Papers
Topics
Authors
Recent
Search
2000 character limit reached

A splitting-based KPIK method for eddy current optimal control problems in an all-at-once approach

Published 18 Mar 2024 in math.NA and cs.NA | (2403.11611v1)

Abstract: In this paper, we focus on efficient methods to solve discretized linear systems obtained from eddy current optimal control problems in an all-at-once approach. We construct a new low-rank matrix equation method based on a special splitting of the coefficient matrix and the Krylov-plus-inverted-Krylov (KPIK) algorithm. Firstly, we rewrite the resulting discretized linear system in a matrix-equation form. Then using the KPIK algorithm, we can obtain the low-rank approximation solution. The new method is named the splitting-based Krylov-plus-inverted-Krylov (SKPIK) method. The SKPIK method can not only solve the large and sparse discretized systems fast but also overcomes the storage problem. Theoretical results about the existence of the low-rank solutions are given. Numerical experiments are used to illustrate the performance of the new low-rank matrix equation method by compared with some existing classical efficient methods.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (56)
  1. I. Anjam and J. Valdman. Fast MATLAB assembly of FEM matrices in 2D and 3D: Edge elements. Applied Mathematics and Computation, 267:252–263, 2015.
  2. O. Axelsson and Z.-Z. Liang. A note on preconditioning methods for time-periodic eddy current optimal control problems. Journal of Computational and Applied Mathematics, 352:262–277, 2019.
  3. O. Axelsson and D. Lukáš. Preconditioners for time-harmonic optimal control eddy–current problems. In International Conference on Large-Scale Scientific Computing, pages 47–54. Springer, 2017.
  4. O. Axelsson and D. Lukáš. Preconditioning methods for eddy-current optimally controlled time-harmonic electromagnetic problems. Journal of Numerical Mathematics, 27(1):1–21, 2019.
  5. Efficient solvers for nonlinear time-periodic eddy current problems. Computing and Visualization in Science, 9(4):197–207, 2006.
  6. Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM Journal on Matrix Analysis and Applications, 24(3):603–626, 2003.
  7. P. Benner and T. Breiten. Low rank methods for a class of generalized Lyapunov equations and related issues. Numerische Mathematik, 124(3):441–470, 2013.
  8. Low-rank solvers for unsteady Stokes–Brinkman optimal control problem with random data. Computer Methods in Applied Mechanics and Engineering, 304:26–54, 2016.
  9. A low-rank solution method for Riccati equations with indefinite quadratic terms. Numerical Algorithms, 92(2):1083–1103, 2023.
  10. Low-rank solution of unsteady diffusion equations with stochastic coefficients. SIAM/ASA Journal on Uncertainty Quantification, 3(1):622–649, 2015.
  11. Block-diagonal preconditioning for optimal control problems constrained by PDEs with uncertain inputs. SIAM Journal on Matrix Analysis and Applications, 37(2):491–518, 2016.
  12. Numerical solution of saddle point problems. Acta Numerica, 14:1–137, 2005.
  13. A preconditioning technique for a class of PDE–constrained optimization problems. Advances in Computational Mathematics, 35:149–173, 2011.
  14. Fast iterative solvers for fractional differential equations. Electronic Transactions on Numerical Analysis, 45:751–776, 2016.
  15. A justification of eddy currents model for the Maxwell equations. SIAM Journal on Applied Mathematics, 60(5):1805–1823, 2000.
  16. A low-rank matrix equation method for solving PDE–constrained optimization problems. SIAM Journal on Scientific Computing, 43(5):S637–S654, 2021.
  17. Tensor networks for dimensionality reduction and large-scale optimization: Part 2 applications and future perspectives. Foundations and Trends® in Machine Learning, 9(6):431–673, 2017.
  18. Fast tensor product solvers for optimization problems with fractional differential equations as constraints. Applied Mathematics and Computation, 273:604–623, 2016.
  19. S. Dolgov and M. Stoll. Low-rank solution to an optimization problem constrained by the Navier–Stokes equations. SIAM Journal on Scientific Computing, 39(1):A255–A280, 2017.
  20. Parallel time integration with multigrid. SIAM Journal on Scientific Computing, 36(6):C635–C661, 2014.
  21. M. J. Gander. 50 years of time parallel time integration. In Multiple Shooting and Time Domain Decomposition Methods: MuS-TDD, Heidelberg, May 6-8, 2013, pages 69–113. Springer, 2015.
  22. A direct solver for time parallelization. In Domain decomposition methods in science and engineering XXII, pages 491–499. Springer, 2016.
  23. M. J. Gander and M. Neumüller. Analysis of a new space-time parallel multigrid algorithm for parabolic problems. SIAM Journal on Scientific Computing, 38(4):A2173–A2208, 2016.
  24. M. J. Gander and S. Vandewalle. Analysis of the parareal time-parallel time-integration method. SIAM Journal on Scientific Computing, 29(2):556–578, 2007.
  25. A. Goddard and A. Wathen. A note on parallel preconditioning for all-at-once evolutionary PDEs. Electronic Transactions on Numerical Analysis, 51:135–150, 2019.
  26. Matrix Computations. The Johns Hopkins UniversityPress,, Baltimore, MD.
