Stochastic discrete Hamiltonian variational integrators
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Abstract
Variational integrators are derived for structurepreserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a typeII stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structurepreserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge–Kutta methods as a special case. Several new lowstage stochastic symplectic methods of meansquare order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior longtime numerical stability and energy behavior compared to nonsymplectic methods.
Keywords
Stochastic Hamiltonian systems Variational integrators Geometric numerical integration methods Geometric mechanics Stochastic differential equationsMathematics Subject Classification
65C301 Introduction
As occurs for other SDEs, most Hamiltonian SDEs cannot be solved analytically and one must resort to numerical simulations to obtain approximate solutions. In principle, general purpose stochastic numerical schemes for SDEs can be applied to stochastic Hamiltonian systems. However, as for their deterministic counterparts, stochastic Hamiltonian systems possess several important geometric features. In particular, their phase space flows (almost surely) preserve the symplectic structure. When simulating these systems numerically, it is therefore advisable that the numerical scheme also preserves such geometric features. Geometric integration of deterministic Hamiltonian systems has been thoroughly studied (see [18, 41, 55] and the references therein) and symplectic integrators have been shown to demonstrate superior performance in longtime simulations of Hamiltonian systems, compared to nonsymplectic methods; so it is natural to pursue a similar approach for stochastic Hamiltonian systems. This is a relatively recent pursuit. Stochastic symplectic integrators were first proposed in [43, 44]. Stochastic generalizations of symplectic partitioned Runge–Kutta methods were analyzed in [13, 35, 36]. A stochastic generating function approach to constructing stochastic symplectic methods, based on approximately solving a corresponding stochastic Hamilton–Jacobi equation satisfied by the generating function, was proposed in [65, 66], and this idea was further pursued in [2, 4, 16]. Stochastic symplectic integrators constructed via composition methods were proposed and analyzed in [45]. A first order weak symplectic numerical scheme and an extrapolation method were proposed and their global error was analyzed in [3]. More recently, an approach based on Padé approximations has been used to construct stochastic symplectic methods for linear stochastic Hamiltonian systems (see [60]). Higherorder strong and weak symplectic partitioned Runge–Kutta methods have been proposed in [67, 68]. Highorder conformal symplectic and ergodic schemes for the stochastic Langevin equation have been introduced in [25]. Other structurepreserving methods for stochastic Hamiltonian systems have also been investigated, see, e.g., [1, 15, 26], and the references therein.
Longtime accuracy and near preservation of the Hamiltonian by symplectic integrators applied to deterministic Hamiltonian systems have been rigorously studied using the socalled backward error analysis (see, e.g., [18] and the references therein). To the best of our knowledge, such rigorous analysis has not been attempted in the stochastic context as yet. However, the numerical evidence presented in the papers cited above is promising and suggests that stochastic symplectic integrators indeed possess the property of very accurately capturing the evolution of the Hamiltonian H over exponentially long time intervals (note that the Hamiltonian H in general does not stay constant for stochastic Hamiltonian systems).
An important class of geometric integrators are variational integrators. This type of numerical schemes is based on discrete variational principles and provides a natural framework for the discretization of Lagrangian systems, including forced, dissipative, or constrained ones. These methods have the advantage that they are symplectic, and in the presence of a symmetry, satisfy a discrete version of Noether’s theorem. For an overview of variational integration for deterministic systems see [40]; see also [21, 32, 33, 47, 48, 53, 63, 64]. Variational integrators were introduced in the context of finitedimensional mechanical systems, but were later generalized to Lagrangian field theories (see [39]) and applied in many computations, for example in elasticity, electrodynamics, or fluid dynamics; see [34, 49, 59, 62].
Stochastic variational integrators were first introduced in [8] and further studied in [7]. However, those integrators were restricted to the special case when the Hamiltonian function \(h=h(q)\) was independent of p, and only loworder Runge–Kutta types of discretization were considered. In the present work we extend the idea of stochastic variational integration to general stochastic Hamiltonian systems by generalizing the variational principle introduced in [33] and applying a Galerkin type of discretization (see [32, 33, 40, 47, 48]), which leads to a more general class of stochastic symplectic integrators than those presented in [7, 8, 35, 36]. Our approach consists in approximating a generating function for the stochastic flow of the Hamiltonian system, but unlike in [65, 66], we do make this discrete approximation by exploiting its variational characterization, rather than solving the corresponding Hamilton–Jacobi equation.

In Sect. 2 we introduce a stochastic variational principle and a stochastic generating function suitable for considering stochastic Hamiltonian systems, and we discuss their properties.

In Sect. 3 we present a general framework for constructing stochastic Galerkin variational integrators, prove the symplecticity and conservation properties of such integrators, show they contain the stochastic symplectic Runge–Kutta methods of [35, 36] as a special case, and finally present several particularly interesting examples of new lowstage stochastic symplectic integrators of meansquare order 1.0 derived with our general methodology.

In Sect. 4 we present the results of our numerical tests, which verify the theoretical convergence rates and the excellent longtime performance of our integrators compared to some popular nonsymplectic methods.

Section 5 contains the summary of our work.
