markovian iteration for coupled FBSDEs, Ann. Appl. Probab., 18 (2008), 143-177. doi: 10.1214/07-AAP448.

[3]

J. M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., 44 (1973), 384-404. doi: 10.1016/0022-247X(73)90066-8.

[4]

B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stoch. Proc. Appl., 111 (2004), 175-206. doi: 10.1016/j.spa.2004.01.001.

[5]

J. F. Chassagneux and D. Crisan, Runge-Kutta schemes for backward stochastic differential equations, Ann. Appl. Probab., 24 (2014), 679-720. doi: 10.1214/13-AAP933.

[6]

P. CheriditoH. M. SonerN. Touzi and N. Victoir, Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs, Commun. Pur. Appl. Math., 60 (2007), 1081-1110. doi: 10.1002/cpa.20168.

[7]

D. Crisan and K. Manolarakis, Solving backward stochastic differential equations using the cubature method: application to nonlinear pricing, SIAM J. Financial Math., 3 (2012), 534-571. doi: 10.1137/090765766.

[8]

J. DouglasJ. Ma and P. Protter, Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab., 6 (1996), 940-968. doi: 10.1214/aoap/1034968235.

[9]

A. FahimN. Touzi and X. Warin, A probabilistic numerical method for fully nonlinear parabolic PDEs, Ann. Appl. Probab., 21 (2011), 1322-1364. doi: 10.1214/10-AAP723.

[10]

Y. FuW. Zhao and T. Zhou, Multistep schemes for forward backward stochastic differential equations with jumps, J. Sci. Comput., 69 (2016), 651-672. doi: 10.1007/s10915-016-0212-y.

[11]

W. GuoJ. Zhang and J. Zhuo, A monotone scheme for high-dimensional fully nonlinear PDEs, Ann. Appl. Probab., 25 (2015), 1540-1580. doi: 10.1214/14-AAP1030.

[12]

N. El KarouiC. KapoudjianE. PardouxS. Peng and M. C. Quenez, Reflected solutions of backward SDE's and related obstacle problems for PDE's, Ann. Probab., 25 (1997), 702-737. doi: 10.1214/aop/1024404416.

[13]

N. El KarouiS. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Financ., 7 (1997), 1-71. doi: 10.1111/1467-9965.00022.

[14]

T. KongW. Zhao and T. Zhou, High order probabilistic numerical schemes for fully nonlinear parabolic PDEs, Commun. Comput. Phys., 18 (2015), 1482-1503. doi: 10.4208/cicp.240515.280815a.

[15]

T. KongW. Zhao and T. Zhou, High order numerical schemes for second order FBSDEs with applications to stochastic optimal control, Commun. Comput. Phys., 21 (2017), 808-834. doi: 10.4208/cicp.OA-2016-0056.

[16]

J. P. LemorE. Gobet and X. Warin, A regression-based Monte Carlo method to solve backward stochastic differential equations, Ann. Appl. Probab., 15 (2005), 2172-2202. doi: 10.1214/105051605000000412.

[17]

Y. LiJ. Yang and W. Zhao, Convergence error estimates of the {C}rank-{N}icolson scheme for solving decoupled {FBSDE}s, Sci. China Math., 60 (2017), 923-948. doi: 10.1007/s11425-016-0178-8.

[18]

J. MaP. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitly -a four step scheme, Probab. Theory Related Fields, 98 (1994), 339-359. doi: 10.1007/BF01192258.

[19]

J. MaJ. Shen and Y. Zhao, On numerical approximations of forward-backward stochastic differential equations, SIAM J. Numer. Anal., 46 (2008), 2636-2661. doi: 10.1137/06067393X.

[20]

G. N. Milstein and M. V. Tretyakov, Numerical algorithms for forward-backward stochastic differential equations, SIAM J. Sci. Comput., 28 (2006), 561-582. doi: 10.1137/040614426.

[21]

F. NobileR. Tempone and C. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal., 46 (2008), 2309-2345. doi: 10.1137/060663660.

[22]

A. Narayan and T. Zhou, Stochastic collocation on unstructured multivariate meshes, Commun. Comput. Phys., 18 (2015), 1-36. doi: 10.4208/cicp.020215.070515a.

[23]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Syst. Control Lett., 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6.

[24]

E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Related Fields, 114 (1999), 123-150. doi: 10.1007/s004409970001.

[25]

S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stoch. Stoch. Repts., 37 (1991), 61-74. doi: 10.1080/17442509108833727.

[26]

M. J. Ruijter and C. W. Oosterlee, Fourier-cosine method for an efficient computation of solutions to BSDEs, SIAM J. Sci. Comput., 37 (2015), A859-A889. doi: 10.1137/130913183.

[27]

J. Shen and H. Yu, Efficient spectral sparse grid methods and applications to high dimensional elliptic problems, SIAM J. Sci. Comput., 32 (2010), 3228-3250. doi: 10.1137/100787842.

