Fast Solution Methods for Convex Quadratic Optimization of Fractional Differential Equations

In this paper, we present numerical methods suitable for solving convex quadratic fractional differential equation (FDE) constrained optimization problems, with box constraints on the state and/or ...

[1]  N. Levinson The Wiener (Root Mean Square) Error Criterion in Filter Design and Prediction , 1946 .

[2]  R. Glowinski Lectures on Numerical Methods for Non-Linear Variational Problems , 1981 .

[3]  R. Koeller Applications of Fractional Calculus to the Theory of Viscoelasticity , 1984 .

[4]  J. L. Hock,et al.  An exact recursion for the composite nearest‐neighbor degeneracy for a 2×N lattice space , 1984 .

[5]  S. Reddi,et al.  Eigenvector properties of Toeplitz matrices and their application to spectral analysis of time series , 1984 .

[6]  James R. Bunch,et al.  Stability of Methods for Solving Toeplitz Systems of Equations , 1985 .

[7]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[8]  T. Chan An Optimal Circulant Preconditioner for Toeplitz Systems , 1988 .

[9]  G. Strang,et al.  Toeplitz equations by conjugate gradients with circulant preconditioner , 1989 .

[10]  Eugene E. Tyrtyshnikov,et al.  Optimal and Superoptimal Circulant Preconditioners , 1992, SIAM J. Matrix Anal. Appl..

[11]  R. Chan,et al.  A Family of Block Preconditioners for Block Systems , 1992, SIAM J. Sci. Comput..

[12]  Stevens,et al.  Self-similar transport in incomplete chaos. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  Jonathan Eckstein,et al.  Nonlinear Proximal Point Algorithms Using Bregman Functions, with Applications to Convex Programming , 1993, Math. Oper. Res..

[14]  J. Nagy,et al.  FFT-based preconditioners for Toeplitz-block least squares problems , 1993 .

[15]  J. Nagy,et al.  Circulant Preconditioned Toeplitz Least Squares Iterations , 1994, SIAM J. Matrix Anal. Appl..

[16]  A. Wathen,et al.  Fast iterative solution of stabilised Stokes systems part II: using general block preconditioners , 1994 .

[17]  Raymond H. Chan,et al.  Best-conditioned circulant preconditioners , 1995 .

[18]  Xiao-Qing Jin,et al.  A preconditioner for constrained and weighted least squares problems with Toeplitz structure , 1996 .

[19]  Paolo Tilli,et al.  Locally Toeplitz sequences: spectral properties and applications , 1998 .

[20]  I. Podlubny Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[21]  V. Simoncini,et al.  Block--diagonal and indefinite symmetric preconditioners for mixed finite element formulations , 1999 .

[22]  Gene H. Golub,et al.  A Note on Preconditioning for Indefinite Linear Systems , 1999, SIAM J. Sci. Comput..

[23]  Stefano Serra Capizzano,et al.  Any Circulant-Like Preconditioner for Multilevel Matrices Is Not Superlinear , 2000, SIAM J. Matrix Anal. Appl..

[24]  Ilse C. F. Ipsen A Note on Preconditioning Nonsymmetric Matrices , 2001, SIAM J. Sci. Comput..

[25]  Vickie E. Lynch,et al.  Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model , 2001 .

[26]  M. Meerschaert,et al.  Finite difference approximations for fractional advection-dispersion flow equations , 2004 .

[27]  Michael K. Ng Iterative Methods for Toeplitz Systems (Numerical Mathematics and Scientific Computation) , 2004 .

[28]  Yury F. Luchko,et al.  Algorithms for the fractional calculus: A selection of numerical methods , 2005 .

[29]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[30]  Tony F. Chan,et al.  Circulant preconditioners for Toeplitz-block matrices , 1994, Numerical Algorithms.

[31]  M. Meerschaert,et al.  Finite difference methods for two-dimensional fractional dispersion equation , 2006 .

[32]  M. Meerschaert,et al.  Finite difference approximations for two-sided space-fractional partial differential equations , 2006 .

[33]  Eric de Sturler,et al.  Preconditioners for Generalized Saddle-Point Problems , 2006, SIAM J. Numer. Anal..

[34]  Shivkumar Chandrasekaran,et al.  A Superfast Algorithm for Toeplitz Systems of Linear Equations , 2007, SIAM J. Matrix Anal. Appl..

[35]  Norbert Heuer,et al.  Numerical Approximation of a Time Dependent, Nonlinear, Space-Fractional Diffusion Equation , 2007, SIAM J. Numer. Anal..

[36]  Weihua Deng,et al.  Finite Element Method for the Space and Time Fractional Fokker-Planck Equation , 2008, SIAM J. Numer. Anal..