2 Variational principle for stochastic Hamiltonian systems
The stochastic variational integrators proposed in [7, 8] were formulated for dynamical systems which are described by a Lagrangian and which are subject to noise whose magnitude depends only on the position q. Therefore, these integrators are applicable to (1.1) only when the Hamiltonian function \(h=h(q)\) is independent of p and the Hamiltonian H is nondegenerate (i.e., the associated Legendre transform is invertible). However, in the case of general \(h=h(q,p)\) the paths q(t) of the system become almost surely nowhere differentiable, which poses a difficulty in interpreting the meaning of the corresponding Lagrangian. Therefore, we need a different sort of action functional and variational principle to construct stochastic symplectic integrators for (1.1). To this end, we will generalize the approach taken in [33]. To begin, in the next section, we will introduce an appropriate stochastic action functional and show that it can be used to define a typeII generating function for the stochastic flow of the system (1.1).
2.1 Stochastic variational principle
 (H1)
H and h are \(C^2\) functions of their arguments
 (H2)
A and B are globally Lipschitz
Theorem 2.1
Proof
Remark It is natural to expect that the converse theorem, that is, if \(\big ( q(\cdot ), p(\cdot ) \big )\) is a critical point of the stochastic action functional (2.4), then the curve \(\big ( q(t), p(t) \big )\) satisfies (1.1), should also hold, although a larger class of variations \((\delta q, \delta p)\) may be necessary. A variant of such a theorem, although for a slightly different variational principle and in a different setting, was proved in LázaroCamí and Ortega [30]. Another variant for Lagrangian systems was proved by BouRabee and Owhadi [8] in the special case when \(h=h(q)\) is independent of p. In that case, one can assume that q(t) is continuously differentiable. In the general case, however, q(t) is not differentiable, and the ideas of [8] cannot be applied directly. We leave this as an open question. Here, we will use the action functional (2.4) and the variational principle (2.5) to construct numerical schemes, and we will directly verify that these numerical schemes converge to solutions of (1.1).
2.2 Stochastic typeII generating function
Theorem 2.2
Proof
We can consider \(S(q_a,p_b)\) as a function of time if we let \(t_b\) vary. Let us denote this function as \(S_t(q_a,p)\). Below we show that \(S_t(q_a,p)\) satisfies a certain stochastic partial differential equation, which is a stochastic generalization of the Hamilton–Jacobi equation considered in [33].
Proposition 2.1
Proof
Remark Similar stochastic Hamilton–Jacobi equations were introduced in [65, 66], where they were used for constructing stochastic symplectic integrators by considering series expansions of generating functions in terms of multiple Stratonovich integrals. This was a direct generalization of a similar technique for deterministic Hamiltonian systems (see [18]). In this work we explore the generalized Galerkin framework for constructing approximations of the generating function \(S(q_a,p_b)\) in (2.13) by using its variational characterization (2.12). Our approach is a stochastic generalization of the techniques proposed in [33, 48] for deterministic Hamiltonian and Lagrangian systems.
2.3 Stochastic Noether’s theorem
Theorem 2.3
Proof
3 Stochastic Galerkin Hamiltonian variational integrators
3.1 Construction of the integrator
Although the scheme (3.8) formally appears to be a straightforward generalization of its deterministic counterpart, it should be emphasized that the main difference lies in the fact that the increments \(\Delta W\) are random variables such that \(E(\Delta W^2)=\Delta t\), which makes the convergence analysis significantly more challenging than in the deterministic case. The main difficulty is in the choice of the parameters s, r, \(\alpha _i\), \(\beta _i\), \(c_i\), so that the resulting numerical scheme converges in some sense to the solutions of (1.1). The number of parameters and order conditions grows rapidly, when terms approximating multiple Stratonovich integrals are added (see Sect. 3.6 and [10, 11, 12, 14]). In Sects. 3.2 and 3.3 we study the geometric properties of the family of schemes described by (3.8), whereas in Sects. 3.4 and 3.5 we provide concrete choices of the coefficients that lead to convergent methods.
3.2 Properties of stochastic Galerkin variational integrators
The key features of variational integrators are their symplecticity and exact preservation of the discrete counterparts of conserved quantities (momentum maps) related to the symmetries of the Lagrangian or Hamiltonian (see [40]). These properties carry over to the stochastic case, as was first demonstrated in [8] for Lagrangian systems. In what follows, we will show that the stochastic Galerkin Hamiltonian variational integrators constructed in Sect. 3.1 also possess these properties.
Theorem 3.1
Proof
The discrete counterpart of stochastic Noether’s theorem readily generalizes from the corresponding theorem in the deterministic case.
Theorem 3.2
Proof
See the proof of Theorem 4 in [33]. In our case the result holds almost surely, because Eq. (3.7) holds almost surely. \(\square \)
Theorem 3.3
Proof
Remark One can easily verify that the interpolating polynomial (3.3) is in particular equivariant with respect to linear actions (translations, rotations, etc.), therefore the stochastic Galerkin variational integrator (3.8) preserves quadratic momentum maps (such as linear and angular momentum) related to linear symmetries of the Hamiltonians H and h.
3.3 Stochastic symplectic partitioned Runge–Kutta methods
for \(i,j=1,\ldots ,s\). We now prove that in the special case when \(r=s\), the stochastic Galerkin variational integrator (3.8) is equivalent to the stochastic symplectic partitioned Runge–Kutta method (3.18).
