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Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging

Molei Tao, Houman Owhadi, Jerrold E. Marsden

arXiv:0908.1241v2math.NAcs.NAmath.DSphysics.comp-ph

TL;DR

Explicit symplectic integration for stiff Hamiltonian systems with non-quadratic potentials is a nontrivial problem, and large time steps can bias molecular-clipper results. FLAVORS averages flows without requiring hidden slow variables to be identified; its approximation error is bounded by a function tending to zero, and it can outperform its associated legacy integrator for sufficiently small ϵ.

  • Problem

    Explicit symplectic integrators for non-quadratic stiff potentials are nontrivial to obtain, while large time steps that freeze bond lengths produce biased molecular-clipper results.

    Source: §2 · p. 15, §6.5 · p. 42

  • Method

    FLAVORS averages the instantaneous flow of systems with hidden slow and fast variables, using splitting and re-synchronization without identifying hidden slow variables.

    Source: §1.1 · p. 4, §1.4 · p. 7

  • Results

    FLAVOR's approximation error is bounded by a function of γ that tends to zero as γ decreases, and the speedup can grow arbitrarily large as ϵ decreases.

    Source: §1.5 · pp. 9–10

  • Takeaways & Limitations

    For sufficiently small ϵ, FLAVORS can outperform their associated legacy integrators at the same microstep by slowing the fast clock and limiting error accumulation.

    Source: §1.5 · p. 10

    The accuracy guarantee relies on averaging and is not assured when ϵ is not small; the method also requires ϵ to be clearly identified.

    Source: §1.8 · p. 12

Abstract

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We introduce a new class of integrators for stiff ODEs as well as SDEs. These integrators are (i) {\it Multiscale}: they are based on flow averaging and so do not fully resolve the fast variables and have a computational cost determined by slow variables (ii) {\it Versatile}: the method is based on averaging the flows of the given dynamical system (which may have hidden slow and fast processes) instead of averaging the instantaneous drift of assumed separated slow and fast processes. This bypasses the need for identifying explicitly (or numerically) the slow or fast variables (iii) {\it Nonintrusive}: A pre-existing numerical scheme resolving the microscopic time scale can be used as a black box and easily turned into one of the integrators in this paper by turning the large coefficients on over a microscopic timescale and off during a mesoscopic timescale (iv) {\it Convergent over two scales}: strongly over slow processes and in the sense of measures over fast ones. We introduce the related notion of two-scale flow convergence and analyze the convergence of these integrators under the induced topology (v) {\it Structure preserving}: for stiff Hamiltonian systems (possibly on manifolds), they can be made to be symplectic, time-reversible, and symmetry preserving (symmetries are group actions that leave the system invariant) in all variables. They are explicit and applicable to arbitrary stiff potentials (that need not be quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy and stability over four orders of magnitude of time scales. For stiff Langevin equations, they are symmetry preserving, time-reversible and Boltzmann-Gibbs reversible, quasi-symplectic on all variables and conformally symplectic with isotropic friction.

1 Overview of the integrator on ODEs

FLAVORS average the flow of a stiff system with hidden slow and fast variables, avoiding explicit identification of the slow variables. They combine short integrations of the full system with intermediate integrations of its non-stiff part, and are strongly accurate for slow variables and accurate in the sense of measures for fast variables.

1.1 FLAVORS

  • FLAVORS average the flow of a system with hidden slow and fast variables rather than the instantaneous drift of an already separated slow process.This avoids explicitly or numerically identifying the hidden slow variables.Source: §1.1 · p. 4
  • A pre-existing microscopic-scale numerical scheme can be used as a black box by switching stiff parameters on for τ and off for δ.The method composes short integrations of the complete system with intermediate integrations of its non-stiff part.Source: §1.1 · p. 4, Figure 1 · p. 5
  • FLAVORS inherit structure-preserving properties of the legacy integrator, including symplecticity and group-action symmetries for Hamiltonian and Langevin systems.Source: §1.1 · p. 5
  • The approach is strongly accurate for hidden slow variables and weakly accurate, in the sense of measures, for hidden fast variables under the stated step-size conditions.The paper calls the related two-scale flow convergence notion F-convergence.Source: §1.1 · p. 5

1.2 Two-scale flow convergence

The section formulates two-scale flow convergence to describe approximations that track slow behavior strongly while capturing fast behavior through measures. Under the stated assumptions, the paper establishes convergence of the FLAVOR approximation in this topology.