[28]

J. Shen and H. Yu, Efficient spectral sparse grid methods and applications to high dimensional elliptic problems Ⅱ unbounded domains, SIAM J. Sci. Comput., 34 (2012), A1141-A1164. doi: 10.1137/110834950.

[29]

S. A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR, 4 (1963), 240-243.

[30]

H. M. SonerN. Touzi and J. Zhang, Wellposedness of second order backward SDEs, Probab. Theory Related Fields,, 153 (2012), 149-190. doi: 10.1007/s00440-011-0342-y.

[31]

T. TangW. Zhao and T. Zhou, Deferred correction methods for forward backward stochastic differential equations, Numer. Math. Theory Methods Appl., 10 (2017), 222-242. doi: 10.4208/nmtma.2017.s02.

[32]

D. Xiu and J. S. Hesthaven, High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), 1118-1139. doi: 10.1137/040615201.

[33]

G. ZhangM. Gunzburger and W. Zhao, A sparse-grid method for multi-dimensional backward stochastic differential equations, J. Comput. Math., 31 (2013), 221-248. doi: 10.4208/jcm.1212-m4014.

[34]

J. Zhang, A numerical scheme for BSDEs, Ann. Appl. Probab., 14 (2004), 459-488. doi: 10.1214/aoap/1075828058.

[35]

W. ZhaoL. Chen and S. Peng, A new kind of accurate numerical method for backward stochastic differential equations, SIAM J. Sci. Comput., 28 (2006), 1563-1581. doi: 10.1137/05063341X.

[36]

W. ZhaoY. Fu and T. Zhou, New kinds of high-order multistep schemes for coupled forward backward stochastic differential equations, SIAM J. Sci. Comput., 36 (2014), A1731-A1751. doi: 10.1137/130941274.

[37]

W. ZhaoY. Li and G. Zhang, A generalized θ-scheme for solving backward stochastic differential equations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1585-1603. doi: 10.3934/dcdsb.2012.17.1585.

[38]

W. ZhaoJ. Wang and S. Peng, Error estimates of the θ-scheme for backward stochastic differential equations, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 905-924. doi: 10.3934/dcdsb.2009.12.905.

[39]

W. ZhaoG. Zhang and L. Ju, A stable multistep scheme for solving backward stochastic differential equations, SIAM J. Numer. Anal., 48 (2010), 1369-1394. doi: 10.1137/09076979X.

[40]

W. ZhaoW. Zhang and L. Ju, A numerical method and its error estimates for the decoupled forward-backward stochastic differential equations, Commun. Comput. Phys., 15 (2014), 618-646. doi: 10.4208/cicp.280113.190813a.

[41]

W. ZhaoW. Zhang and L. Ju, A multistep scheme for decoupled forward-backward stochastic differential equations, Numer. Math. Theory Methods Appl., 9 (2016), 262-288. doi: 10.4208/nmtma.2016.m1421.

show all references

References:
[1]

V. BarthelmannE. Novak and K. Ritter, High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math., 12 (2000), 273-288. doi: 10.1023/A:1018977404843.

[2]

C. Bender and J. Zhang, Time discretization and Markovian iteration for coupled FBSDEs, Ann. Appl. Probab., 18 (2008), 143-177. doi: 10.1214/07-AAP448.

[3]

J. M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., 44 (1973), 384-404. doi: 10.1016/0022-247X(73)90066-8.

[4]

B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stoch. Proc. Appl., 111 (2004), 175-206. doi: 10.1016/j.spa.2004.01.001.

[5]

J. F. Chassagneux and D. Crisan, Runge-Kutta schemes for backward stochastic differential equations, Ann. Appl. Probab., 24 (2014), 679-720. doi: 10.1214/13-AAP933.

[6]

P. CheriditoH. M. SonerN. Touzi and N. Victoir, Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs, Commun. Pur. Appl. Math., 60 (2007), 1081-1110. doi: 10.1002/cpa.20168.

[7]

D. Crisan and K. Manolarakis, Solving backward stochastic differential equations using the cubature method: application to nonlinear pricing, SIAM J. Financial Math., 3 (2012), 534-571. doi: 10.1137/090765766.

[8]

J. DouglasJ. Ma and P. Protter, Numerical methods for forward-backward stochastic differential equations, Ann. Appl. Probab., 6 (1996), 940-968. doi: 10.1214/aoap/1034968235.

[9]

A. FahimN. Touzi and X. Warin, A probabilistic numerical method for fully nonlinear parabolic PDEs, Ann. Appl. Probab., 21 (2011), 1322-1364. doi: 10.1214/10-AAP723.

[10]

Y. FuW. Zhao and T. Zhou, Multistep schemes for forward backward stochastic differential equations with jumps, J. Sci. Comput., 69 (2016), 651-672. doi: 10.1007/s10915-016-0212-y.

[11]

W. GuoJ. Zhang and J. Zhuo, A monotone scheme for high-dimensional fully nonlinear PDEs, Ann. Appl. Probab., 25 (2015), 1540-1580. doi: 10.1214/14-AAP1030.