Theorem 3.4
for \(i,j=1,\ldots ,s\).
Proof
3.4 Examples

Gauss–Legendre quadratures (Gau): midpoint rule (\(r=1\)), etc.

Lobatto quadratures (Lob): trapezoidal rule (\(r=2\)), Simpson’s rule (\(r=3\)), etc.

Open trapezoidal rule (Otr; \(r=2\))

Milne’s rule (Mil; \(r=3\))

Rectangle rule (Rec; \(r=1\))—only in the case when \(h=h(q)\).
3.4.1 General Hamiltonian function h(q, p)
 1.
P1N1Q2Gau (Stochastic midpoint method)
Using the midpoint rule (\(r=1\), \(c_1=1/2\), \(\alpha _1=\beta _1=1\)) together with polynomials of degree \(s=1\) gives a stochastic Runge–Kutta method (3.18) with \(a_{11}={\bar{a}}_{11}=b_{11}={\bar{b}}_{11}=1/2\). Noting that \(Q_1=(q_k+q_{k+1})/2\) and \(P_1=(p_k+p_{k+1})/2\), this method can be written as$$\begin{aligned} q_{k+1}&= q_k + \frac{\partial H}{\partial p} \bigg (\frac{q_k+q_{k+1}}{2},\frac{p_k+p_{k+1}}{2} \bigg )\Delta t + \frac{\partial h}{\partial p} \bigg (\frac{q_k+q_{k+1}}{2},\frac{p_k+p_{k+1}}{2} \bigg )\Delta W, \nonumber \\ p_{k+1}&= p_k  \frac{\partial H}{\partial q} \bigg (\frac{q_k+q_{k+1}}{2},\frac{p_k+p_{k+1}}{2} \bigg )\Delta t  \frac{\partial h}{\partial q} \bigg (\frac{q_k+q_{k+1}}{2},\frac{p_k+p_{k+1}}{2} \bigg )\Delta W. \end{aligned}$$(3.28)The stochastic midpoint method was considered in [36, 44]. It is an implicit method and in general one has to solve 2N equations for 2N unknowns. However, if the Hamiltonians are separable, that is, \(H(q,p)=T_0(p)+U_0(q)\) and \(h(q,p)=T_1(p)+U_1(q)\), then \(p_{k+1}\) from the second equation can be substituted into the first one. In that case only N nonlinear equations have to be solved for \(q_{k+1}\).
 2.
P2N2Q2Lob (Stochastic Störmer–Verlet method)
If the trapezoidal rule (\(r=2\), \(c_1=0\), \(c_2=1\), \(\alpha _1=\beta _1=1/2\), \(\alpha _2=\beta _2=1/2\)) is used with polynomials of degree \(s=2\), we obtain another partitioned Runge–Kutta method (3.18) with \(a_{11}=a_{12}=0\), \(a_{21}=a_{22}=1/2\), \({\bar{a}}_{11}={\bar{a}}_{21}=1/2\), \({\bar{a}}_{12}={\bar{a}}_{22}=0\), \((b_{ij})=(a_{ij})\), \(({\bar{b}}_{ij})=({\bar{a}}_{ij})\). Noting that \(Q_1=q_k\), \(Q_2=q_{k+1}\), and \(P_1=P_2\), this method can be more efficiently written as$$\begin{aligned} P_1&= p_k  \frac{1}{2} \frac{\partial H}{\partial q} \big (q_k, P_1 \big )\Delta t  \frac{1}{2} \frac{\partial h}{\partial q} \big (q_k, P_1 \big )\Delta W, \nonumber \\ q_{k+1}&= q_k + \frac{1}{2} \frac{\partial H}{\partial p} \big (q_k, P_1 \big )\Delta t + \frac{1}{2} \frac{\partial H}{\partial p} \big (q_{k+1}, P_1 \big )\Delta t \nonumber \\&\quad + \frac{1}{2} \frac{\partial h}{\partial p} \big (q_k, P_1 \big )\Delta W + \frac{1}{2} \frac{\partial h}{\partial p} \big (q_{k+1}, P_1 \big )\Delta W, \nonumber \\ p_{k+1}&= P_1  \frac{1}{2} \frac{\partial H}{\partial q} \big (q_{k+1}, P_1 \big )\Delta t  \frac{1}{2} \frac{\partial h}{\partial q} \big (q_{k+1}, P_1 \big )\Delta W. \end{aligned}$$(3.29)This method is a stochastic generalization of the Störmer–Verlet method (see [18]) and was considered in [36]. It is particularly efficient, because the first equation can be solved separately from the second one, and the last equation is an explicit update. Moreover, if the Hamiltonians are separable, this method becomes fully explicit.
 3.
