Two meshless methods for Dirichlet boundary optimal control problem governed by elliptic PDEs
In this paper, finite point method (FPM) and meshless weighted least squares (MWLS) method are proposed for solving Dirichlet boundary optimal control problems governed by elliptic equations. The FPM scheme uses shape function constructed by moving least square (MLS) approximation to discretize the...
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| Published in | Computers & mathematics with applications (1987) Vol. 82; pp. 113 - 129 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Ltd
15.01.2021
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| Online Access | Get full text |
| ISSN | 0898-1221 1873-7668 |
| DOI | 10.1016/j.camwa.2020.10.026 |
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| Abstract | In this paper, finite point method (FPM) and meshless weighted least squares (MWLS) method are proposed for solving Dirichlet boundary optimal control problems governed by elliptic equations. The FPM scheme uses shape function constructed by moving least square (MLS) approximation to discretize the equations, while the MWLS scheme employs both MLS approximation and penalty terms to solve the same problem. Error estimates for the FPM scheme are presented and numerical results are provided to examine the impact of parameters and validate the efficiency of the proposed schemes. The extended model (Navier–Stokes equations) shows the ability of our algorithm to handle complex problems. Our explorative work shows the flexibility and great potential of the meshless methods in optimal control problems. |
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| AbstractList | In this paper, finite point method (FPM) and meshless weighted least squares (MWLS) method are proposed for solving Dirichlet boundary optimal control problems governed by elliptic equations. The FPM scheme uses shape function constructed by moving least square (MLS) approximation to discretize the equations, while the MWLS scheme employs both MLS approximation and penalty terms to solve the same problem. Error estimates for the FPM scheme are presented and numerical results are provided to examine the impact of parameters and validate the efficiency of the proposed schemes. The extended model (Navier–Stokes equations) shows the ability of our algorithm to handle complex problems. Our explorative work shows the flexibility and great potential of the meshless methods in optimal control problems. |
| Author | Cheng, Ai-Jie Liu, Yang |
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| Cites_doi | 10.1016/j.jprocont.2017.11.009 10.1155/2016/3824835 10.1016/S0045-7949(01)00067-0 10.1016/j.jmaa.2019.04.031 10.1016/S0045-7825(96)01132-2 10.4208/aamm.OA-2018-0186 10.1016/j.cam.2019.112431 10.1090/S0025-5718-1981-0616367-1 10.1016/j.apnum.2007.04.003 10.1016/j.aml.2019.06.025 10.1016/j.camwa.2018.05.031 10.1137/100795632 10.1016/j.enganabound.2019.08.008 10.1016/j.aml.2019.05.021 10.1007/s00211-018-01021-7 10.1016/j.compfluid.2018.03.039 10.1016/j.mbs.2019.06.001 10.1016/j.jcp.2018.03.044 |
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| References | Kamranian, Dehghan, Tatari (b12) 2017; 30 Casanova, Gout, Zavaleta (b15) 2019; 108 Liu, Wang (b4) 2019; 314 Kumar, Ruiz-Baier, Sandilya (b10) 2018; 76 Oñate, Perazzo, Miquel (b22) 2001; 79 Zhang (b11) 2019; 97 Liu, Li, Belytschko (b19) 1997; 143 Golub, Van Loan (b21) 2012 Nita, Vandewalle, Meyers (b3) 2018; 366 Lapin, Zhang, Lapin (b5) 2019; 361 Lancaster, Salkauskas (b16) 1981; 37 Guan, Wang, Zhu (b17) 2019; 98 Lapin, Zhang, Lapin, Yan (b6) 2019; 11 Cheng, Cheng (b18) 2008; 58 Shakya, Sinha (b8) 2020; 367 Gong, Yan (b23) 2011; 49 Xie, Xie, Yang, Gui, Wang (b1) 2018; 61 Yao, Li, Zhang, Zhang (b13) 2016; 2016 Yang, Wang, Xie (b9) 2018 Yu, Huang, Liu (b7) 2019; 477 Künemund, Narcowich, Ward, Wendland (b14) 2019; 142 Tamellini, Parolini, Verani (b2) 2018; 172 Brenner, Scott (b20) 2007 Cheng (10.1016/j.camwa.2020.10.026_b18) 2008; 58 Yu (10.1016/j.camwa.2020.10.026_b7) 2019; 477 Guan (10.1016/j.camwa.2020.10.026_b17) 2019; 98 Liu (10.1016/j.camwa.2020.10.026_b4) 2019; 314 Brenner (10.1016/j.camwa.2020.10.026_b20) 2007 Yang (10.1016/j.camwa.2020.10.026_b9) 2018 Kumar (10.1016/j.camwa.2020.10.026_b10) 2018; 76 Casanova (10.1016/j.camwa.2020.10.026_b15) 2019; 108 Künemund (10.1016/j.camwa.2020.10.026_b14) 2019; 142 Nita (10.1016/j.camwa.2020.10.026_b3) 2018; 366 Liu (10.1016/j.camwa.2020.10.026_b19) 1997; 143 Xie (10.1016/j.camwa.2020.10.026_b1) 2018; 61 Tamellini (10.1016/j.camwa.2020.10.026_b2) 2018; 172 Yao (10.1016/j.camwa.2020.10.026_b13) 2016; 2016 Lancaster (10.1016/j.camwa.2020.10.026_b16) 1981; 37 Oñate (10.1016/j.camwa.2020.10.026_b22) 2001; 79 Zhang (10.1016/j.camwa.2020.10.026_b11) 2019; 97 Kamranian (10.1016/j.camwa.2020.10.026_b12) 2017; 30 Lapin (10.1016/j.camwa.2020.10.026_b5) 2019; 361 Shakya (10.1016/j.camwa.2020.10.026_b8) 2020; 367 Golub (10.1016/j.camwa.2020.10.026_b21) 2012 Gong (10.1016/j.camwa.2020.10.026_b23) 2011; 49 Lapin (10.1016/j.camwa.2020.10.026_b6) 2019; 11 |
| References_xml | – volume: 79 start-page: 2151 year: 2001 end-page: 2163 ident: b22 article-title: A finite point method for elasticity problems publication-title: Comput. Struct. – volume: 37 start-page: 141 year: 1981 end-page: 158 ident: b16 article-title: Surfaces generated by moving least squares methods publication-title: Math. Comput. – volume: 11 start-page: 1358 year: 2019 end-page: 1375 ident: b6 article-title: Analysis of finite difference approximations of an optimal control problem in economics publication-title: Adv. Appl. Math. Mech. – volume: 367 year: 2020 ident: b8 article-title: Finite element method for parabolic optimal control problems with a bilinear state equation publication-title: J. Comput. Appl. Math. – volume: 361 start-page: 715 year: 2019 end-page: 729 ident: b5 article-title: Numerical solution of a parabolic optimal control problem arising in economics and management publication-title: Appl. Math. Comput. – volume: 142 start-page: 383 year: 2019 end-page: 419 ident: b14 article-title: A high-order meshless Galerkin method for semilinear parabolic equations on spheres publication-title: Numer. Math. – volume: 30 year: 2017 ident: b12 article-title: Study of the two-dimensional sine-Gordon equation arising in josephson junctions using meshless finite point method publication-title: Int. J. Numer. Modelling, Electron. Netw. Devices Fields – volume: 366 start-page: 14 year: 2018 end-page: 32 ident: b3 article-title: Multigrid optimization for DNS-based optimal control in turbulent channel flows publication-title: J. Comput. Phys. – volume: 143 start-page: 113 year: 1997 end-page: 154 ident: b19 article-title: Moving least-square reproducing kernel methods (I) methodology and convergence publication-title: Comput. Methods Appl. Mech. Engrg. – volume: 108 start-page: 201 year: 2019 end-page: 209 ident: b15 article-title: Radial basis function methods for optimal control of the convection–diffusion equation: A numerical study publication-title: Eng. Anal. Bound. Elem. – volume: 61 start-page: 47 year: 2018 end-page: 57 ident: b1 article-title: Distributed parameter modeling and optimal control of the oxidation rate in the iron removal process publication-title: J. Process Control – volume: 2016 year: 2016 ident: b13 article-title: Prediction of chloride diffusion in concrete structure using meshless methods