Fractional PDE Constrained Optimization: Box and Sparse Constrained Problems

  • Fabio DurastanteEmail author
  • Stefano Cipolla
Part of the Springer INdAM Series book series (SINDAMS, volume 29)


In this paper we address the numerical solution of two Fractional Partial Differential Equation constrained optimization problems: the two-dimensional semilinear Riesz Space Fractional Diffusion equation with box or sparse constraints. Both a theoretical and experimental analysis of the problems is carried out. The algorithmic framework is based on the L-BFGS-B method coupled with a Krylov subspace solver for the box constrained problem within an optimize-then-discretize approach and on the semismooth Newton–Krylov method for the sparse one. Suitable preconditioning strategies by approximate inverses and Generalized Locally Toeplitz sequences are taken into account. The numerical experiments are performed with benchmarked software/libraries enforcing the reproducibility of the results.


Fractional differential equation Constrained optimization Preconditioner Saddle matrix 


  1. 1.
    Annunziato, M., Borzì, A., Magdziarz, M., Weron, A.: A fractional Fokker–Planck control framework for subdiffusion processes. Optim. Control. Appl. Methods 37(2), 290–304 (2016). MathSciNetCrossRefGoogle Scholar
  2. 2.
    Antil, H., Otarola, E.: A FEM for an optimal control problem of fractional powers of elliptic operators. SIAM J. Control. Optim. 53(6), 3432–3456 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bell, N., Garland, M.: Cusp: Generic Parallel Algorithms for Sparse Matrix and Graph Computations (2015). Version 0.5.1
  4. 4.
    Bellavia, S., Bertaccini, D., Morini, B.: Nonsymmetric preconditioner updates in Newton–Krylov methods for nonlinear systems. SIAM J. Sci. Comput. 33(5), 2595–2619 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Benzi, M., Bertaccini, D.: Approximate inverse preconditioning for shifted linear systems. BIT 43(2), 231–244 (2003)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Benzi, M., Tuma, M.: A sparse approximate inverse preconditioner for nonsymmetric linear systems. SIAM J. Sci. Comput. 19(3), 968–994 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bertaccini, D.: Efficient preconditioning for sequences of parametric complex symmetric linear systems. Electron. Trans. Numer. Anal. 18, 49–64 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bertaccini, D., Durastante, F.: Interpolating preconditioners for the solution of sequence of linear systems. Comput. Math. Appl. 72(4), 1118–1130 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bertaccini, D., Durastante, F.: Solving mixed classical and fractional partial differential equations using short-memory principle and approximate inverses. Numer. Algorithms 74(4), 1061–1082 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bertaccini, D., Durastante, F.: Iterative Methods and Preconditioning for Large and Sparse Linear Systems with Applications. Chapman & Hall/CRC Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2018)CrossRefGoogle Scholar
  12. 12.
    Bertaccini, D., Filippone, S.: Sparse approximate inverse preconditioners on high performance GPU platforms. Comput. Math. Appl. 71(3), 693–711 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Byrd, R.H., Lu, P., Nocedal, J., Zhu, C.: A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Comput. 16(5), 1190–1208 (1995)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cipolla, S., Durastante, F.: Fractional PDE constrained optimization: an optimize-then-discretize approach with L-BFGS and approximate inverse preconditioning. Appl. Numer. Math. 123, 43–57 (2018). MathSciNetCrossRefGoogle Scholar
  15. 15.
    De los Reyes, J.C.: Numerical PDE-Constrained Optimization. Springer, Cham (2015)Google Scholar
  16. 16.
    Dolgov, S., Pearson, J.W., Savostyanov, D.V., Stoll, M.: Fast tensor product solvers for optimization problems with fractional differential equations as constraints. Appl. Math. Comput. 273, 604–623 (2016)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Donatelli, M., Mazza, M., Serra-Capizzano, S.: Spectral analysis and structure preserving preconditioners for fractional diffusion equations. J. Comput. Phys. 307, 262–279 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Duff, I.S., Erisman, A.M., Reid, J.K.: Direct Methods for Sparse Matrices. Oxford University Press, Oxford (2017)CrossRefGoogle Scholar
  19. 19.
    Garoni, C., Serra-Capizzano, S.: Generalized Locally Toeplitz Sequences: Theory and Applications, vol. 1. Springer, Berlin (2017). CrossRefGoogle Scholar
  20. 20.
    Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints, vol. 23. Springer Science & Business Media, Berlin (2008)zbMATHGoogle Scholar
  21. 21.
    Jin, X.Q., Lin, F.R., Zhao, Z.: Preconditioned iterative methods for two-dimensional space-fractional diffusion equations. Commun. Comput. Phys. 18(2), 469–488 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kelley, C.T.: Iterative Methods for Optimization. SIAM, Philadelphia (1999)CrossRefGoogle Scholar
  23. 23.
    Lei, S.L., Sun, H.W.: A circulant preconditioner for fractional diffusion equations. J. Comput. Phys. 242, 715–725 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Liu, D.C., Nocedal, J.: On the limited memory BFGS method for large scale optimization. Math. Program. 45(1), 503–528 (1989)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Moghaderi, H., Dehghan, M., Donatelli, M., Mazza, M.: Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations. J. Comput. Phys. 350(Suppl. C), 992–1011 (2017).
  26. 26.
    Ortigueira, M.D.: Riesz potential operators and inverses via fractional centred derivatives. Int. J. Math. Math. Sci. 2006, 12 pp. (2006)Google Scholar
  27. 27.
    Pan, J., Ke, R., Ng, M.K., Sun, H.W.: Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations. SIAM J. Sci. Comput. 36(6), A2698–A2719 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Pan, J., Ng, M., Wang, H.: Fast preconditioned iterative methods for finite volume discretization of steady-state space-fractional diffusion equations. Numer. Algorithms 74(1), 153–173 (2017)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Pang, H.K., Sun, H.W.: Multigrid method for fractional diffusion equations. J. Comput. Phys. 231(2), 693–703 (2012)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198. Academic, London (1998)zbMATHGoogle Scholar
  31. 31.
    Porcelli, M., Simoncini, V., Stoll, M.: Preconditioning PDE-constrained optimization with L 1-sparsity and control constraints. Comput. Math. Appl. 74(5), 1059–1075 (2017)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Stadler, G.: Elliptic optimal control problems with \(\mathbb {L}^1\)-control cost and applications for the placement of control devices. Comput. Optim. Appl. 44(2), 159 (2009)Google Scholar
  33. 33.
    Ulbrich, M.: Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. SIAM, Philadelphia (2011)CrossRefGoogle Scholar
  34. 34.
    Van der Vorst, H.A.: Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Comput. 13(2), 631–644 (1992)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Dipartimento di InformaticaUniversità di PisaPisa (PI)Italy
  2. 2.Dipartimento di MatematicaUniversità di PadovaPadova (PD)Italy

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