P1N2Q2Lob (Stochastic trapezoidal method)
This integrator is based on polynomials of degree \(s=1\) with control points \(d_0=0\), \(d_1=1\), and the trapezoidal rule. Equations (3.8) take the form$$\begin{aligned} p_k&= \frac{1}{2}(P_1+P_2) + \frac{1}{2} \frac{\partial H}{\partial q} \big (q_k, P_1 \big )\Delta t + \frac{1}{2} \frac{\partial h}{\partial q} \big (q_k, P_1 \big )\Delta W, \nonumber \\ p_{k+1}&= \frac{1}{2}(P_1+P_2)  \frac{1}{2} \frac{\partial H}{\partial q} \big (q_{k+1}, P_2 \big )\Delta t  \frac{1}{2} \frac{\partial h}{\partial q} \big (q_{k+1}, P_2 \big )\Delta W, \nonumber \\ q_{k+1}&= q_k + \frac{\partial H}{\partial p} \big (q_k,P_1 \big )\Delta t + \frac{\partial h}{\partial p} \big (q_k,P_1 \big )\Delta W, \nonumber \\ q_{k+1}&= q_k + \frac{\partial H}{\partial p} \big (q_{k+1},P_2 \big )\Delta t + \frac{\partial h}{\partial p} \big (q_{k+1},P_2 \big )\Delta W. \end{aligned}$$(3.30)This integrator is a stochastic generalization of the trapezoidal method for deterministic systems (see [40]). One may easily verify that if the Hamiltonians are separable, that is, \(H(q,p)=T_0(p)+U_0(q)\) and \(h(q,p)=T_1(p)+U_1(q)\), then \(P_1=P_2\) and (3.30) is equivalent to the Störmer–Verlet method (3.29) and is fully explicit.
 4.
P1N3Q4Lob
If we use Simpson’s rule (\(r=3\), \(c_1=0\), \(c_2=1/2\), \(c_3=1\), \(\alpha _1=1/6\), \(\alpha _2=2/3\), \(\alpha _3=1/6\), \(\beta _i=\alpha _i\)), the resulting integrator (3.8) requires solving simultaneously 4N nonlinear equations, so it is computationally expensive in general. However, if the Hamiltonians H and h are separable, then (3.8d) implies \(P_1=P_2=P_3\), and the integrator can be rewritten as$$\begin{aligned} q_{k+1}&= q_k + \frac{\partial T_0}{\partial p} \big (P_1 \big )\Delta t + \frac{\partial T_1}{\partial p} \big (P_1 \big )\Delta W, \nonumber \\ p_{k+1}&= P_1  \frac{1}{3} \frac{\partial U_0}{\partial q} \bigg (\frac{q_k+q_{k+1}}{2}\bigg )\Delta t  \frac{1}{6} \frac{\partial U_0}{\partial q} \big (q_{k+1}\big )\Delta t \nonumber \\&\quad  \frac{1}{3} \frac{\partial U_1}{\partial q} \bigg (\frac{q_k+q_{k+1}}{2}\bigg )\Delta W  \frac{1}{6} \frac{\partial U_1}{\partial q} \big (q_{k+1}\big )\Delta W, \end{aligned}$$(3.31)where$$\begin{aligned} P_1&= p_k  \frac{1}{6} \frac{\partial U_0}{\partial q} \big (q_k\big )\Delta t  \frac{1}{3} \frac{\partial U_0}{\partial q} \bigg (\frac{q_k+q_{k+1}}{2}\bigg )\Delta t \nonumber \\&\quad  \frac{1}{6} \frac{\partial U_1}{\partial q} \big (q_k\big )\Delta W  \frac{1}{3} \frac{\partial U_1}{\partial q} \bigg (\frac{q_k+q_{k+1}}{2}\bigg )\Delta W, \end{aligned}$$(3.32)and \(H(q,p)=T_0(p)+U_0(q)\) and \(h(q,p)=T_1(p)+U_1(q)\). In this case only the first equation needs to be solved for \(q_{k+1}\), and then the second equation is an explicit update.
 5.
P1N2Q2Otr
Like the method (3.30), this integrator is based on polynomials of degree \(s=1\) with control points \(d_0=0\), \(d_1=1\), but uses the open trapezoidal rule (\(r=2\), \(c_1=1/3\), \(c_2=2/3\), \(\alpha _1=1/2\), \(\alpha _2=1/2\), \(\beta _i=\alpha _i\)). Equations (3.8) take the form$$\begin{aligned} p_k&= \frac{1}{2}(P_1+P_2) + \frac{1}{3} \frac{\partial H}{\partial q} \bigg (\frac{q_{k+1}+2q_k}{3}, P_1 \bigg )\Delta t + \frac{1}{6} \frac{\partial H}{\partial q} \bigg (\frac{2q_{k+1}+q_k}{3}, P_2 \bigg )\Delta t \nonumber \\&\quad + \frac{1}{3} \frac{\partial h}{\partial q} \bigg (\frac{q_{k+1}+2q_k}{3}, P_1 \bigg )\Delta W + \frac{1}{6} \frac{\partial h}{\partial q} \bigg (\frac{2q_{k+1}+q_k}{3}, P_2 \bigg )\Delta W, \nonumber \\ p_{k+1}&= \frac{1}{2}(P_1+P_2)  \frac{1}{6} \frac{\partial H}{\partial q} \bigg (\frac{q_{k+1}+2q_k}{3}, P_1 \bigg )\Delta t  \frac{1}{3} \frac{\partial H}{\partial q} \bigg (\frac{2q_{k+1}+q_k}{3}, P_2 \bigg )\Delta t \nonumber \\&\quad  \frac{1}{6} \frac{\partial h}{\partial q} \bigg (\frac{q_{k+1}+2q_k}{3}, P_1 \bigg )\Delta W  \frac{1}{3} \frac{\partial h}{\partial q} \bigg (\frac{2q_{k+1}+q_k}{3}, P_2 \bigg )\Delta W, \nonumber \\ q_{k+1}&= q_k + \frac{\partial H}{\partial p} \bigg (\frac{q_{k+1}+2q_k}{3}, P_1 \bigg )\Delta t + \frac{\partial h}{\partial p} \bigg (\frac{q_{k+1}+2q_k}{3}, P_1 \bigg )\Delta W, \nonumber \\ q_{k+1}&= q_k + \frac{\partial H}{\partial p} \bigg (\frac{2q_{k+1}+q_k}{3}, P_2 \bigg )\Delta t + \frac{\partial h}{\partial p} \bigg (\frac{2q_{k+1}+q_k}{3}, P_2 \bigg )\Delta W. \end{aligned}$$(3.33)In general one has to solve the first, third, and fourth equation simultaneously (3N equations for 3N variables). In case of separable Hamiltonians we have \(P_1=P_2\) and one only needs to solve N nonlinear equations: \(P_1\) can be explicitly calculated from the first equation and substituted into the third one, and the resulting nonlinear equation then has to be solved for \(q_{k+1}\).
