Convergence of a Low-Rank Lie-Trotter Splitting for Stiff Matrix Differential Equations

We propose a numerical integrator for determining low-rank approximations to solutions of large-scale matrix differential equations. The considered differential equations are semilinear and stiff. Our method consists of first splitting the differential equation into a stiff and a non-stiff part, respectively, and then following a dynamical low-rank approach. We conduct an error analysis of the proposed procedure, which is independent of the stiffness and robust with respect to possibly small singular values in the approximation matrix. Following the proposed method, we show how to obtain low-rank approximations for differential Lyapunov and for differential Riccati equations. Our theory is illustrated by numerical experiments.

[1]  Peter Benner,et al.  Numerical solution of the infinite-dimensional LQR problem and the associated Riccati differential equations , 2018, J. Num. Math..

[2]  Tony Stillfjord,et al.  Convergence Analysis for Splitting of the Abstract Differential Riccati Equation , 2014, SIAM J. Numer. Anal..

[3]  Y. Zhou,et al.  On the decay rate of Hankel singular values and related issues , 2002, Syst. Control. Lett..

[4]  Chiara Piazzola,et al.  Numerical low-rank approximation of matrix differential equations , 2017, J. Comput. Appl. Math..

[5]  L. Dieci,et al.  Positive definiteness in the numerical solution of Riccati differential equations , 1994 .

[6]  C. Lubich,et al.  A projector-splitting integrator for dynamical low-rank approximation , 2013, BIT Numerical Mathematics.

[7]  H. Abou-Kandil,et al.  Matrix Riccati Equations in Control and Systems Theory , 2003, IEEE Transactions on Automatic Control.

[8]  Alexander Ostermann,et al.  Exponential integrators on graphic processing units , 2013, 2013 International Conference on High Performance Computing & Simulation (HPCS).

[9]  G. Quispel,et al.  Splitting methods , 2002, Acta Numerica.

[10]  C. Lubich,et al.  Error Bounds for Exponential Operator Splittings , 2000 .

[11]  Awad H. Al-Mohy,et al.  Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators , 2011, SIAM J. Sci. Comput..

[12]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[13]  S. McKee,et al.  Weakly Singular Discrete Gronwall Inequalities , 1986 .

[14]  Thorsten Gerber,et al.  Semigroups Of Linear Operators And Applications To Partial Differential Equations , 2016 .

[15]  Hanna Walach,et al.  Discretized Dynamical Low-Rank Approximation in the Presence of Small Singular Values , 2016, SIAM J. Numer. Anal..

[16]  Tony Stillfjord,et al.  Low-Rank Second-Order Splitting of Large-Scale Differential Riccati Equations , 2015, IEEE Transactions on Automatic Control.

[17]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[18]  L. Tucker,et al.  Some mathematical notes on three-mode factor analysis , 1966, Psychometrika.

[19]  Christian Lubich,et al.  Time Integration in the Multiconfiguration Time-Dependent Hartree Method of Molecular Quantum Dynamics , 2015 .

[20]  Tony Stillfjord,et al.  Adaptive high-order splitting schemes for large-scale differential Riccati equations , 2016, Numerical Algorithms.

[21]  Tatjana Stykel,et al.  Balanced truncation model reduction for linear time-varying systems , 2016 .

[22]  I. Petersen,et al.  Robust Control Design Using H-? Methods , 2012 .

[23]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[24]  Reinhold Schneider,et al.  Dynamical Approximation by Hierarchical Tucker and Tensor-Train Tensors , 2013, SIAM J. Matrix Anal. Appl..

[25]  Othmar Koch,et al.  Dynamical Low-Rank Approximation , 2007, SIAM J. Matrix Anal. Appl..

[26]  Daniel J. Arrigo,et al.  An Introduction to Partial Differential Equations , 2017, An Introduction to Partial Differential Equations.

[27]  Alexander Ostermann,et al.  Overcoming Order Reduction in Diffusion-Reaction Splitting. Part 2: Oblique Boundary Conditions , 2015, SIAM J. Sci. Comput..

[28]  R. Nagel,et al.  A Short Course on Operator Semigroups , 2006 .

[29]  Ivan V. Oseledets,et al.  Time Integration of Tensor Trains , 2014, SIAM J. Numer. Anal..

[30]  Alexander Ostermann,et al.  Overcoming Order Reduction in Diffusion-Reaction Splitting. Part 1: Dirichlet Boundary Conditions , 2014, SIAM J. Sci. Comput..

[31]  Marco Caliari,et al.  The Leja Method Revisited: Backward Error Analysis for the Matrix Exponential , 2015, SIAM J. Sci. Comput..

[32]  Akira Ichikawa,et al.  Remarks on the time-varying H∞ Riccati equations , 1999 .

[33]  Eugene E. Tyrtyshnikov,et al.  Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions , 2009, SIAM J. Sci. Comput..

[34]  Tony Stillfjord,et al.  Singular Value Decay of Operator-Valued Differential Lyapunov and Riccati Equations , 2018, SIAM J. Control. Optim..

[35]  Christian Lubich,et al.  Dynamical low-rank approximation: applications and numerical experiments , 2008, Math. Comput. Simul..

[36]  Y. Saad.  Analysis of some Krylov subspace approximations to the matrix exponential operator , 1992 .

[37]  Ivan Oseledets,et al.  Tensor-Train Decomposition , 2011, SIAM J. Sci. Comput..

[38]  U. Helmke,et al.  Optimization and Dynamical Systems , 1994, Proceedings of the IEEE.

[39]  Thilo Penzl.  Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case , 2000 .