publication-title: Adv. Mater. Sci. Eng. – volume: 97 start-page: 93 year: 2019 end-page: 98 ident: b11 article-title: An accurate and stable RBF method for solving partial differential equations publication-title: Appl. Math. Lett. – volume: 76 start-page: 923 year: 2018 end-page: 937 ident: b10 article-title: Mixed and discontinuous finite volume element schemes for the optimal control of immiscible flow in porous media publication-title: Comput. Math. Appl. – year: 2007 ident: b20 article-title: The Mathematical Theory of Finite Element Methods, Vol. 15 – volume: 477 start-page: 250 year: 2019 end-page: 271 ident: b7 article-title: Finite element approximations of impulsive optimal control problems for heat equations publication-title: J. Math. Anal. Appl. – volume: 49 start-page: 984 year: 2011 end-page: 1014 ident: b23 article-title: Mixed finite element method for Dirichlet boundary control problem governed by elliptic PDEs publication-title: SIAM J. Control Optim. – volume: 314 start-page: 28 year: 2019 end-page: 42 ident: b4 article-title: Numerical optimal control of a size-structured PDE model for metastatic cancer treatment publication-title: Math. Biosci. – year: 2018 ident: b9 article-title: An interface-unfitted finite element method for elliptic interface optimal control problem – volume: 98 start-page: 438 year: 2019 end-page: 445 ident: b17 article-title: Meshless methods for solving Dirichlet boundary optimal control problems governed by elliptic PDEs publication-title: Appl. Math. Lett. – year: 2012 ident: b21 article-title: Matrix Computations, Vol. 3 – volume: 172 start-page: 538 year: 2018 end-page: 548 ident: b2 article-title: An optimal control problem for two-phase compressible–incompressible flows publication-title: Comput. & Fluids – volume: 58 start-page: 884 year: 2008 end-page: 898 ident: b18 article-title: Error estimates for the finite point method publication-title: Appl. Numer. Math. – volume: 361 start-page: 715 year: 2019 ident: 10.1016/j.camwa.2020.10.026_b5 article-title: Numerical solution of a parabolic optimal control problem arising in economics and management publication-title: Appl. Math. Comput. – volume: 61 start-page: 47 year: 2018 ident: 10.1016/j.camwa.2020.10.026_b1 article-title: Distributed parameter modeling and optimal control of the oxidation rate in the iron removal process publication-title: J. Process Control doi: 10.1016/j.jprocont.2017.11.009 – volume: 2016 year: 2016 ident: 10.1016/j.camwa.2020.10.026_b13 article-title: Prediction of chloride diffusion in concrete structure using meshless methods publication-title: Adv. Mater. Sci. Eng. doi: 10.1155/2016/3824835 – volume: 79 start-page: 2151 issue: 22–25 year: 2001 ident: 10.1016/j.camwa.2020.10.026_b22 article-title: A finite point method for elasticity problems publication-title: Comput. Struct. doi: 10.1016/S0045-7949(01)00067-0 – volume: 477 start-page: 250 issue: 1 year: 2019 ident: 10.1016/j.camwa.2020.10.026_b7 article-title: Finite element approximations of impulsive optimal control problems for heat equations publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2019.04.031 – volume: 143 start-page: 113 issue: 1 year: 1997 ident: 10.1016/j.camwa.2020.10.026_b19 article-title: Moving least-square reproducing kernel methods (I) methodology and convergence publication-title: Comput. Methods Appl. Mech. Engrg. doi: 10.1016/S0045-7825(96)01132-2 – year: 2007 ident: 10.1016/j.camwa.2020.10.026_b20 – year: 2012 ident: 10.1016/j.camwa.2020.10.026_b21 – volume: 11 start-page: 1358 issue: 6 year: 2019 ident: 10.1016/j.camwa.2020.10.026_b6 article-title: Analysis of finite difference approximations of an optimal control problem in economics publication-title: Adv. Appl. Math. Mech. doi: 10.4208/aamm.OA-2018-0186 – volume: 367 year: 2020 ident: 10.1016/j.camwa.2020.10.026_b8 article-title: Finite element method for parabolic optimal control problems with a bilinear state equation publication-title: J. Comput. Appl. Math. doi: 10.1016/j.cam.2019.112431 – volume: 37 start-page: 141 issue: 155 year: 1981 ident: 10.1016/j.camwa.2020.10.026_b16 article-title: Surfaces generated by moving least squares methods publication-title: Math. Comput. doi: 10.1090/S0025-5718-1981-0616367-1 – volume: 58 start-page: 884 issue: 6 year: 2008 ident: 10.1016/j.camwa.2020.10.026_b18 article-title: Error estimates for the finite point method publication-title: Appl. Numer. Math. doi: 10.1016/j.apnum.2007.04.003 – volume: 30 issue: 6 year: 2017 ident: 10.1016/j.camwa.2020.10.026_b12 article-title: Study of the two-dimensional sine-Gordon equation arising in josephson junctions using meshless finite point method publication-title: Int. J. Numer. Modelling, Electron. Netw. Devices Fields – volume: 98 start-page: 438 year: 2019 ident: 10.1016/j.camwa.2020.10.026_b17 article-title: Meshless methods for solving Dirichlet boundary optimal control problems governed by elliptic PDEs publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2019.06.025 – volume: 76 start-page: 923 issue: 4 year: 2018 ident: 10.1016/j.camwa.2020.10.026_b10 article-title: Mixed and discontinuous finite volume element schemes for the optimal control of immiscible flow in porous media publication-title: Comput. Math. Appl. doi: 10.1016/j.camwa.2018.05.031 – volume: 49 start-page: 984 issue: 3 year: 2011 ident: 10.1016/j.camwa.2020.10.026_b23 article-title: Mixed finite element method for Dirichlet boundary control problem governed by elliptic PDEs publication-title: SIAM J. Control Optim. doi: 10.1137/100795632 – volume: 108 start-page: 201 year: 2019 ident: 10.1016/j.camwa.2020.10.026_b15 article-title: Radial basis function methods for optimal control of the convection–diffusion equation: A numerical study publication-title: Eng. Anal. Bound. Elem. doi: 10.1016/j.enganabound.2019.08.008 – volume: 97 start-page: 93 year: 2019 ident: 10.1016/j.camwa.2020.10.026_b11 article-title: An accurate and stable RBF method for solving partial differential equations publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2019.05.021 – volume: 142 start-page: 383 issue: 2 year: 2019 ident: 10.1016/j.camwa.2020.10.026_b14 article-title: A high-order meshless Galerkin method for semilinear parabolic equations on spheres publication-title: Numer. Math. doi: 10.1007/s00211-018-01021-7 – volume: 172 start-page: 538 year: 2018 ident: 10.1016/j.camwa.2020.10.026_b2 article-title: An optimal control problem for two-phase compressible–incompressible flows publication-title: Comput. & Fluids doi: 10.1016/j.compfluid.2018.03.039 – volume: 314 start-page: 28 year: 2019 ident: 10.1016/j.camwa.2020.10.026_b4 article-title: Numerical optimal control of a size-structured PDE model for metastatic cancer treatment publication-title: Math. Biosci. doi: 10.1016/j.mbs.2019.06.001 – year: 2018 ident: 10.1016/j.camwa.2020.10.026_b9 – volume: 366 start-page: 14 year: 2018 ident: 10.1016/j.camwa.2020.10.026_b3 article-title: Multigrid optimization for DNS-based optimal control in turbulent channel flows publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2018.03.044 |
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| SubjectTerms | Dirichlet boundary condition Elliptic equations Finite point method Meshless weighted least squares method Optimal control |
| Title | Two meshless methods for Dirichlet boundary optimal control problem governed by elliptic PDEs |
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