Drift-preserving numerical integrators for stochastic Hamiltonian systems

The paper deals with numerical discretizations of separable nonlinear Hamiltonian systems with additive noise. For such problems, the expected value of the total energy, along the exact solution, drifts linearly with time. We present and analyze a time integrator having the same property for all times. Furthermore, strong and weak convergence of the numerical scheme along with efficient multilevel Monte Carlo estimators are studied. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.

[1]  David Cohen,et al.  On the numerical discretisation of stochastic oscillators , 2012, Math. Comput. Simul..

[2]  E. Hairer.  Energy-preserving variant of collocation methods 1 , 2010 .

[3]  G. Quispel,et al.  A new class of energy-preserving numerical integration methods , 2008 .

[4]  Jingjing Zhang,et al.  Discrete Gradient Approach to Stochastic Differential Equations with a Conserved Quantity , 2011, SIAM J. Numer. Anal..

[5]  Henri Schurz.  Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise , 2008 .

[6]  Xiaohua Ding,et al.  High-order stochastic symplectic partitioned Runge-Kutta methods for stochastic Hamiltonian systems with additive noise , 2019, Appl. Math. Comput..

[7]  David Cohen,et al.  Linear energy-preserving integrators for Poisson systems , 2011 .

[8]  Modified averaged vector field methods preserving multiple invariants for conservative stochastic differential equations , 2018, 1810.10737.

[9]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[10]  M. V. Tretyakov,et al.  Stochastic Numerics for Mathematical Physics , 2004, Scientific Computation.

[11]  Erwan Faou,et al.  Conservative stochastic differential equations: Mathematical and numerical analysis , 2009, Math. Comput..

[12]  Luigi Brugnano,et al.  Line integral methods which preserve all invariants of conservative problems , 2012, J. Comput. Appl. Math..

[13]  Yuto Miyatake,et al.  A Characterization of Energy-Preserving Methods and the Construction of Parallel Integrators for Hamiltonian Systems , 2015, SIAM J. Numer. Anal..

[14]  Jialin Hong,et al.  Conservative methods for stochastic differential equations with a conserved quantity , 2014, 1411.1819.

[15]  O. Gonzalez.  Time integration and discrete Hamiltonian systems , 1996 .

[16]  Yuto Miyatake,et al.  An energy-preserving exponentially-fitted continuous stage Runge–Kutta method for Hamiltonian systems , 2014 .

[17]  Luigi Brugnano,et al.  Analysis of Energy and QUadratic Invariant Preserving (EQUIP) methods , 2017, J. Comput. Appl. Math..

[18]  F. Iavernaro,et al.  High-order Symmetric Schemes for the Energy Conservation of Polynomial Hamiltonian Problems 1 2 , 2009 .

[19]  M. J. Senosiain,et al.  A review on numerical schemes for solving a linear stochastic oscillator , 2015 .

[20]  Jialin Hong,et al.  Preservation of quadratic invariants of stochastic differential equations via Runge–Kutta methods , 2015 .

[21]  Liying Sun,et al.  Exponential integrators for stochastic Maxwell's equations driven by Itô noise , 2019, J. Comput. Phys..

[22]  Stig Larsson,et al.  A Trigonometric Method for the Linear Stochastic Wave Equation , 2012, SIAM J. Numer. Anal..

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

[24]  Desmond J. Higham,et al.  Numerical simulation of a linear stochastic oscillator with additive noise , 2004 .

[25]  Elena Celledoni,et al.  The minimal stage, energy preserving Runge-Kutta method for polynomial Hamiltonian systems is the averaged vector field method , 2012, Math. Comput..

[26]  Michael B. Giles,et al.  Multilevel Monte Carlo Path Simulation , 2008, Oper. Res..

[27]  Kevin Burrage,et al.  Low rank Runge-Kutta methods, symplecticity and stochastic Hamiltonian problems with additive noise , 2012, J. Comput. Appl. Math..

[28]  David Cohen,et al.  Energy-preserving integrators for stochastic Poisson systems , 2014 .

[29]  Bin Wang,et al.  Efficient energy-preserving integrators for oscillatory Hamiltonian systems , 2013, J. Comput. Phys..

[30]  Xuli Han.  Direction-consistent tangent vectors for generating interpolation curves , 2019, J. Comput. Appl. Math..

[31]  David Cohen,et al.  Drift-preserving numerical integrators for stochastic Poisson systems , 2020, ArXiv.

[32]  David Cohen,et al.  Exponential Integrators for Stochastic Schrödinger Equations Driven by Itô Noise , 2016, Journal of Computational Mathematics.

[33]  Ernst Hairer,et al.  Energy Conservation with Non-Symplectic Methods: Examples and Counter-Examples , 2004 .

[34]  Annika Lang,et al.  A Note on the Importance of Weak Convergence Rates for SPDE Approximations in Multilevel Monte Carlo Schemes , 2015, MCQMC.

[35]  G. Quispel,et al.  Geometric integration using discrete gradients , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[36]  Xiaojie Wang,et al.  Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise , 2015, 1503.00073.

[37]  H. Kojima,et al.  Invariants preserving schemes based on explicit Runge–Kutta methods , 2016, BIT Numerical Mathematics.

[38]  J. C. Jimenez,et al.  Locally Linearized methods for the simulation of stochastic oscillators driven by random forces , 2017 .

[39]  Jialin Hong,et al.  Projection methods for stochastic differential equations with conserved quantities , 2016, 1601.04157.

[40]  David Cohen,et al.  Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations , 2012, Numerische Mathematik.

[41]  Stefan Heinrich,et al.  Multilevel Monte Carlo Methods , 2001, LSSC.

[42]  Kevin Burrage,et al.  Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise , 2013, Numerical Algorithms.