1.2 Two-scale flow convergence

  • Two-scale flow convergence compares time-averaged tests of the approximating process with averages over a fast-variable probability measure indexed by the slow state.The definition tests bounded, uniformly Lipschitz-continuous functions.Source: §1.2 · p. 6

1.3 Asymptotic convergence result

  • The asymptotic convergence result assumes Lipschitz vector fields and uniformly bounded trajectories for both the exact and approximate processes.The approximate trajectories are required to remain bounded across the specified δ and τ ranges.Source: §1.3 · p. 6
  • Under the stated assumptions, the FLAVOR approximation converges to the solution of the original system in the two-scale flow sense.The result is expressed for bounded, uniformly Lipschitz-continuous test functions and all t > 0.Source: §1.3 · p. 6

1.4 Rationale and mechanism behind FLAVORS

FLAVORS work by splitting and re-synchronizing the integration so that switching stiff parameters averages the flow over fast degrees of freedom. The error analysis also identifies a condition involving the slow-fast coordinate map that matters when it is nonlinear.

1.4 Rationale and mechanism behind FLAVORS

  • Splitting and re-synchronization average the flow over fast degrees of freedom while advancing the fast variables’ internal clock by τ during each interval δ.The method composes short-time integration of the complete system with integration of the non-stiff system.Source: §1.4 · p. 7
  • The mesoscopic switching averages the slow drift along the fast-variable trajectory, which can mix with respect to its local ergodic measure over a coarse step.The passage describes this mechanism when the coarse step is of order one and the specified step-size conditions hold.Source: §1.4 · p. 8
  • For nonlinear coordinate maps, the analysis says flow averaging works without identifying slow variables when condition (1.22) is satisfied; otherwise Hessian-related error can accumulate on slow variables.For a linear map, its Hessian is zero and condition (1.21) suffices for accuracy.Source: §1.4 · p. 9

1.5 Non asymptotic convergence result

The non-asymptotic analysis bounds the approximation error and relates fixed-accuracy speedup to the stiffness parameter. It also explains that error accumulation on fast variables can limit that speedup when the accumulation is exponential.

1.5 Non asymptotic convergence result

  • At fixed accuracy, the speedup is of order 1/ϵ unless error accumulation on fast variables is exponential.In the latter case, the passage says the speedup does not scale like 1/ϵ.Source: §1.5 · p. 10
  • For sufficiently small ϵ, FLAVORS can outperform their associated legacy integrator over large simulation times, while slowing the fast clock prevents its error accumulation from growing as ϵ ↓ 0.The simulations limited speedup to 200× for comparison, though the passage says it can be arbitrarily large as ϵ ↓ 0.Source: §1.5 · p. 10
  • With τ = 10−4 and δ = 2 · 10−3, FLAVORS remain accurate on the Fermi-Pasta-Ulam example where Velocity Verlet with h = 10−5 fails to capture O(ϵ−1) dynamics.Source: §1.5 · p. 10

1.6 Natural FLAVORS

Natural FLAVORS extend the method beyond legacy integrators by defining it through two mappings that approximate flows over different time intervals. The construction retains a legacy-integrator option, but does not require it.

1.6 Natural FLAVORS

  • Legacy integrators are convenient but not necessary for FLAVORS, and the stated theorems remain valid under the alternative discrete-process definition.In that construction, θ_G is obtained from the legacy integrator by setting α to zero.Source: §1.6 · pp. 10–11
  • Natural FLAVORS can be defined using two mappings: one approximates the full system’s flow for time τ, and the other approximates the flow of ẋ = G(x) for time δ − τ.The mappings are θ^ε_τ and θ^G_{δ−τ}.Source: §1.6 · p. 11
  • The mappings are required to satisfy the conditions collected as Condition 1.4.The supplied passage identifies the condition but does not give its full contents.Source: §1.6 · p. 11

1.7 Related work

Multiscale simulation is costly when fast motions force small time steps over slow observation windows. The passage contrasts existing methods by stiffness type and presents FLAVORS as a unified approach.