[12]

N. El KarouiC. KapoudjianE. PardouxS. Peng and M. C. Quenez, Reflected solutions of backward SDE's and related obstacle problems for PDE's, Ann. Probab., 25 (1997), 702-737. doi: 10.1214/aop/1024404416.

[13]

N. El KarouiS. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Financ., 7 (1997), 1-71. doi: 10.1111/1467-9965.00022.

[14]

T. KongW. Zhao and T. Zhou, High order probabilistic numerical schemes for fully nonlinear parabolic PDEs, Commun. Comput. Phys., 18 (2015), 1482-1503. doi: 10.4208/cicp.240515.280815a.

[15]

T. KongW. Zhao and T. Zhou, High order numerical schemes for second order FBSDEs with applications to stochastic optimal control, Commun. Comput. Phys., 21 (2017), 808-834. doi: 10.4208/cicp.OA-2016-0056.

[16]

J. P. LemorE. Gobet and X. Warin, A regression-based Monte Carlo method to solve backward stochastic differential equations, Ann. Appl. Probab., 15 (2005), 2172-2202. doi: 10.1214/105051605000000412.

[17]

Y. LiJ. Yang and W. Zhao, Convergence error estimates of the {C}rank-{N}icolson scheme for solving decoupled {FBSDE}s, Sci. China Math., 60 (2017), 923-948. doi: 10.1007/s11425-016-0178-8.

[18]

J. MaP. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitly -a four step scheme, Probab. Theory Related Fields, 98 (1994), 339-359. doi: 10.1007/BF01192258.

[19]

J. MaJ. Shen and Y. Zhao, On numerical approximations of forward-backward stochastic differential equations, SIAM J. Numer. Anal., 46 (2008), 2636-2661. doi: 10.1137/06067393X.

[20]

G. N. Milstein and M. V. Tretyakov, Numerical algorithms for forward-backward stochastic differential equations, SIAM J. Sci. Comput., 28 (2006), 561-582. doi: 10.1137/040614426.

[21]

F. NobileR. Tempone and C. Webster, A sparse grid stochastic collocation method for partial differential equations with random input data, SIAM J. Numer. Anal., 46 (2008), 2309-2345. doi: 10.1137/060663660.

[22]

A. Narayan and T. Zhou, Stochastic collocation on unstructured multivariate meshes, Commun. Comput. Phys., 18 (2015), 1-36. doi: 10.4208/cicp.020215.070515a.

[23]

E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Syst. Control Lett., 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6.

[24]

E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Probab. Theory Related Fields, 114 (1999), 123-150. doi: 10.1007/s004409970001.

[25]

S. Peng, Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stoch. Stoch. Repts., 37 (1991), 61-74. doi: 10.1080/17442509108833727.

[26]

M. J. Ruijter and C. W. Oosterlee, Fourier-cosine method for an efficient computation of solutions to BSDEs, SIAM J. Sci. Comput., 37 (2015), A859-A889. doi: 10.1137/130913183.

[27]

J. Shen and H. Yu, Efficient spectral sparse grid methods and applications to high dimensional elliptic problems, SIAM J. Sci. Comput., 32 (2010), 3228-3250. doi: 10.1137/100787842.

[28]

J. Shen and H. Yu, Efficient spectral sparse grid methods and applications to high dimensional elliptic problems Ⅱ unbounded domains, SIAM J. Sci. Comput., 34 (2012), A1141-A1164. doi: 10.1137/110834950.

[29]

S. A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR, 4 (1963), 240-243.

[30]

H. M. SonerN. Touzi and J. Zhang, Wellposedness of second order backward SDEs, Probab. Theory Related Fields,, 153 (2012), 149-190. doi: 10.1007/s00440-011-0342-y.

[31]

T. TangW. Zhao and T. Zhou, Deferred correction methods for forward backward stochastic differential equations, Numer. Math. Theory Methods Appl., 10 (2017), 222-242. doi: 10.4208/nmtma.2017.s02.

[32]

D. Xiu and J. S. Hesthaven, High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), 1118-1139. doi: 10.1137/040615201.

[33]

G. ZhangM. Gunzburger and W. Zhao, A sparse-grid method for multi-dimensional backward stochastic differential equations, J. Comput. Math., 31 (2013), 221-248. doi: 10.4208/jcm.1212-m4014.

[34]

J. Zhang, A numerical scheme for BSDEs, Ann. Appl. Probab., 14 (2004), 459-488. doi: 10.1214/aoap/1075828058.

[35]

W. ZhaoL. Chen and S. Peng, A new kind of accurate numerical method for backward stochastic differential equations, SIAM J. Sci. Comput., 28 (2006), 1563-1581. doi: 10.1137/05063341X.

[36]

W. ZhaoY. Fu and T. Zhou, New kinds of high-order multistep schemes for coupled forward backward stochastic differential equations, SIAM J. Sci. Comput., 36 (2014), A1731-A1751. doi: 10.1137/130941274.

