Control and Optimization in Applied Mathematics

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A Meshless Method for Optimal Control of Parabolic PDEs Using Rational Radial Basis Functions

Articles in Press

Document Type : Research Article

Authors

1 Department of Applied Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎University of Qom‎, ‎Qom‎, ‎Iran.

2 Department of Mathematics‎, ‎Behbahan Khatam Alanbia University of Technology‎, ‎Khouzestan‎, ‎Iran.

10.30473/coam.2025.73126.1277

Abstract

This study introduces an innovative approach for addressing optimal control problems related to parabolic partial differential equations (PDEs) through the application of rational radial basis functions (RBFs). Parabolic PDEs, which are instrumental in modeling time-dependent processes such as heat transfer and diffusion, pose significant computational challenges in optimal control due to the requirement for precise approximations of both state and adjoint equations. The proposed approach exploits the adaptability and spectral accuracy of rational RBFs within a meshless framework, effectively addressing the limitations of traditional discretization methods. By enhancing the accuracy and efficiency of control strategies, this method significantly contributes to advancing the theory and application of optimal control in dynamic systems. The tunable shape parameters of rational RBFs allow for accurate representation of solution characteristics, including steep gradients and localized behaviors. Additionally, their meshless framework adeptly accommodates complex geometries and boundary conditions, ensuring computational efficiency through the generation of sparse and well-conditioned system matrices. This paper also introduces a novel hybrid rational RBF, termed the Gaussian rational hybrid RBF. The efficacy of the proposed approach is validated through a series of benchmark tests and practical applications, highlighting its ability to achieve high accuracy with reduced computational effort. The findings illustrate the potential of rational RBFs as a robust and versatile tool for solving optimal control problems governed by parabolic PDEs, paving the way for further exploration of advanced rational RBF-based techniques in the field of computational optimal control.

Highlights

  • Introduced a novel hybrid rational Radial Basis Function (Gaussian rational hybrid RBF) specifically designed to address the challenges of optimal control problems governed by parabolic PDEs, leveraging the spectral accuracy and adaptability of rational RBFs.
  • Developed a fully meshless computational framework that exploits the spectral accuracy of rational RBFs, effectively capturing steep gradients, localized behaviors, and complex solution features inherent in dynamic parabolic systems.
  • Demonstrated the method’s high accuracy and computational efficiency through extensive benchmark tests and practical applications, highlighting its ability to produce solutions with small ( L2 ) and ( L_infty ) errors while reducing computational effort compared to traditional discretization techniques.
  • Showcased the flexibility of the hybrid RBF approach in handling intricate geometries and boundary conditions within a sparse, well-conditioned system matrix, emphasizing its robustness for complex optimal control scenarios.
  • Highlighted promising directions for future research by exploring advanced RBF-based techniques, including adaptive methods and extensions to nonlinear and multi-physics PDEs, to further enhance the scalability, precision, and applicability of meshless spectral approaches in computational optimal control.

Keywords

Main Subjects

References
[1] Buhmann, M.D., De Marchi, S., Perracchione, E. (2020). “Analysis of a new class of rational RBF expansions”, IMA Journal of Numerical Analysis, 40(3), 1972-1993, doi:10.1093/imanum/drz015.
[2] Darehmiraki, M., Rakhshan, S.A. (2023). “Radial basis functions for the zero sum differential game with fractional derivatives”, International Journal of Applied and Computational Mathematics, 9(5), 90, doi:10.1007/s40819-023-01587-3.
[3] Darehmiraki, M., Rezazadeh, A., Ahmadian, A., Salahshour, S. (2022). “An interpolation method for the optimal control problem governed by the elliptic convection–diffusion equation”, Numerical Methods for Partial Differential Equations, 38(2), 137-159, doi:10.1002/num.22625.
[4] Dehghan, M., Abbaszadeh, M., Mohebbi, A. (2015). “A meshless technique based on the local radial basis functions collocation method for solving parabolic–parabolic Patlak–Keller–Segel chemotaxis model”, Engineering Analysis with Boundary Elements, 56, 129-144, doi:10.1016/j.enganabound.2015.02.005.
[5] Fabien, M.S. (2014). “A radial basis function (RBF) method for the fully nonlinear 1D Serre GreenNaghdi equations”, arXiv preprint arXiv:1407.4300, doi:10.48550/arXiv.1407.4300.
[6] Farazandeh, E., Mirzaei, D. (2021). “A rational RBF interpolation with conditionally positive definite kernels”,  Advances in Computational Mathematics, 47, 1-21, doi:10.1007/s10444-021-09900-8.
[7] Fasshauer, G.E., Zhang, J.G. (2007). “On choosing optimal shape parameters for RBF approximation”, Numerical Algorithms, 45, 345-368, doi:10.1007/s11075-007-9072-8.
[8] Fuhrer, T., Karkulik, M. (2022). “Space-time finite element methods for parabolic distributed optimal control problems”, arXiv preprint arXiv:2208.09879, doi:10.48550/arXiv.2208.09879.
[9] Glusa, C., Otarola, E. (2021). “Error estimates for the optimal control of a parabolic fractional PDE”, SIAM Journal on Numerical Analysis, 59(2), 1140-1165, doi:10.1137/19M1267581.