 6.
P2N2Q2Otr
If the open trapezoidal rule is used with polynomials of degree \(s=2\), we obtain yet another partitioned Runge–Kutta method (3.18) with \(a_{11}={\bar{a}}_{22}=1/2\), \(a_{12}={\bar{a}}_{12}=1/6\), \(a_{21}={\bar{a}}_{21}=2/3\), \(a_{22}={\bar{a}}_{11}=0\), \((b_{ij})=(a_{ij})\), \(({\bar{b}}_{ij})=({\bar{a}}_{ij})\). Inspecting Eq. (3.18) we see that, for example, \(Q_2\) is explicitly given in terms of \(Q_1\) and \(P_1\), therefore one only needs to solve 3N equations for the 3N variables \(Q_1\), \(P_1\), \(P_2\), and the remaining equations are explicit updates. This method further simplifies for separable Hamiltonians H and h: \(Q_1\) and \(Q_2\) are now explicitly given in terms of \(P_1\) and \(P_2\), and the nonlinear equation for \(P_1\) can be solved separately from the nonlinear equation for \(P_2\).
 7.
P1N3Q4Mil
A method similar to (3.31) is obtained if we use Milne’s rule (\(r=3\), \(c_1=1/4\), \(c_2=1/2\), \(c_3=3/4\), \(\alpha _1=2/3\), \(\alpha _2=1/3\), \(\alpha _3=2/3\), \(\beta _i=\alpha _i\)) instead of Simpson’s rule. The resulting integrator is also computationally expensive in general, but if the Hamiltonians H and h are separable, then (3.8d) implies \(P_1=P_2=P_3\), and the integrator can be rewritten as$$\begin{aligned} q_{k+1}&= q_k + \frac{\partial T_0}{\partial p} \big (P_1 \big )\Delta t + \frac{\partial T_1}{\partial p} \big (P_1 \big )\Delta W, \nonumber \\ p_{k+1}&= p_k  \frac{2}{3} \frac{\partial U_0}{\partial q} \bigg (\frac{3 q_k+q_{k+1}}{4}\bigg )\Delta t + \frac{1}{3} \frac{\partial U_0}{\partial q} \bigg (\frac{q_k+q_{k+1}}{2}\bigg )\Delta t \nonumber \\&\quad  \frac{2}{3} \frac{\partial U_0}{\partial q} \bigg (\frac{q_k+3 q_{k+1}}{4}\bigg )\Delta t  \frac{2}{3} \frac{\partial U_1}{\partial q} \bigg (\frac{3 q_k+q_{k+1}}{4}\bigg )\Delta W \nonumber \\&\quad + \frac{1}{3} \frac{\partial U_1}{\partial q} \bigg (\frac{q_k+q_{k+1}}{2}\bigg )\Delta W  \frac{2}{3} \frac{\partial U_1}{\partial q} \bigg (\frac{q_k+3 q_{k+1}}{4}\bigg )\Delta W, \end{aligned}$$(3.34)where$$\begin{aligned} P_1&= p_k  \frac{1}{2} \frac{\partial U_0}{\partial q} \bigg (\frac{3 q_k+q_{k+1}}{4}\bigg )\Delta t + \frac{1}{6} \frac{\partial U_0}{\partial q} \bigg (\frac{q_k+q_{k+1}}{2}\bigg )\Delta t \nonumber \\&\quad  \frac{1}{6} \frac{\partial U_0}{\partial q} \bigg (\frac{q_k+3 q_{k+1}}{4}\bigg )\Delta t  \frac{1}{2} \frac{\partial U_1}{\partial q} \bigg (\frac{3 q_k+q_{k+1}}{4}\bigg )\Delta W \nonumber \\&\quad + \frac{1}{6} \frac{\partial U_1}{\partial q} \bigg (\frac{q_k+q_{k+1}}{2}\bigg )\Delta W  \frac{1}{6} \frac{\partial U_1}{\partial q} \bigg (\frac{q_k+3 q_{k+1}}{4}\bigg )\Delta W, \end{aligned}$$(3.35)and \(H(q,p)=T_0(p)+U_0(q)\) and \(h(q,p)=T_1(p)+U_1(q)\). In this case only the first equation needs to be solved for \(q_{k+1}\), and then the second equation is an explicit update.