[37]

W. ZhaoY. Li and G. Zhang, A generalized θ-scheme for solving backward stochastic differential equations, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1585-1603. doi: 10.3934/dcdsb.2012.17.1585.

[38]

W. ZhaoJ. Wang and S. Peng, Error estimates of the θ-scheme for backward stochastic differential equations, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 905-924. doi: 10.3934/dcdsb.2009.12.905.

[39]

W. ZhaoG. Zhang and L. Ju, A stable multistep scheme for solving backward stochastic differential equations, SIAM J. Numer. Anal., 48 (2010), 1369-1394. doi: 10.1137/09076979X.

[40]

W. ZhaoW. Zhang and L. Ju, A numerical method and its error estimates for the decoupled forward-backward stochastic differential equations, Commun. Comput. Phys., 15 (2014), 618-646. doi: 10.4208/cicp.280113.190813a.

[41]

W. ZhaoW. Zhang and L. Ju, A multistep scheme for decoupled forward-backward stochastic differential equations, Numer. Math. Theory Methods Appl., 9 (2016), 262-288. doi: 10.4208/nmtma.2016.m1421.

Figure 1.  Sparse grid for CGL $(x_1, x_2)\in\mathcal{C}_2^6$ (Left) and sparse grid for GH $(x_1, x_2)\in\mathcal{G}_2^6$ (Right)
Figure 2.  Dimensions v.s. the running time for Example 3 with $N=128$
Table 1.  Main techniques used in the SSG and LTG methods for solving FBSDEs. TP: tensor product. SG: sparse grid. GH: Gaussian-Hermite
MethodMeshesConditional expectationsApproximation & interpolation
SSGsparse gridSG GH quadratureSG interpolation
LTGTP uniform meshTP GH quadratureLagrangian
MethodMeshesConditional expectationsApproximation & interpolation
SSGsparse gridSG GH quadratureSG interpolation
LTGTP uniform meshTP GH quadratureLagrangian
Table 2.  Errors and convergence rates for Example 1 by SSG (Top) and LTG (Bottom)
step numbererrorsN=8N=16N=32N=64N=128CR
1-step$\mathcal{E}_Y$3.991E-022.050E-021.039E-025.232E-032.625E-030.982
$\mathcal{E}_Z$5.186E-022.649E-021.339E-026.733E-033.377E-030.986
RT2.5194.6389.52119.19938.679
2-step$\mathcal{E}_Y$5.620E-031.456E-033.670E-049.182E-052.286E-051.987
$\mathcal{E}_Z$6.978E-031.847E-034.813E-041.225E-043.090E-051.955
RT6.85215.59633.40168.541143.552
3-step$\mathcal{E}_Y$9.748E-041.342E-041.728E-052.264E-068.196E-072.632
$\mathcal{E}_Z$3.091E-033.850E-044.757E-056.009E-068.834E-072.955
RT6.89919.40343.83595.902197.608
step numbererrorsN=8N=16N=32N=64N=128CR
1-step$\mathcal{E}_Y$7.204E-013.989E-011.849E-017.321E-022.871E-021.174
$\mathcal{E}_Z$2.670E-011.534E-017.409E-023.152E-021.332E-021.093
RT1.0474.60924.120141.209844.974
2-step$\mathcal{E}_Y$4.873E-011.105E-012.220E-024.165E-037.708E-042.333
$\mathcal{E}_Z$1.999E-014.706E-021.174E-022.494E-034.159E-042.205
RT3.86112.97060.661363.0522139.540
3-step$\mathcal{E}_Y$2.656E-012.252E-022.295E-032.247E-042.024E-053.401
$\mathcal{E}_Z$1.136E-019.841E-031.306E-031.417E-041.244E-053.243
RT13.01043.490193.784968.7884929.977
step numbererrorsN=8N=16N=32N=64N=128CR
1-step$\mathcal{E}_Y$3.991E-022.050E-021.039E-025.232E-032.625E-030.982
$\mathcal{E}_Z$5.186E-022.649E-021.339E-026.733E-033.377E-030.986
RT2.5194.6389.52119.19938.679
2-step$\mathcal{E}_Y$5.620E-031.456E-033.670E-049.182E-052.286E-051.987