[37]  Hong Wang,et al.  A direct O(N log2 N) finite difference method for fractional diffusion equations , 2010, J. Comput. Phys..

[38]  Jianfei Huang,et al.  The Grünwald-Letnikov method for fractional differential equations , 2011, Comput. Math. Appl..

[39]  R. Chan Circulant preconditioners for Hermitian Toeplitz systems , 2011 .

[40]  M. Neytcheva On element-by-element Schur complement approximations , 2011 .

[41]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[42]  Andrew J. Wathen,et al.  A new approximation of the Schur complement in preconditioners for PDE‐constrained optimization , 2012, Numer. Linear Algebra Appl..

[43]  Martin Stoll,et al.  Regularization-Robust Preconditioners for Time-Dependent PDE-Constrained Optimization Problems , 2012, SIAM J. Matrix Anal. Appl..

[44]  Jacek Gondzio,et al.  Interior point methods 25 years later , 2012, Eur. J. Oper. Res..

[45]  Han Xiao,et al.  Covariance matrix estimation for stationary time series , 2011, 1105.4563.

[46]  Jianlin Xia,et al.  A Superfast Structured Solver for Toeplitz Linear Systems via Randomized Sampling , 2012, SIAM J. Matrix Anal. Appl..

[47]  A. Wathen,et al.  FAST ITERATIVE SOLVERS FOR CONVECTION-DIFFUSION CONTROL PROBLEMS ∗ , 2013 .

[48]  Siu-Long Lei,et al.  A circulant preconditioner for fractional diffusion equations , 2013, J. Comput. Phys..

[49]  Richard G. Baraniuk,et al.  Fast Alternating Direction Optimization Methods , 2014, SIAM J. Imaging Sci..

[50]  Yvan Notay,et al.  A New Analysis of Block Preconditioners for Saddle Point Problems , 2014, SIAM J. Matrix Anal. Appl..

[51]  Maya Neytcheva,et al.  Numerical and computational aspects of some block-preconditioners for saddle point systems , 2015, Parallel Comput..

[52]  G. Wang,et al.  Alternating Direction Method of Multipliers for Separable Convex Optimization of Real Functions in Complex Variables , 2015 .

[53]  Wotao Yin,et al.  On the Global and Linear Convergence of the Generalized Alternating Direction Method of Multipliers , 2016, J. Sci. Comput..

[54]  Stefano Serra Capizzano,et al.  Spectral analysis and structure preserving preconditioners for fractional diffusion equations , 2016, J. Comput. Phys..

[55]  Maya Neytcheva,et al.  Spectral analysis of coupled PDEs and of their Schur complements via Generalized Locally Toeplitz sequences in 2D , 2016 .

[56]  Martin Stoll,et al.  Fast tensor product solvers for optimization problems with fractional differential equations as constraints , 2016, Appl. Math. Comput..

[57]  Lek-Heng Lim,et al.  Algorithms for structured matrix-vector product of optimal bilinear complexity , 2016, 2016 IEEE Information Theory Workshop (ITW).

[58]  Siu-Long Lei,et al.  Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations , 2016 .

[59]  Mehdi Dehghan,et al.  Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations , 2017, J. Comput. Phys..

[60]  Michael K. Ng,et al.  A Splitting Preconditioner for Toeplitz-Like Linear Systems Arising from Fractional Diffusion Equations , 2017, SIAM J. Matrix Anal. Appl..

[61]  Carlo Garoni,et al.  Generalized Locally Toeplitz Sequences: Theory and Applications: Volume I , 2017 .

[62]  Stefano Serra Capizzano,et al.  Block Generalized Locally Toeplitz Sequences: From the Theory to the Applications , 2018, Axioms.

[63]  Daniele Bertaccini,et al.  Limited Memory Block Preconditioners for Fast Solution of Fractional Partial Differential Equations , 2018, J. Sci. Comput..

[64]  Fabio Durastante,et al.  Fractional PDE constrained optimization , 2018 .

[65]  Stefano Serra Capizzano,et al.  Spectral Analysis and Multigrid Methods for Finite Volume Approximations of Space-Fractional Diffusion Equations , 2018, SIAM J. Sci. Comput..

[66]  Michael K. Ng,et al.  Circulant preconditioners for a kind of spatial fractional diffusion equations , 2018, Numerical Algorithms.

[67]  Guo-Feng Zhang,et al.  An MHSS-like iteration method for two-by-two linear systems with application to FDE optimization problems , 2019, J. Comput. Appl. Math..

[68]  B. Khoromskij,et al.  Tensor product method for fast solution of optimal control problems with fractional multidimensional Laplacian in constraints , 2018, J. Comput. Phys..