[10] Golbabai, A., Safaei, N., Molavi-Arabshahi, M. (2022). “Numerical solution of optimal control problem for economic growth model using RBF collocation method”, Computational Methods for Differential Equations, 10(2), 327-337, doi:10.22034/cmde.2021.40223.1757.
[11] Gotschel, Minion, M.L. (2019). “An efficient parallel-in-time method for optimization with parabolic PDEs”, SIAM Journal on Scientific Computing, 41(6), C603-C626, doi:10.1137/19M1239313.
[12] Grubisic, L., Lazar, M., Nakic, I., Tautenhahn, M. (2023). “Optimal control of parabolic equations–a spectral calculus based approach”, SIAM Journal on Control and Optimization, 61(5), 2802-2825, doi:10.1137/21M1449 762..
[13] Guth, P.A., Kaarnioja, V., Kuo, F.Y., Schillings, C., Sloan, I.H. (2024). “Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration”, Numerische Mathematik, 156(2), 565-608, doi:10.1007/s00211-024-01397-9.
[14] Hashemi Borzabadi, A., Gholami Baladezaei, M., Ghachpazan, M. (2024). “A sub-ordinary approach to achieve near-exact solutions for a class of optimal control problems”, Control and Optimization in Applied Mathematics, 9(2), 1-19, doi:10.30473/coam.2024.70834.1254.
[15] Heidari, M., Ghovatmand, M., Skandari, M.N., Baleanu, D. (2023). “Numerical solution of reaction–diffusion equations with convergence analysis”, Journal of Nonlinear Mathematical Physics, 30(2), 384-399, doi:10.1007/s44198-022-00086-1.
[16] Herzog, R., Kunisch, K. (2010). “Algorithms for PDE constrained optimization”, GAMM Mitteilungen, 33(2), doi:10.1002/gamm.201010013.
[17] Huang, Y., Mohammadi Zadeh, F., Hadi Noori Skandari, M., Ahsani Tehrani, H., Tohidi, E. (2021). “Space–time Chebyshev spectral collocation method for nonlinear timefractional Burgers equations based on efficient basis functions”, Mathematical Methods in the Applied Sciences, 44(5), 4117-4136, doi:10.1002/mma.7015.
[18] Langer, U., Schafelner, A. (2022). “Adaptive space–time finite element methods for parabolic optimal control problems”, Journal of Numerical Mathematics, 30(4), 247-266, doi:10.1515/jnma-2021-0059.
[19] Langer, U., Steinbach, O., Yang, H. (2024). “Robust space-time finite element methods for parabolic distributed optimal control problems with energy regularization”, Advances in Computational Mathematics, 50(2), 1-30, doi:10.1007/s10444-024-10123-w.
[20] Loscher, R., Reichelt, M., Steinbach, O. (2024). “Efficient solution of state-constrained distributed parabolic optimal control problems”, arXiv preprint arXiv:2410.06021, doi:10.48550/arXiv2410.06021.
[21] Luo, D., O’Leary-Roseberry, T., Chen, P., Ghattas, O. (2023). “Efficient PDE-constrained optimization under high-dimensional uncertainty using derivative-informed neural operators”, arXiv preprint arXiv:2305.20053, doi:10.48550/arXiv.2305.20053.
[22] Mahmoudi, M., Shojaeizadeh, T., Darehmiraki, M. (2021). “Optimal control of time-fractional convection–diffusion–reaction problem employing compact integrated RBF method”, Mathematical Sciences, 1-14, doi:10.1007/s40096-021-00434-0.
[23] Mesrizadeh, M., Shanazari, K. (2021). “Meshless Galerkin method based on RBFs and reproducing Kernel for quasi-linear parabolic equations with Dirichlet boundary conditions”, Mathematical Modelling and Analysis, 26(2), 318-336, doi:10.3846/mma.2021.11436.
[24] Micchelli, C.A. (1986). “Interpolation of scattered data: Distance matrices and conditionally positive definite functions”, Constructive Approximation, 2(1), 11-22, doi:10.1007/978-94-009-6466-2-7.