$\mathcal{E}_Z$6.978E-031.847E-034.813E-041.225E-043.090E-051.955
RT6.85215.59633.40168.541143.552
3-step$\mathcal{E}_Y$9.748E-041.342E-041.728E-052.264E-068.196E-072.632
$\mathcal{E}_Z$3.091E-033.850E-044.757E-056.009E-068.834E-072.955
RT6.89919.40343.83595.902197.608
step numbererrorsN=8N=16N=32N=64N=128CR
1-step$\mathcal{E}_Y$7.204E-013.989E-011.849E-017.321E-022.871E-021.174
$\mathcal{E}_Z$2.670E-011.534E-017.409E-023.152E-021.332E-021.093
RT1.0474.60924.120141.209844.974
2-step$\mathcal{E}_Y$4.873E-011.105E-012.220E-024.165E-037.708E-042.333
$\mathcal{E}_Z$1.999E-014.706E-021.174E-022.494E-034.159E-042.205
RT3.86112.97060.661363.0522139.540
3-step$\mathcal{E}_Y$2.656E-012.252E-022.295E-032.247E-042.024E-053.401
$\mathcal{E}_Z$1.136E-019.841E-031.306E-031.417E-041.244E-053.243
RT13.01043.490193.784968.7884929.977
Table 3.  Errors and convergence rates for Example 2
step numbererrorsN=8N=16N=32N=64N=128CR
1-step$\mathcal{E}_Y$6.229E-033.173E-031.547E-037.695E-043.849E-041.008
$\mathcal{E}_Z$8.693E-024.335E-022.165E-021.082E-025.411E-031.001
RT4.2689.03617.72035.74076.049
2-step$\mathcal{E}_Y$4.126E-051.148E-052.251E-063.986E-071.051E-072.208
$\mathcal{E}_Z$5.470E-041.630E-044.222E-058.722E-062.061E-062.032
RT6.98615.92634.53072.690153.092
3-step$\mathcal{E}_Y$4.078E-058.408E-061.107E-061.370E-071.837E-082.817
$\mathcal{E}_Z$4.438E-045.143E-057.673E-061.049E-061.514E-072.865
RT7.98020.65048.610102.595212.189
step numbererrorsN=8N=16N=32N=64N=128CR
1-step$\mathcal{E}_Y$6.229E-033.173E-031.547E-037.695E-043.849E-041.008
$\mathcal{E}_Z$8.693E-024.335E-022.165E-021.082E-025.411E-031.001
RT4.2689.03617.72035.74076.049
2-step$\mathcal{E}_Y$4.126E-051.148E-052.251E-063.986E-071.051E-072.208
$\mathcal{E}_Z$5.470E-041.630E-044.222E-058.722E-062.061E-062.032
RT6.98615.92634.53072.690153.092
3-step$\mathcal{E}_Y$4.078E-058.408E-061.107E-061.370E-071.837E-082.817
$\mathcal{E}_Z$4.438E-045.143E-057.673E-061.049E-061.514E-072.865
RT7.98020.65048.610102.595212.189
Table 4.  Errors and convergence rates for Example 3
schemesparse gridN=8N=16N=32N=64N=128CR
1-step$q=3$
$\mathcal{C}_3^4$ & $\mathcal{G}_3^4$
$\mathcal{E}_Y$5.717E-022.999E-021.557E-028.021E-034.104E-030.950
$\mathcal{E}_Z$3.129E-021.575E-027.988E-034.057E-032.057E-030.981
RT0.1070.2260.3890.6271.268
2-step$q=3$
$\mathcal{C}_3^4$ & $\mathcal{G}_3^4$
$\mathcal{E}_Y$2.766E-037.861E-042.091E-045.396E-051.371E-051.918
$\mathcal{E}_Z$4.833E-031.197E-033.006E-047.588E-051.916E-051.994
RT0.1310.2350.5091.0672.187
3-step$q=3$
$\mathcal{C}_3^4$ & $\mathcal{G}_3^4$
$\mathcal{E}_Y$1.244E-041.183E-051.061E-069.522E-088.587E-093.460
$\mathcal{E}_Z$3.425E-044.375E-055.575E-067.067E-078.922E-082.976
RT0.1730.3700.7501.4783.072
1-step$q=4$
$\mathcal{C}_4^5$ & $\mathcal{G}_4^5$
$\mathcal{E}_Y$1.196E-016.208E-023.187E-021.626E-028.259E-030.965
$\mathcal{E}_Z$3.445E-021.735E-028.772E-034.439E-032.244E-030.985
RT1.6823.1536.17812.93426.182
2-step$q=4$
$\mathcal{C}_4^5$ & $\mathcal{G}_4^5$
$\mathcal{E}_Y$9.114E-032.817E-037.851E-042.087E-045.426E-051.854
$\mathcal{E}_Z$6.769E-031.628E-034.038E-041.012E-042.547E-052.012
RT2.5134.4429.60620.16041.504
3-step$q=4$
$\mathcal{C}_4^5$ & $\mathcal{G}_4^5$
