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Mohsin Mohammed Ali, Maha. (1405). A Localized RBF-Partition of Unity Collocation Method for Elliptic PDE-Constrained Optimal Control. فصلنامه علمی زیست فناوری گیاهان زراعی, (), -. doi: 10.30473/coam.2026.78551.1436
Maha Mohsin Mohammed Ali. "A Localized RBF-Partition of Unity Collocation Method for Elliptic PDE-Constrained Optimal Control". فصلنامه علمی زیست فناوری گیاهان زراعی, , , 1405, -. doi: 10.30473/coam.2026.78551.1436
Mohsin Mohammed Ali, Maha. (1405). 'A Localized RBF-Partition of Unity Collocation Method for Elliptic PDE-Constrained Optimal Control', فصلنامه علمی زیست فناوری گیاهان زراعی, (), pp. -. doi: 10.30473/coam.2026.78551.1436
Mohsin Mohammed Ali, Maha. A Localized RBF-Partition of Unity Collocation Method for Elliptic PDE-Constrained Optimal Control. فصلنامه علمی زیست فناوری گیاهان زراعی, 1405; (): -. doi: 10.30473/coam.2026.78551.1436

A Localized RBF-Partition of Unity Collocation Method for Elliptic PDE-Constrained Optimal Control

Control and Optimization in Applied Mathematics
مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 01 شهریور 1405    اصل مقاله (1.04 M)
نوع مقاله: Applied Article
شناسه دیجیتال (DOI): 10.30473/coam.2026.78551.1436
نویسنده
Maha Mohsin Mohammed Ali*
Department of Information and Communication Technology, Polytechnic College of Engineering Specializations, Middle Technical University, Baghdad, Iraq
چکیده
A core challenge in numerically solving partial differential equations (PDEs) is achieving accuracy without sacrificing computational tractability. Globally supported methods provide high-order precision but at a computational cost that grows prohibitively with problem size, while locally supported schemes scale more efficiently, though often with a reduction in solution accuracy. The proposed localized meshless framework resolves this tension through the integration of Radial Basis Function (RBF) approximation with the Partition of Unity (PU) decomposition, with applications to elliptic, convection-diffusion, and Helmholtz boundary value problems and their corresponding PDE-constrained optimal control problems. The domain is partitioned into overlapping subdomains, each accommodating an independent RBF expansion that is subsequently blended by compactly supported PU weight functions. This localization strategy replaces the single ill-conditioned dense system of global RBF collocation with a collection of smaller, well-conditioned local systems, reducing the theoretical computational complexity from O(N3) to O(N log N). Three canonical benchmark problems validate the framework: Poisson’s equation, a singularly perturbed convection-diffusion equation, and the Helmholtz equation on a circular domain, each embedded in a distributed optimal control formulation. The proposed scheme achieves L2-norm errors of order 10-4 at N = 1,000 nodes, outperforming both standard finite difference methods and global RBF collocation by factors of 28 to 49 in accuracy. Optimal control problems are handled through a Lagrange multiplier formulation yielding coupled state and adjoint systems, both discretized within the same RBF-PU architecture. The framework is illustrated through a detailed engineering case study — optimal heat-source regulation in a thermally loaded plate — in which the complete optimality system is derived, physically interpreted, and numerically solved. Numerical stability, current limitations, and directions for future investigation are discussed.

تازه های تحقیق
  • RBF-Partition of Unity collocation framework is developed for elliptic PDE-constrained optimal control problems.
  • L₂-norm errors of order 10⁻⁴ were achieved at N = 1,000 nodes across three benchmark PDE problems.
  • Accuracy surpasses unstabilized FDM by 28–49× and global RBF collocation by 3.5–7.1×.
  • Computational complexity is reduced from O(N³) to O(N log N) via localized partition-of-unity decomposition.
  • Local patch condition numbers remain below 10³, versus 10¹⁰–10¹² for global RBF at the same N.
  • Lagrange multiplier formulation yields coupled state–adjoint systems solved within a unified meshless architecture.
  • Fixed-point iteration converges in 14 steps for the thermal plate case study with γ = 10⁻³.
  • Optimal heat-source regulation in a thermally loaded plate is demonstrated with full physical interpretation.
کلیدواژه‌ها
Radial basis functions؛ Partition of unity؛ Meshless methods؛ PDE-constrained optimization؛ Optimal control؛ Convection-diffusion equations؛ Lagrange multiplier method
مراجع