3.4.2 Hamiltonian function h(q) independent of momentum
 1.
P1N1Q1Rec (Stochastic symplectic Euler method)
The rectangle rule (\({\bar{r}}=1\), \({\bar{c}}_1=1\), \({\bar{\alpha }}_1=1\)) does not yield a convergent numerical scheme in the general case, but when \(h=h(q)\), the Itô and Stratonovich interpretations of (1.1) are equivalent, and the rectangle rule can be used to construct efficient integrators. In fact, applying the rectangle rule to both the Lebesgue and Stratonovich integrals and using polynomials of degree \(s=1\) yield a method which can be written as$$\begin{aligned} q_{k+1}&= q_k + \frac{\partial H}{\partial p} \big (q_{k+1}, p_k \big )\Delta t, \nonumber \\ p_{k+1}&= p_k  \frac{\partial H}{\partial q} \big (q_{k+1}, p_k \big )\Delta t  \frac{\partial h}{\partial q} \big (q_{k+1}\big )\Delta W. \end{aligned}$$(3.36)This method is a straightforward generalization of the symplectic Euler scheme (see [18, 40]) and is particularly computationally efficient, as only the first equation needs to be solved for \(q_{k+1}\), and then the second equation is an explicit update. Moreover, in case the Hamiltonian H is separable, the method becomes explicit.
 2.
P1N1Q1RecN2Q2Lob
The accuracy of the stochastic symplectic Euler scheme above can be improved by applying the trapezoidal rule to the Stratonovich integral instead of the rectangle rule. The resulting integrator takes the form$$\begin{aligned} q_{k+1}&= q_k + \frac{\partial H}{\partial p} \big (q_{k+1}, P_1 \big )\Delta t, \nonumber \\ p_{k+1}&= p_k  \frac{\partial H}{\partial q} \big (q_{k+1},P_1 \big )\Delta t  \frac{1}{2} \frac{\partial h}{\partial q} \big (q_k\big )\Delta W  \frac{1}{2} \frac{\partial h}{\partial q} \big (q_{k+1} \big )\Delta W, \end{aligned}$$(3.37)where$$\begin{aligned} P_1= p_k  \frac{1}{2} \frac{\partial h}{\partial q} \big (q_k \big ) \Delta W. \end{aligned}$$(3.38)While having a similar computational cost, this method yields a more accurate solution than (3.36) (see Sect. 4 for numerical tests). Moreover, in case the Hamiltonian H is separable, the method becomes explicit.
 3.
P1N1Q1RecN1Q2Gau
Similarly, if we apply the midpoint rule instead of the trapezoidal rule, we obtain the following modification of the stochastic symplectic Euler method:$$\begin{aligned} q_{k+1}&= q_k + \frac{\partial H}{\partial p} \big (q_{k+1}, P_1 \big )\Delta t, \nonumber \\ p_{k+1}&= p_k  \frac{\partial H}{\partial q} \big (q_{k+1},P_1 \big )\Delta t  \frac{\partial h}{\partial q} \bigg ( \frac{q_k+q_{k+1}}{2}\bigg )\Delta W, \end{aligned}$$(3.39)where$$\begin{aligned} P_1= p_k  \frac{1}{2} \frac{\partial h}{\partial q} \bigg ( \frac{q_k+q_{k+1}}{2}\bigg ) \Delta W. \end{aligned}$$(3.40)This method demonstrates a similar performance as (3.37) (see Sect. 4 for numerical tests). It becomes explicit if the Hamiltonian H is separable and the noise is additive, i.e., \(\partial h / \partial q = \text {const}\).
 4.
P2N2Q2LobN1Q1Rec
A modification of the stochastic Störmer–Verlet method (3.29) is obtained if we use the rectangle rule to approximate the Stratonovich integral:$$\begin{aligned} P_1&= p_k  \frac{1}{2} \frac{\partial H}{\partial q} \big (q_k, P_1 \big )\Delta t, \nonumber \\ q_{k+1}&= q_k + \frac{1}{2} \frac{\partial H}{\partial p} \big (q_k, P_1 \big )\Delta t + \frac{1}{2} \frac{\partial H}{\partial p} \big (q_{k+1}, P_1 \big )\Delta t, \nonumber \\ p_{k+1}&= P_1  \frac{1}{2} \frac{\partial H}{\partial q} \big (q_{k+1}, P_1 \big )\Delta t  \frac{\partial h}{\partial q} \big (q_{k+1}\big )\Delta W. \end{aligned}$$(3.41)This integrator has a similar computational cost as the stochastic Störmer–Verlet method (see Sect. 4), but it yields a slightly less accurate solution (see Sect. 4). Moreover, in case the Hamiltonian H is separable, the method becomes explicit.
 5.