  27. L. Grasedyck. Existence of a low rank or ℋℋ\mathcal{H}caligraphic_H-matrix approximant to the solution of a Sylvester equation. Numerical Linear Algebra with Applications, 11(4):371–389, 2004.
  28. E. Haber and U. M. Ascher. Preconditioned all-at-once methods for large, sparse parameter estimation problems. Inverse Problems, 17(6):1847, 2001.
  29. M. Hinze. A hierarchical space-time solver for distributed control of the Stokes equation. Univ., Department Mathematik, 2008.
  30. A space–time multigrid method for optimal flow control. Constrained Optimization and Optimal Control for Partial Differential Equations, pages 147–170, 2012.
  31. G. Horton and S. Vandewalle. A space-time multigrid method for parabolic partial differential equations. SIAM Journal on Scientific Computing, 16(4):848–864, 1995.
  32. M. Kolmbauer. Efficient solvers for multiharmonic eddy current optimal control problems with various constraints and their analysis. IMA Journal of Numerical Analysis, 33(3):1063–1094, 2013.
  33. M. Kolmbauer and U. Langer. A frequency-robust solver for the time-harmonic eddy current problem. In Scientific Computing in Electrical Engineering SCEE 2010, pages 97–105. Springer, 2011.
  34. M. Kolmbauer and U. Langer. A robust preconditioned MinRes solver for distributed time-periodic eddy current optimal control problems. SIAM Journal on Scientific Computing, 34(6):B785–B809, 2012.
  35. M. Kolmbauer and U. Langer. A robust preconditioned MinRes solver for time-periodic eddy current problems. Computational Methods in Applied Mathematics, 13(1):1–20, 2013.
  36. F. Kwok and B. W. Ong. Schwarz waveform relaxation with adaptive pipelining. SIAM Journal on Scientific Computing, 41(1):A339–A364, 2019.
  37. Résolution d’edp par un schéma en temps <<<pararéel>>>. Comptes Rendus de l’Académie des Sciences-Series I-Mathematics, 332(7):661–668, 2001.
  38. Y. Maday and G. Turinici. A parareal in time procedure for the control of partial differential equations. Comptes Rendus Mathematique, 335(4):387–392, 2002.
  39. I. Markovsky. Low rank approximation: algorithms, implementation, applications, volume 906. Springer, 2012.
  40. Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations. SIAM Journal on Scientific Computing, 40(2):A1012–A1033, 2018.
  41. J.-C. Nédélec. Mixed finite elements in ℝ3superscriptℝ3\mathbb{R}^{3}blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. Numerische Mathematik, 35:315–341, 1980.
  42. J.-C. Nédélec. A new family of mixed finite elements in ℝ3superscriptℝ3\mathbb{R}^{3}blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. Numerische Mathematik, 50:57–81, 1986.
  43. Solution of sparse indefinite systems of linear equations. SIAM journal on numerical analysis, 12(4):617–629, 1975.
  44. Regularization-robust preconditioners for time-dependent PDE-constrained optimization problems. SIAM Journal on Matrix Analysis and Applications, 33(4):1126–1152, 2012.
  45. A new approximation of the Schur complement in preconditioners for PDE-constrained optimization. Numerical Linear Algebra with Applications, 19(5):816–829, 2012.
  46. All-at-once preconditioning in PDE–constrained optimization. Kybernetika, 46(2):341–360, 2010.
  47. S. D. Shank and V. Simoncini. Krylov subspace methods for large-scale constrained Sylvester equations. SIAM Journal on Matrix Analysis and Applications, 34(4):1448–1463, 2013.
  48. Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Oxford University Press, United Kingdom, first edition, 2005.
  49. V. Simoncini. A new iterative method for solving large-scale Lyapunov matrix equations. SIAM Journal on Scientific Computing, 29(3):1268–1288, 2007.
  50. M. Stoll and T. Breiten. A low–rank in time approach to PDE–constrained optimization. SIAM Journal on Scientific Computing, 37(1):B1–B29, 2015.
  51. M. Stoll and A. Wathen. All–at–once solution of time–dependent Stokes control. Journal of Computational Physics, 232(1):498–515, 2013.
  52. M. Wolfmayr. A posteriori error estimation for the optimal control of time-periodic eddy current problems. Computational Methods in Applied Mathematics, 2023.
  53. S.-L. Wu. Toward parallel coarse grid correction for the parareal algorithm. SIAM Journal on Scientific Computing, 40(3):A1446–A1472, 2018.
  54. S.-L. Wu and T. Zhou. Acceleration of the two-level MGRIT algorithm via the diagonalization technique. SIAM Journal on Scientific Computing, 41(5):A3421–A3448, 2019.
  55. M.-L. Zeng. Respectively scaled splitting iteration method for a class of block 4-by-4 linear systems from eddy current electromagnetic problems. Japan Journal of Industrial and Applied Mathematics, 38:489–501, 2021.
  56. A preconditioning technique for an all-at-once system from Volterra subdiffusion equations with graded time steps. Journal of Scientific Computing, 88(1):11, 2021.

Summary

No one has generated a summary of this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Sign Up to Summarize

Paper to Video (Beta)

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Generate Now

Authors (2)

  1. Min-Li Zeng 
  2. Martin Stoll 

Collections

Sign up for free to add this paper to one or more collections.

User Circle Single Streamline Icon: https://streamlinehq.com Sign Up