[1] Babuška, I., Melenk, J.M. (1997). “The partition of unity method”. International Journal for Numerical Methods in Engineering, 40(4), 727–758. https://doi.org/10.1002/ (SICI)1097-0207(19970228)40:4%3C727::AID-NME86%3E3.0.CO;2-N

[2] Beatson, R.K., Powell, M.J.D. (1992). “Univariate multiquadric approximation: Quasiinterpolation to scattered data”. Constructive Approximation, 8(3), 275–288. https:// doi.org/10.1007/BF01279020

[3] Biranvand, N., Ebrahimijahan, A. (2024). “Utilizing differential quadrature-based RBF partition of unity collocation method to simulate distributed-order time fractional Cable equation”. Computational and Applied Mathematics, 43, 52. https://doi.org/10. 1007/s40314-023-02507-3

[4] Buhmann, M.D. (2003). Radial Basis Functions: Theory and Implementations. Cambridge University Press. https://doi.org/10.1017/CBO9780511543241

[5] Cavoretto, R., De Rossi, A. (2014). “A meshless interpolation algorithm using a cell-based searching procedure”. Computers and Mathematics with Applications, 67(5), 1024–1038. https://doi.org/10.1016/j.camwa.2014.01.007

[6] Cavoretto, R., De Marchi, S., De Rossi, A., Perracchione, E., Santin, G. (2017). “Partition of unity interpolation using stable kernel-based techniques”. Applied Numerical Mathematics, 116, 95–107. https://doi.org/10.1016/j.apnum.2016.07.005

[7] Cavoretto, R., De Rossi, A. (2026). “Radial basis function partition of unity methods for scattered data interpolation: A review”. Mathematics and Computers in Simulation, 250, 376–394. https://doi.org/10.1016/j.matcom.2026.07.006

[8] Fardi, M., Azarnavid, B. (2024). “A study on the numerical solution of the Sobolev equation with a Burgers-type nonlinearity on two-dimensional irregular domains using the local RBF partition of unity method”. Computational and Applied Mathematics, 44(17). https://doi.org/10.1007/s40314-024-02968-0

[9] Fasshauer, G.E. (2007). Meshfree Approximation Methods with MATLAB. World Scientific, Interdisciplinary Mathematical Sciences, Vol. 6. https://doi.org/10.1142/ 6437

[10] Fasshauer, G.E. (2015). Kernel-Based Approximation Methods Using MATLAB. World Scientific, Interdisciplinary Mathematical Sciences, Vol. 9. https://doi.org/10.1142/9335

[11] Flyer, N., Fornberg, B., Bayona, V., Barnett, G.A. (2016). “On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy”. Journal of Computational Physics, 321, 21–38. https://doi.org/10.1016/j.jcp.2016.05.026

[12] Fornberg, B., Flyer, N. (2015). A Primer on Radial Basis Functions with Applications to the Geosciences. SIAM. https://doi.org/10.1137/1.9781611974041

[13] Gonzalez-Casanova, P., Gout, C., Zavaleta, J. (2019). “Radial basis function methods for optimal control of the convection-diffusion equation”. Engineering Analysis with Boundary Elements, 108, 201–209. https://doi.org/10.1016/j.enganabound.2019.08. 008

[14] Griebel, M., Schweitzer, M.A. (2001). “A particle-partition of unity method for the solution of elliptic, parabolic, and hyperbolic PDEs”. SIAM Journal on Scientific Computing, 22(3), 853–890. https://doi.org/10.1137/S1064827599355840