1.7 Related work

  • Fast motions can require small stable time steps, making simulations costly when observing slow degrees of freedom.The passage describes this as a major challenge in multiscale simulation.Source: §1.7 · p. 11
  • Prior integrators are described separately for stiff systems with fast transients and those with rapid oscillations.Methods cited for transients include implicit schemes, Chebyshev methods, and projective integrators; oscillatory systems have used filtering and Poincaré map techniques.Source: §1.7 · p. 11
  • The passage says no unified strategy existed for both stiffness classes, whereas FLAVORS apply to both, with or without noise, using a mesoscopic step chosen independently of stiffness.Averaging instantaneous drift is described for cases where slow variables can be identified.Source: §1.7 · p. 11

1.8 Limitations of the method

The accuracy argument relies on small stiffness parameter ε and on identifying the stiff parameter. When these conditions do not hold, the stated accuracy guarantee is limited.

1.8 Limitations of the method

  • The accuracy proof for the method’s theorems 1.1 and 1.2 relies on an averaging principle; when ϵ is not small, the method may be stable yet accuracy is not guaranteed.The global error increases with ϵ, δ, and τ.Source: §1.8 · p. 12
  • The computational-gain condition requires ε^(2/3) ≪ γ ≪ 1, which can be satisfied only when ε is small.This scaling requirement restricts the accuracy argument to sufficiently stiff cases.Source: §1.8 · p. 12
  • The method requires the stiff parameter 1/ε to be clearly identified, although stiffness may instead arise from nonlinearity, initial conditions, or boundary conditions.The passage also notes that molecular dynamics can produce widely separated time scales through nonlinear effects.Source: §1.8 · p. 12

1.9 Generic stiff ODEs

For generic stiff ODEs, the method uses averaging with probability measures in transformed coordinates. Its stated convergence and accuracy results depend on scale conditions and assumptions on the dynamics.

1.9 Generic stiff ODEs

  • The setup assumes an ε- and α-independent diffeomorphism into slow and fast coordinates, with a family of probability measures used in the averaged dynamics.The passage also assumes g is Lipschitz continuous in x and y on bounded sets.Source: §1.9 · p. 12
  • The slow variable is not frozen when estimating local invariant-measure error; convergence requires T→0 while Tα^ν→∞.The passage attributes control of the error induced by slow drift to E2.Source: §1.9 · p. 13
  • The accuracy statement requires δ ≪ h0 and τ ≪ ε^ν.It is stated under Conditions 1.5 and 1.6.Source: §1.9 · p. 13
  • The theorem states F-convergence as ε→0 under vanishing τ ε^-ν, δ ε^ν/τ, and τ² ε^(2ν)/δ.The passage gives the convergence target as η^-1_*; its notation is truncated in the supplied text.Source: §1.9 · p. 13

2 Deterministic mechanical systems: Hamiltonian equations

FLAVORS extend flow averaging to stiff Hamiltonian systems, including systems on configuration manifolds, while inheriting structure-preserving properties from their legacy integrators. The section also develops artificial and variational constructions, with accuracy conditions and a caveat for constrained dynamics.

2 Deterministic mechanical systems: Hamiltonian equations

  • FLAVORS arise by composing averaged flows, extending the method to stiff Hamiltonian systems with mechanical phase spaces on Euclidean space or cotangent bundles.The artificial variant freezes and unfreezes degrees of freedom associated with fast potentials; constrained solvers such as SHAKE or RATTLE require care because symplecticity may be lost.Source: §2 · p. 14, §2.1.4 · p. 17
  • Existing stiff-Hamiltonian approaches include methods limited to identified slow variables, explicit stiff-flow maps, or quadratic potentials.The section positions FLAVORS against methods that may require fine steps throughout the evolution or have long-time stability and accuracy issues.Source: §2 · pp. 14–15

2.1 FLAVORS for mechanical systems on manifolds

  • Theorem 1.3 establishes the integrator’s accuracy under Conditions 1.5 and 1.6, provided τ ≪ √ϵ ≪ δ.Source: §2.1 · p. 15