$\mathcal{E}_Y$4.879E-047.611E-051.052E-051.378E-061.763E-072.865
$\mathcal{E}_Z$1.176E-031.420E-041.767E-052.215E-062.786E-073.009
RT1.9935.72213.29628.61459.671
1-step$q=5$
$\mathcal{C}_5^6$ & $\mathcal{G}_5^6$
$\mathcal{E}_Y$2.269E-011.177E-016.023E-023.061E-021.549E-020.969
$\mathcal{E}_Z$4.345E-022.196E-021.110E-025.608E-032.829E-030.985
RT23.19647.56699.318203.587411.991
2-step$q=5$
$\mathcal{C}_5^6$ & $\mathcal{G}_5^6$
$\mathcal{E}_Y$2.941E-028.713E-032.376E-036.223E-041.600E-041.885
$\mathcal{E}_Z$9.182E-032.216E-035.534E-041.393E-043.510E-052.005
RT33.61080.876177.315372.765766.094
3-step$q=5$
$\mathcal{C}_5^6$ & $\mathcal{G}_5^6$
$\mathcal{E}_Y$2.549E-034.157E-046.185E-058.531E-061.127E-062.789
$\mathcal{E}_Z$2.364E-032.687E-043.254E-054.031E-065.037E-073.045
RT36.049105.507245.986530.8531106.870
1-step$q=6$
$\mathcal{C}_6^7$ & $\mathcal{G}_6^7$
$\mathcal{E}_Y$4.068E-012.117E-011.084E-015.500E-022.779E-020.969
$\mathcal{E}_Z$5.899E-022.996E-021.516E-027.652E-033.855E-030.984
RT368.274792.7581640.7743346.5296758.066
2-step$q=6$
$\mathcal{C}_6^7$ & $\mathcal{G}_6^7$
$\mathcal{E}_Y$7.236E-022.130E-025.783E-031.510E-033.869E-041.891
$\mathcal{E}_Z$1.307E-023.277E-038.382E-042.134E-045.405E-051.978
RT1112.5242813.6416114.39012814.47326454.702
3-step$q=6$
$\mathcal{C}_6^7$ & $\mathcal{G}_6^7$
$\mathcal{E}_Y$1.037E-021.742E-032.528E-043.407E-054.433E-062.806
$\mathcal{E}_Z$3.924E-034.391E-045.339E-056.647E-068.328E-073.045
RT594.0601767.9384110.8768853.13118481.311
schemesparse gridN=8N=16N=32N=64N=128CR
1-step$q=3$
$\mathcal{C}_3^4$ & $\mathcal{G}_3^4$
$\mathcal{E}_Y$5.717E-022.999E-021.557E-028.021E-034.104E-030.950
$\mathcal{E}_Z$3.129E-021.575E-027.988E-034.057E-032.057E-030.981
RT0.1070.2260.3890.6271.268
2-step$q=3$
$\mathcal{C}_3^4$ & $\mathcal{G}_3^4$
$\mathcal{E}_Y$2.766E-037.861E-042.091E-045.396E-051.371E-051.918
$\mathcal{E}_Z$4.833E-031.197E-033.006E-047.588E-051.916E-051.994
RT0.1310.2350.5091.0672.187
3-step$q=3$
$\mathcal{C}_3^4$ & $\mathcal{G}_3^4$
$\mathcal{E}_Y$1.244E-041.183E-051.061E-069.522E-088.587E-093.460
$\mathcal{E}_Z$3.425E-044.375E-055.575E-067.067E-078.922E-082.976
RT0.1730.3700.7501.4783.072
1-step$q=4$
$\mathcal{C}_4^5$ & $\mathcal{G}_4^5$
$\mathcal{E}_Y$1.196E-016.208E-023.187E-021.626E-028.259E-030.965
$\mathcal{E}_Z$3.445E-021.735E-028.772E-034.439E-032.244E-030.985
RT1.6823.1536.17812.93426.182
2-step$q=4$
$\mathcal{C}_4^5$ & $\mathcal{G}_4^5$
$\mathcal{E}_Y$9.114E-032.817E-037.851E-042.087E-045.426E-051.854
$\mathcal{E}_Z$6.769E-031.628E-034.038E-041.012E-042.547E-052.012
RT2.5134.4429.60620.16041.504
3-step$q=4$
$\mathcal{C}_4^5$ & $\mathcal{G}_4^5$
$\mathcal{E}_Y$4.879E-047.611E-051.052E-051.378E-061.763E-072.865
$\mathcal{E}_Z$1.176E-031.420E-041.767E-052.215E-062.786E-073.009
RT1.9935.72213.29628.61459.671
1-step$q=5$
$\mathcal{C}_5^6$ & $\mathcal{G}_5^6$
$\mathcal{E}_Y$2.269E-011.177E-016.023E-023.061E-021.549E-020.969
$\mathcal{E}_Z$4.345E-022.196E-021.110E-025.608E-032.829E-030.985
RT23.19647.56699.318203.587411.991
2-step$q=5$
$\mathcal{C}_5^6$ & $\mathcal{G}_5^6$
$\mathcal{E}_Y$2.941E-028.713E-032.376E-036.223E-041.600E-041.885