P1N1Q2GauN2Q2Lob
This integrator is a modification of the stochastic midpoint method (3.28). We apply the midpoint rule (\({\bar{r}}=1\), \({\bar{c}}_1=1/2\), \({\bar{\alpha }}_1=1\)) to the Lebesgue integral in (3.4), and the trapezoidal rule (\(\tilde{r}=2\), \(\tilde{c}_1=0\), \(\tilde{c}_2=1\), \(\tilde{\beta }_1=1/2\), \(\tilde{\beta }_2=1/2\)) to the Stratonovich integral. The resulting numerical scheme can be written as$$\begin{aligned} q_{k+1}&= q_k + \frac{\partial H}{\partial p} \bigg (\frac{q_k+q_{k+1}}{2},P_1 \bigg )\Delta t, \nonumber \\ p_{k+1}&= p_k  \frac{\partial H}{\partial q} \bigg (\frac{q_k+q_{k+1}}{2},P_1 \bigg )\Delta t  \frac{1}{2} \frac{\partial h}{\partial q} \big (q_k\big )\Delta W  \frac{1}{2} \frac{\partial h}{\partial q} \big (q_{k+1} \big )\Delta W, \end{aligned}$$(3.42)where$$\begin{aligned} P_1=\frac{p_k+p_{k+1}}{2}+\frac{1}{4} \Delta W \left[ \frac{\partial h}{\partial q} \big (q_{k+1} \big )\frac{\partial h}{\partial q} \big (q_k \big ) \right] . \end{aligned}$$(3.43)This method is fully implicit, but similar to (3.28), simplifies when the Hamiltonian H is separable.
 6.
P1N2Q2LobN1Q2Gau
If instead we apply the trapezoidal rule to the Lebesgue integral and the midpoint rule to the Stratonovich integral in (3.4), we obtain a modification of the stochastic trapezoidal rule (3.30):$$\begin{aligned} p_k&= \frac{1}{2}(P_1+P_2) + \frac{1}{2} \frac{\partial H}{\partial q} \big (q_k, P_1 \big )\Delta t + \frac{1}{2} \frac{\partial h}{\partial q} \bigg (\frac{q_k+q_{k+1}}{2} \bigg )\Delta W, \nonumber \\ p_{k+1}&= \frac{1}{2}(P_1+P_2)  \frac{1}{2} \frac{\partial H}{\partial q} \big (q_{k+1}, P_2 \big )\Delta t  \frac{1}{2} \frac{\partial h}{\partial q} \bigg (\frac{q_k+q_{k+1}}{2} \bigg )\Delta W, \nonumber \\ q_{k+1}&= q_k + \frac{\partial H}{\partial p} \big (q_k,P_1 \big )\Delta t, \nonumber \\ q_{k+1}&= q_k + \frac{\partial H}{\partial p} \big (q_{k+1},P_2 \big )\Delta t. \end{aligned}$$(3.44)This method becomes explicit when the Hamiltonian H is separable and the noise is additive, i.e., \(\partial h / \partial q = \text {const}\).
3.5 Convergence
Remark

Hamiltonian functions \(h_i\) linear in q and p for all \(i=1,\ldots ,M\), i.e. additive noise

Hamiltonian functions \(h_i\) simultaneously independent of one of the variables q or p for all \(i=1,\ldots ,M\)
3.6 Methods of order 3 / 2
4 Numerical experiments
In this section we present the results of our numerical experiments. We verify numerically the convergence results from Sect. 3.5 and investigate the conservation properties of our integrators. In particular, we show that our stochastic variational integrators demonstrate superior behavior in longtime simulations compared to some popular general purpose nonsymplectic stochastic algorithms.
4.1 Numerical convergence analysis
4.1.1 Kubo oscillator
4.1.2 Synchrotron oscillations of particles in storage rings
4.2 Longtime energy behavior
4.2.1 Kubo oscillator
One can easily check that in the case of the Kubo oscillator the Hamiltonian H(q, p) stays constant for almost all sample paths, i.e., \(H({\bar{q}}(t), {\bar{p}}(t))=H(q_0,p_0)\) almost surely. We used this example to test the performance of the integrators from Sect. 3.4.1. Simulations with the initial conditions \(q_0=0\), \(p_0=1\), the noise intensity \(\beta =0.1\), and the relatively large time step \(\Delta t = 0.25\) were carried out until the time \(T=1000\) (approximately 160 periods of the oscillator in the absence of noise) for a single realization of the Wiener process. For comparison, similar simulations were carried out using nonsymplectic explicit methods like Milstein’s scheme and the order 3/2 strong Taylor scheme (see [28]). The numerical value of the Hamiltonian H(q, p) as a function of time for each of the integrators is depicted in Fig. 3. We find that the nonsymplectic schemes do not preserve the Hamiltonian well, even if small time steps are used. For example, we find that Milstein’s scheme does not give a satisfactory solution even with \(\Delta t = 0.001\), and though the Taylor scheme with \(\Delta t=0.05\) yields a result comparable to the variational integrators, the growing trend of the numerical Hamiltonian is evident. On the other hand, the variational integrators give numerical solutions for which the Hamiltonian oscillates around the true value (one can check via a direct calculation that the stochastic midpoint method (3.28) in this case preserves the Hamiltonian exactly; of course this does not necessarily hold in the general case).