2.1.1 Structure preserving properties of FLAVORS

  • FLAVORS inherit structure preservation from their legacy integrators, including symplecticity on cotangent bundles and symmetry under the same group action.A symplectic legacy integrator and its adjoint can produce a symplectic, symmetric, time-reversible FLAVOR, including for constrained mechanical systems on manifolds.Source: §2.1.1 · p. 16

2.1.2 An example of a symplectic FLAVOR

  • Symplectic Euler provides an example of a symplectic FLAVOR for Euclidean phase space.The section also describes a symplectic and time-reversible FLAVOR using symplectic Euler.Source: §2.1.2 · p. 16, §2.1.3 · p. 17

2.1.4 An artificial FLAVOR

  • Artificial FLAVOR generally overperforms nonintrusive FLAVOR in long-time FPU simulations, but its constrained-dynamics implementation may lose symplecticity.The full-space property is preserved only when the evolution on the constrained manifold preserves its inherited symplectic form.Source: §2.1.4 · pp. 17–18

2.2 Variational derivation of FLAVORS

  • A variational legacy integrator yields a variational FLAVOR through a corresponding FLAVORized discrete Lagrangian.The construction is illustrated for Variational Euler, with Velocity Verlet FLAVORizations following similarly.Source: §2.2 · pp. 18–19

3 SDEs

The SDE section extends flow-averaging FLAVORS to stochastic systems with hidden slow and fast processes, including generic stiff SDEs. Under stated assumptions and scale conditions, the integrators converge to effective slow stochastic dynamics, while the convergence result accommodates noise coupled to fast variables.

3 SDEs

  • Flow-averaging FLAVORS apply to SDEs with hidden mixed slow and fast variables without identifying slow variables and can use a pre-existing scheme by switching stiff parameters on and off.This extends the flow-averaging approach from ODEs to stochastic systems.Source: §3 · p. 19

3.1 Two-scale flow convergence for SDEs

  • The SDE section introduces two-scale flow convergence for stochastic processes and extends the corresponding definition from the deterministic setting.The supplied passages identify the process sequence, a limiting process, and a family of probability measures as elements of the definition.Source: §3 · p. 20, §3.1 · p. 20

3.3 Convergence theorem

  • For the stated SDE setting, Theorem 3.1 gives an effective slow SDE with drift averaged over the fast-variable measure and an effective diffusion term.The result assumes equation (3.6) and Conditions 3.1.Source: §3.3 · p. 21
  • FLAVOR converges in distribution for SDEs with hidden slow and fast processes even when noise on the hidden slow variables is not null or independent of fast variables.The proof uses generator convergence because the slow process converges only weakly in this setting.Source: §3.3 · pp. 21–22

3.4 Natural FLAVORS

  • Natural FLAVORS need not be built from legacy integrators; Theorem 3.1 remains valid for algorithms satisfying the stated flow-approximation conditions.The conditions distinguish mappings approximating the full flow over short steps from mappings approximating the slow component.Source: §3.4 · p. 22

3.5 FLAVORS for generic stiff SDEs

  • The paper extends FLAVORS to generic stiff SDEs and describes a random flow map as a black box for acceleration without prior identification of slow variables.The stated setup includes Lipschitz coefficients and conditions on transformed slow and fast processes and their probability measures.Source: §3.5 · p. 23

FLAVORS

  • For generic stiff SDEs, the flow-averaging integrator is accurate in F-convergence when τ ≪ ϵ^ν ≪ δ under Conditions 3.3 and 3.4.The passage states the scale condition and assumptions for Theorem 3.2.Source: FLAVORS · p. 24

4 Stochastic mechanical systems: Langevin equations

The section constructs FLAVORS for stiff Langevin systems and shows how they inherit structure-preserving properties from their Hamiltonian integrators. It also describes reversible Metropolis-adjusted variants, while noting that their weak accuracy after metropolization remains unresolved.