$\mathcal{E}_Z$9.182E-032.216E-035.534E-041.393E-043.510E-052.005
RT33.61080.876177.315372.765766.094
3-step$q=5$
$\mathcal{C}_5^6$ & $\mathcal{G}_5^6$
$\mathcal{E}_Y$2.549E-034.157E-046.185E-058.531E-061.127E-062.789
$\mathcal{E}_Z$2.364E-032.687E-043.254E-054.031E-065.037E-073.045
RT36.049105.507245.986530.8531106.870
1-step$q=6$
$\mathcal{C}_6^7$ & $\mathcal{G}_6^7$
$\mathcal{E}_Y$4.068E-012.117E-011.084E-015.500E-022.779E-020.969
$\mathcal{E}_Z$5.899E-022.996E-021.516E-027.652E-033.855E-030.984
RT368.274792.7581640.7743346.5296758.066
2-step$q=6$
$\mathcal{C}_6^7$ & $\mathcal{G}_6^7$
$\mathcal{E}_Y$7.236E-022.130E-025.783E-031.510E-033.869E-041.891
$\mathcal{E}_Z$1.307E-023.277E-038.382E-042.134E-045.405E-051.978
RT1112.5242813.6416114.39012814.47326454.702
3-step$q=6$
$\mathcal{C}_6^7$ & $\mathcal{G}_6^7$
$\mathcal{E}_Y$1.037E-021.742E-032.528E-043.407E-054.433E-062.806
$\mathcal{E}_Z$3.924E-034.391E-045.339E-056.647E-068.328E-073.045
RT594.0601767.9384110.8768853.13118481.311
Table 5.  Errors and convergence rates for Example 4
schemesparse girdN=8N=16N=32N=64N=128CR
1-step$q=2$
$\mathcal{C}_2^3$ & $\mathcal{G}_2^3$
$\mathcal{E}_Y$4.246E-031.925E-039.181E-044.508E-042.243E-041.058
$\mathcal{E}_Z$1.149E-024.898E-032.098E-039.057E-043.948E-041.216
RT0.0300.0450.0830.1600.267
2-step$q=2$
$\mathcal{C}_2^3$ & $\mathcal{G}_2^3$
$\mathcal{E}_Y$4.093E-048.289E-051.646E-053.207E-066.101E-072.347
$\mathcal{E}_Z$5.429E-031.395E-033.568E-049.099E-052.315E-051.968
RT0.0320.0590.0900.1710.327
3-step$q=2$
$\mathcal{C}_2^3$ & $\mathcal{G}_2^3$
$\mathcal{E}_Y$2.459E-052.651E-062.695E-072.658E-082.538E-093.312
$\mathcal{E}_Z$5.266E-055.713E-066.099E-076.276E-086.215E-093.261
RT0.0330.0750.1320.2760.430
1-step$q=3$
$\mathcal{C}_3^4$ & $\mathcal{G}_3^4$
$\mathcal{E}_Y$2.463E-031.068E-034.812E-042.229E-041.072E-041.130
$\mathcal{E}_Z$3.243E-031.363E-035.990E-042.630E-041.154E-041.200
RT0.2950.4220.7621.4202.690
2-step$q=3$
$\mathcal{C}_3^4$ & $\mathcal{G}_3^4$
$\mathcal{E}_Y$2.044E-047.910E-051.676E-053.397E-066.738E-072.103
$\mathcal{E}_Z$1.073E-032.444E-046.285E-051.632E-054.227E-061.988
RT0.3520.7171.3852.6104.946
3-step$q=3$
$\mathcal{C}_3^4$ & $\mathcal{G}_3^4$
$\mathcal{E}_Y$8.103E-061.743E-061.890E-071.942E-081.933E-093.055
$\mathcal{E}_Z$7.540E-061.577E-061.795E-071.933E-081.995E-093.012
RT0.3641.2231.9363.7437.150
1-step$q=4$
$\mathcal{C}_4^5$ & $\mathcal{G}_4^5$
$\mathcal{E}_Y$1.663E-037.351E-043.068E-041.370E-046.294E-051.187
$\mathcal{E}_Z$1.360E-035.763E-042.374E-041.068E-044.838E-051.206
RT3.9168.05015.73829.98755.418
2-step$q=4$
$\mathcal{C}_4^5$ & $\mathcal{G}_4^5$
$\mathcal{E}_Y$6.776E-054.165E-051.569E-053.408E-066.925E-071.684
$\mathcal{E}_Z$3.665E-048.033E-051.727E-054.479E-061.191E-062.070
RT5.06413.27928.97456.345105.155
3-step$q=4$
$\mathcal{C}_4^5$ & $\mathcal{G}_4^5$
$\mathcal{E}_Y$4.932E-061.065E-061.111E-071.115E-081.087E-093.087
$\mathcal{E}_Z$2.474E-065.495E-076.062E-086.348E-096.390E-103.027
RT34.015117.047250.853483.481926.454
1-step$q=5$
$\mathcal{C}_5^6$ & $\mathcal{G}_5^6$
$\mathcal{E}_Y$1.167E-035.489E-042.341E-049.343E-054.184E-051.216
$\mathcal{E}_Z$6.751E-043.055E-041.268E-045.183E-052.408E-051.218