4.2.2 Anharmonic oscillator
Remark
One can verify by a direct calculation that when the P2N2Q2Otr integrator (example 6 in Sect. 3.4.1) is applied to the Kubo oscillator, then the corresponding system of Eqs. (3.18) does not have a solution when \(\Delta t + \beta \Delta W = 3\). To avoid numerical difficulties, one could in principle use the truncated increments (3.9) with, e.g., \(A=(3\Delta t)/(2\beta )\) (for \(\Delta t <3\)). However, given the negligible probability that \(\Delta W>A\) for the parameters used in Sects. 4.1.1 and 4.2.1, we did not observe any numerical issues, even though we did not use truncated increments. In the case of all the other numerical experiments presented in Sect. 4, the applied algorithms either turned out to be explicit, or the corresponding nonlinear systems of equations had solutions for all values of \(\Delta W\). Nonlinear equations were solved using Newton’s method and the previous time step values of the position \(q_k\) and momentum \(p_k\) were used as initial guesses.
5 Summary
In this paper we have presented a general framework for constructing a new class of stochastic symplectic integrators for stochastic Hamiltonian systems. We generalized the approach of Galerkin variational integrators introduced in [33, 40, 48] to the stochastic case, following the ideas underlying the stochastic variational integrators introduced in [8]. The solution of the stochastic Hamiltonian system was approximated by a polynomial of degree s, and the action functional was approximated by a quadrature formula based on r quadrature points. We showed that the resulting integrators are symplectic, preserve integrals of motion related to Lie group symmetries, and include stochastic symplectic Runge–Kutta methods introduced in [35, 36, 44] as a special case when \(r=s\). We pointed out several new lowstage stochastic symplectic methods of meansquare order 1.0 for systems driven by a onedimensional noise, both for the case of a general Hamiltonian function \(h=h(q,p)\) and a Hamiltonian function \(h=h(q)\) independent of p, and demonstrated their superior longtime numerical stability and energy behavior via numerical experiments. We also stated the conditions under which these integrators retain their first order of convergence when applied to systems driven by a multidimensional noise.
Our work can be extended in several ways. In Sect. 3.6 we indicated how higherorder stochastic variational integrators can be constructed and showed that a type of stochastic symplectic partitioned Runge–Kutta methods of meansquare order 3/2 considered in [44] can be recast in that formalism. It would be interesting to derive new stochastic integrators of order 3/2 by choosing appropriate values for the parameters in (3.54) or (3.56). It would also be interesting to apply the Galerkin approach to construct stochastic variational integrators for constrained (see [7]) and dissipative (see [9]) stochastic Hamiltonian systems, and systems defined on Lie groups (see [32]). Another important problem would be stochastic variational error analysis. That is, rather than considering how closely the numerical solution follows the exact trajectory of the system, one could investigate how closely the discrete Hamiltonian matches the exact generating function. In the deterministic setting these two notions of the order of convergence are equivalent (see [40]). It would be instructive to know if a similar result holds in the stochastic case. A further vital task would be to develop higherorder weakly convergent stochastic variational integrators. As mentioned in Sects. 3.1 and 3.6, higherorder methods require inclusion of higherorder multiple Stratonovich integrals, which are cumbersome to simulate in practice. In many cases, though, one is only interested in calculating the probability distribution of the solution rather than precisely approximating each sample path. In such cases weakly convergent methods are much easier to use (see [28, 42]). Finally, one may extend the idea of variational integration to stochastic multisymplectic partial differential equations such as the stochastic Korteweg–de Vries, Camassa–Holm or Hunter–Saxton equations. Theoretical groundwork for such numerical schemes has been recently presented in [24].
Footnotes
 1.
In this work we only consider Hamiltonian functions H and h that are independent of time. In the timedependent case one needs to add a further assumption that the growth of A and B is linearly bounded, i.e. \(\Vert A(z,t)\Vert ^2+\Vert B(z,t)\Vert ^2 \le K (1 + \Vert z\Vert ^2)\) for a constant \(K>0\) (see [5, 27, 28, 29]).
 2.
A generating function for the symplectic transformation \((q_a,p_a)\longrightarrow (q_b,p_b)\) is a function of one of the variables \((q_a,p_a)\) and one of the variables \((q_b,p_b)\). Therefore, there are four basic types of generating functions: \(S=S_1(q_a, q_b)\), \(S=S_2(q_a, p_b)\), \(S=S_3(p_a, q_b)\), and \(S=S_4(p_a, p_b)\). In this work we use the typeII generating function \(S=S_2(q_a, p_b)\).
 3.
In analogy to ordinary calculus, the total stochastic differential is understood as \(S_2(q_a, {\bar{p}}(t_b),t_b)S_2(q_a, {\bar{p}}(t_a),t_a) = \int _{t_a}^{t_b} {\bar{d}}S_2(q_a, {\bar{p}}(t),t)\), whereas the partial stochastic differential means \(S_2(q_a, p_b,t_b)S_2(q_a, p_b,t_a) = \int _{t_a}^{t_b} dS_2(q_a,p_b,t)\).
Notes
Acknowledgements
Open access funding provided by Max Planck Society. We would like to thank Nawaf BouRabee, Mickael Chekroun, Dan Crisan, Nader Ganaba, Melvin Leok, JuanPablo Ortega, Houman Owhadi and Wei Pan for useful comments and references. This work was partially supported by the European Research Council Advanced Grant 267382 FCCA. Parts of this project were completed while the authors were visiting the Institute for Mathematical Sciences, National University of Singapore in 2016.
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