4 Stochastic mechanical systems: Langevin equations

  • FLAVORS are developed as structure-preserving integrators for stiff Langevin equations.The section considers stochastic mechanical systems on Euclidean phase space or cotangent bundles of configuration manifolds.Source: §4 · p. 25

4.2 Structure Preserving properties of FLAVORS for stochastic mechanical systems on manifolds

  • The stochastic mechanical FLAVORS inherit symmetry under a group action when their component maps are symmetric under that action.The resulting method is a symmetric composition of symmetric steps.Source: §4.2 · p. 26
  • When the Hamiltonian component is symplectic, the resulting FLAVORS are quasi-symplectic; with isotropic friction, the specified variant is conformally symplectic.Quasi-symplecticity degenerates to symplecticity when friction is zero and the flow-map Jacobian is independent of (q, p).Source: §4.2 · pp. 26–27

4.2.1 Example of quasi-symplectic FLAVORS

  • Symplectic Euler gives an example of a quasi-symplectic FLAVOR that is also conformally symplectic when friction is slow and isotropic.The example uses the stated symplectic Euler integrator for the Hamiltonian part.Source: §4.2.1 · p. 27

4.2.2 Example of quasi-symplectic and time-reversible FLAVORS

  • A symmetric Strang splitting gives an example of a quasi-symplectic and time-reversible FLAVOR, also conformally symplectic under slow isotropic friction.The example is given for the second specified Langevin form.Source: §4.2.2 · p. 27

4.2.3 Example of Boltzmann-Gibbs reversible Metropolis-adjusted FLAVORS

  • Metropolis-adjusted FLAVORS can be made reversible with respect to the Gibbs distribution, but whether trajectory accuracy transfers weakly remains uninvestigated.The source states that the algorithms are stochastically stable and ergodic when momentum noise is nondegenerate.Source: §4.2.3 · p. 27
  • Figure 2 plots stability domains for non-intrusive and artificial FLAVOR against δ and τ/ϵ at ω = 1000.The caption identifies the plotted system as (5.1).Source: Figure 2 · p. 28

5 Numerical analysis of FLAVOR based on Variational Euler

Across the tested nonlinear system, FLAVOR’s error is first-order in the mesostep and grows linearly with simulation time; its stability and accuracy depend on the integrator and step-size choices. The analysis also finds no resonant mesostep for stability or convergence, while the observed long-time error behavior remains conjectural.

5.1 Stability

  • For δ∈(0, 2), nonintrusive FLAVOR is stable when 1/√τ≫ω≫1, while artificial FLAVOR is stable when 1/τ≫ω≫1.These are the stated stability conditions for the two integrators.Source: §5.1 · p. 28
  • Artificial FLAVOR has a much larger stability domain; at large δ, nonintrusive FLAVOR requires τ=o(ϵ), a more restrictive condition.The comparison concerns stable integrations of the linear Hamiltonian system (5.1).Source: §5.1 · pp. 29–30

5.1 Stability; 5.3 Numerical error analysis for nonlinear systems

  • The stability analysis reports no resonance, and the convergence analysis finds no resonant value of δ.Stability is determined by whether parameters lie below the stability curves; outside them, integration is not stable.Source: §5.1 · p. 30, §5.3 · p. 32

5.2 Error analysis

  • For fixed δ, errors are minimized at particular τ/ϵ values, but nonintrusive FLAVOR is less sensitive to τ/ϵ than artificial FLAVOR.Artificial FLAVOR’s optimal τ/ϵ roughly scales linearly with δ, and its error’s weak dependence on τ/ϵ means tuning is not necessary in the reported experiments.Source: §5.2 · p. 30

5.3 Numerical error analysis for nonlinear systems

  • For M not too small, Figure 5(a) error scales linearly with 1/M, while error grows linearly with simulation time and is independent of fast frequency at fixed δ.The frequency-independent plateau occurs when scale separation is complete; no resonant δ appears in convergence.Source: §5.3 · p. 32, Figure 5 · p. 33
  • The observed linear long-time error growth is conjectured to hold only for a subclass of systems, possibly integrable ones; rigorous analysis remains open.The theoretical error bound grows exponentially with simulation time.Source: §5.3 · p. 33

6 Numerical experiments

The numerical experiments examine FLAVOR in hidden ODE dynamics and stiff, forced mechanical and stochastic systems, including slow and fast processes. They report accelerated computation and comparisons with reference simulations, while assessing long-time behavior, resonance, and strong versus measure-based convergence.