RT58.937134.345289.234570.5871049.247
2-step$q=5$
$\mathcal{C}_5^6$ & $\mathcal{G}_5^6$
$\mathcal{E}_Y$6.572E-054.173E-051.246E-052.548E-065.087E-071.806
$\mathcal{E}_Z$1.464E-042.885E-056.466E-061.763E-064.792E-072.054
RT736.4052004.1854124.2867776.64514561.700
3-step$q=5$
$\mathcal{C}_5^6$ & $\mathcal{G}_5^6$
$\mathcal{E}_Y$1.649E-066.334E-079.009E-089.209E-099.147E-102.774
$\mathcal{E}_Z$5.318E-072.072E-073.079E-083.274E-093.372E-102.723
RT709.5872599.7835916.66011366.63321380.402
schemesparse girdN=8N=16N=32N=64N=128CR
1-step$q=2$
$\mathcal{C}_2^3$ & $\mathcal{G}_2^3$
$\mathcal{E}_Y$4.246E-031.925E-039.181E-044.508E-042.243E-041.058
$\mathcal{E}_Z$1.149E-024.898E-032.098E-039.057E-043.948E-041.216
RT0.0300.0450.0830.1600.267
2-step$q=2$
$\mathcal{C}_2^3$ & $\mathcal{G}_2^3$
$\mathcal{E}_Y$4.093E-048.289E-051.646E-053.207E-066.101E-072.347
$\mathcal{E}_Z$5.429E-031.395E-033.568E-049.099E-052.315E-051.968
RT0.0320.0590.0900.1710.327
3-step$q=2$
$\mathcal{C}_2^3$ & $\mathcal{G}_2^3$
$\mathcal{E}_Y$2.459E-052.651E-062.695E-072.658E-082.538E-093.312
$\mathcal{E}_Z$5.266E-055.713E-066.099E-076.276E-086.215E-093.261
RT0.0330.0750.1320.2760.430
1-step$q=3$
$\mathcal{C}_3^4$ & $\mathcal{G}_3^4$
$\mathcal{E}_Y$2.463E-031.068E-034.812E-042.229E-041.072E-041.130
$\mathcal{E}_Z$3.243E-031.363E-035.990E-042.630E-041.154E-041.200
RT0.2950.4220.7621.4202.690
2-step$q=3$
$\mathcal{C}_3^4$ & $\mathcal{G}_3^4$
$\mathcal{E}_Y$2.044E-047.910E-051.676E-053.397E-066.738E-072.103
$\mathcal{E}_Z$1.073E-032.444E-046.285E-051.632E-054.227E-061.988
RT0.3520.7171.3852.6104.946
3-step$q=3$
$\mathcal{C}_3^4$ & $\mathcal{G}_3^4$
$\mathcal{E}_Y$8.103E-061.743E-061.890E-071.942E-081.933E-093.055
$\mathcal{E}_Z$7.540E-061.577E-061.795E-071.933E-081.995E-093.012
RT0.3641.2231.9363.7437.150
1-step$q=4$
$\mathcal{C}_4^5$ & $\mathcal{G}_4^5$
$\mathcal{E}_Y$1.663E-037.351E-043.068E-041.370E-046.294E-051.187
$\mathcal{E}_Z$1.360E-035.763E-042.374E-041.068E-044.838E-051.206
RT3.9168.05015.73829.98755.418
2-step$q=4$
$\mathcal{C}_4^5$ & $\mathcal{G}_4^5$
$\mathcal{E}_Y$6.776E-054.165E-051.569E-053.408E-066.925E-071.684
$\mathcal{E}_Z$3.665E-048.033E-051.727E-054.479E-061.191E-062.070
RT5.06413.27928.97456.345105.155
3-step$q=4$
$\mathcal{C}_4^5$ & $\mathcal{G}_4^5$
$\mathcal{E}_Y$4.932E-061.065E-061.111E-071.115E-081.087E-093.087
$\mathcal{E}_Z$2.474E-065.495E-076.062E-086.348E-096.390E-103.027
RT34.015117.047250.853483.481926.454
1-step$q=5$
$\mathcal{C}_5^6$ & $\mathcal{G}_5^6$
$\mathcal{E}_Y$1.167E-035.489E-042.341E-049.343E-054.184E-051.216
$\mathcal{E}_Z$6.751E-043.055E-041.268E-045.183E-052.408E-051.218
RT58.937134.345289.234570.5871049.247
2-step$q=5$
$\mathcal{C}_5^6$ & $\mathcal{G}_5^6$
$\mathcal{E}_Y$6.572E-054.173E-051.246E-052.548E-065.087E-071.806
$\mathcal{E}_Z$1.464E-042.885E-056.466E-061.763E-064.792E-072.054
RT736.4052004.1854124.2867776.64514561.700
3-step$q=5$
$\mathcal{C}_5^6$ & $\mathcal{G}_5^6$
$\mathcal{E}_Y$1.649E-066.334E-079.009E-089.209E-099.147E-102.774
$\mathcal{E}_Z$5.318E-072.072E-073.079E-083.274E-093.372E-102.723
RT709.5872599.7835916.66011366.63321380.402
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