[15] Guan, H., Wang, Y., Zhu, H. (2019). “Meshless methods for solving Dirichlet boundary optimal control problems governed by elliptic PDEs”. Applied Mathematics Letters, 98, 438–445. https://doi.org/10.1016/j.aml.2019.06.025

[16] Kansa, E.J. (1990). “Multiquadrics — A scattered data approximation scheme with applications to computational fluid-dynamics — II: Solutions to parabolic, hyperbolic and elliptic partial differential equations”. Computers and Mathematics with Applications, 19(8–9), 147–161. https://doi.org/10.1016/0898-1221(90)90271-K

[17] Lions, J.L. (1971). Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin, Heidelberg.

[18] Ma, Z., Imin, R. (2025). “A high-precision meshless method for time-fractional mixed diffusion and wave equations”. International Journal for Numerical Methods in Engineering, 126, e70020. https://doi.org/10.1002/nme.70020

[19] Mahmoudi, M., Shojaeizadeh, T., Darehmiraki, M. (2023). “Optimal control of time-fractional convection–diffusion–reaction problem employing compact integrated RBF method”. Mathematical Sciences, 17(1). https://doi.org/10.1007/ s40096-021-00434-0

[20] Mirzaei, D. (2021). “The direct radial basis function partition of unity (D-RBF-PU) method for solving PDEs”. SIAM Journal on Scientific Computing, 43(1), A54–A83. https://doi.org/10.1137/19M128911X

[21] Mohammed Ali, M.M., Mahmoudi, M., Darehmiraki, M. (2024). “Applying radial basis functions and partition of unity for solving heating equations optimal control issues”. Mathematical Modelling of Engineering Problems, 11(12), 3402–3410. https: //doi.org/10.18280/mmep.111218

[22] Mohammed Ali, M.M., Mahmoudi, M., Darehmiraki, M. (2025). “Hybrid RBF method for solving fractional PDE-constrained optimal control problems”. Control and Optimization in Applied Mathematics, 10(2), 213–240. https://doi.org/10.30473/coam.2025. 73804.1289

[23] Pearson, J.W. (2013). “A radial basis function method for solving PDE-constrained optimization problems”. Numerical Algorithms, 64(3), 481–506. https://doi.org/10. 1007/s11075-012-9675-6

[24] Raeisi, B., Fardi, M., Ahmadi Darani, M. (2024). “RBF-based partition of unity methods for two-dimensional time-dependent PDEs: Numerical and theoretical aspects”. Mathematics and Computers in Simulation, 226, 152–171. https://doi.org/10.1016/j. matcom.2024.07.001

[25] Raeisi, B., Ahmadi Darani, M., Fardi, M. (2024). “The RBF partition of unity method for a 2D time-fractional parabolic equation”. Computers and Mathematics with Applications, 166, 237–252. https://doi.org/10.1016/j.camwa.2024.05.012

[26] Safdari-Vaighani, A., Heryudono, A., Larsson, E. (2015). “A radial basis function partition of unity collocation method for convection-diffusion equations arising in financial applications”. Journal of Scientific Computing, 64(2), 341–367. https://doi.org/10. 1007/s10915-014-9935-9

[27] Schaback, R., Wendland, H. (2006). “Kernel techniques: From machine learning to meshless methods”. Acta Numerica, 15, 543–639. https://doi.org/10.1017/ S0962492906270016

[28] Tröltzsch, F. (2010). Optimal Control of Partial Differential Equations: Theory, Methods and Applications. American Mathematical Society, Graduate Studies in Mathematics, Vol. 112. https://doi.org/10.1090/gsm/112

[29] Wendland, H. (2002). “Fast evaluation of radial basis functions: Methods based on partition of unity”. In Approximation Theory X: Wavelets, Splines, and Applications, Vanderbilt University Press, Nashville, 473–483.

[30] Wendland, H. (2004). Scattered Data Approximation. Cambridge University Press. https://doi.org/10.1017/CBO9780511617539

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