[25] Mohammadizadeh, F., Tehrani, H. A., Noori Skandari, M.H. (2019). “Chebyshev pseudo-spectral method for optimal control problem of Burgers’ equation”, Iranian Journal of Numerical Analysis and Optimization, 9(2), 77-102, doi:10.22067/ijnao.v9i2.72109.
[26] Mohammadizadeh, F., Tehrani, H.A., Georgiev, S.G., Noori Skandari, M.H. (2020). “An optimal control problem associated to a class of fractional Burgers’ equations”, Asian-European Journal of Mathematics, 13(04), 2050079, doi:10.1142/S1793557120500795.
[27] Mojarrad, F.N., Veiga, M.H., Hesthaven, J.S., Offner, P. (2023). “A new variable shape parameter strategy for RBF approximation using neural networks”, Computers and Mathematics with Applications, 143, 151-168, doi:10.1016/j.camwa.2023.05.005.
[28] Park, J., Sandberg, I.W. (1991). “Universal approximation using radial-basis-function networks”, Neural Computation, 3(2), 246-257, doi:10.1162/neco.1991.3.2.246.
[29] Pearson, J.W., Wathen, A.J. (2013). “Fast iterative solvers for convection-diffusion control problems”, Electronic Transactions on Numerical Analysis, 40, 294-311.
[30] Peykrayegan, N., Ghovatmand, M., Noori Skandari, M.H., Shateyi, S. (2024). “Numerical solution of nonlinear fractional delay integro-differential equations with convergence analysis”, Indian Journal of Pure and Applied Mathematics, 1-21, doi:10.1007/s13226-024-00620-5.
[31] Rezazadeh, A., Darehmiraki, M. (2024). “A fast Galerkin-spectral method based on discrete Legendre polynomials for solving parabolic differential equation”, Computational and Applied Mathematics, 43(6), 315, doi:10.1007/s40314-024-02792-6.
[32] Rezazadeh, A., Mahmoudi, M., Darehmiraki, M. (2020). “A solution for fractional PDE constrained optimization problems using reduced basis method”, Computational and Applied Mathematics, 39(2), 1-17, doi:10.1007/s40314-020-1092-1.
[33] Shiralizadeh, M., Alipanah, A., Mohammadi, M. (2023). “Approximate solutions to the Allen–Cahn equation using rational radial basis functions method”, Iranian Journal of Numerical Analysis and Optimization, 13(2), 187-204, doi:10.22067/ijnao.2022.76010.1123.
[34] Shiralizadeh, M., Alipanah, A., Mohammadi, M. (2023). “A numerical method for KdV equation using rational radial basis functions”, Computational Methods for Differential Equations, 11(2), 303-318, doi:10.22034/cmde.2022.51967.2171.
[35] Shojaeizadeh, T., Mahmoudi, M., Darehmiraki, M. (2021). “Optimal control problem of advection diffusion-reaction equation of kind fractal-fractional applying shifted Jacobi polynomials”, Chaos, Solitons and Fractals, 143, 110568, doi:10.1016/j.chaos.2020.110568.
[36] Song, H., Zhang, J., Hao, Y. (2023). “A splitting algorithm for constrained optimization problems with parabolic equations”, Computational and Applied Mathematics, 42(205), doi:10.1007/s40314-023-02343-5.
[37] Su, L. (2019). “A radial basis function (RBF)-finite difference (FD) method for the backward heat conduction problem”, Applied Mathematics and Computation, 354, 232-247, doi:10.1016/j.amc.2019.02.035.
[38] Zeng, Y., Duan, Y., Liu, B.S. (2021). “Solving 2D parabolic equations by using time parareal coupling with meshless collocation RBFs methods”, Engineering Analysis with Boundary Elements, 127, 102-112, doi:10.1016/j.enganabound.2021.03.008.
Control and Optimization in Applied Mathematics

Articles in Press, Accepted Manuscript
Available Online from 22 May 2025
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