6.1 Hidden Van der Pol oscillator (ODE)

  • A 200x acceleration produced trajectories close to the Forward Euler benchmark and correctly captured the Van der Pol relaxation oscillation’s O(1/ϵ) period.The example applies nonintrusive FLAVOR to the oscillator in hidden slow and fast variables.Source: §6.1 · pp. 33–34, Figure 6 · p. 34

6.2 Hamiltonian system with nonlinear stiff and soft potentials

  • For nonlinear stiff and soft Hamiltonian potentials, FLAVOR was strongly accurate in slow variables and accurate in the sense of measures in fast variables, with roughly 100x acceleration over Symplectic Euler.The plotted x−y variable includes fast modulation, so its FLAVOR trajectory retains slowed local oscillations rather than representing a purely slow variable.Source: Figure 7 · p. 35, §6.2 · p. 35

6.3 Fermi-Pasta-Ulam problem

  • Roughly 40x acceleration accompanied FLAVOR’s capture of the correct period and phase of slow periodic behavior and slower energy transfer on the FPU O(ω) timescale.With exact fast flow, artificial FLAVOR also produced less oscillatory stiff-spring energy curves while preserving a similar transfer pattern.Source: §6.3 · pp. 36–37, Figure 11 · p. 39
  • On the FPU O(ω^2) timescale, FLAVOR alone among the compared methods showed periodic energy exchange, and step-size tests found stable trajectories, though no analytic or provably accurate benchmark was available.A modified quadratic FPU system’s exact energy-exchange pattern resembled FLAVOR’s result for the non-harmonic system; the authors characterize this as numerical evidence, not a quantified error bound.Source: §6.3 · p. 37, Figure 10 · p. 38, Figure 12 · p. 39

6.3.1 On resonances

  • FLAVORS avoided Takens resonance in the FPU problem despite identical stiff-potential eigenfrequencies, capturing trajectories for large ω with mesostep δ ≫ 1/ω independent of ω.The analysis also found no instability from interactions among ϵ, τ, and δ under the stated nonintrusive or artificial FLAVOR step-size requirements.Source: §6.3.1 · p. 39

6.3 Fermi-Pasta-Ulam problem

  • In the FPU comparison across timescales, FLAVOR was 200 times faster than Velocity Verlet, captured the fastest timescale in the sense of measure, and Velocity Verlet missed the slowest O(ω^2) timescale.For the long-time non-harmonic FPU experiment, the authors note that no benchmark was available and rigorous analysis of FLAVOR’s long-time behavior remains future work.Source: §6.3 · p. 38, Figure 13 · p. 40

6.4 Nonlinear 2D primitive molecular dynamics

  • In the nonlinear 2D primitive molecular-dynamics example, FLAVOR reproduced the slow θ trajectory with roughly 50x acceleration while capturing the fast process’ amplitude, though its period was lengthened.The tested system has a fully nonlinear, nonpolynomial potential in Euclidean coordinates.Source: §6.4 · pp. 40–41, Figure 14 · p. 41

6.5 Nonlinear 2D molecular clipper

  • For the fully nonlinear propane model, induced symplectic FLAVOR achieved 10x acceleration compared with Symplectic Euler.The experiment exaggerates the timescale separation between bond stretching and bond-angle bending.Source: §6.5 · pp. 42–43

6.6 Forced nonautonomous mechanical system: Kapitza’s inverted pendulum

  • For Kapitza’s inverted pendulum, FLAVOR captures the rapid force’s contribution to the effective Hamiltonian rather than merely a fraction of that contribution.The passage explains that FLAVOR slows the rapid force by a factor τ/ϵ, and notes that the method remains accurate despite the rapid oscillations’ stabilizing effective contribution.Source: §6.6 · p. 45

6.7 Nonautonomous SDE system with hidden slow variables

  • FLAVOR accelerated integration of the nonautonomous SDE with hidden slow variables by 100x, and was applied without first identifying its slow or fast variables.The comparison used FLAVOR with mesostep δ = 0.01 and Euler–Maruyama with time step 10^-4; expected slow-variable values were empirically averaged over 100 trajectories.Source: §6.7 · pp. 45–46, Figure 18 · p. 46

6.8 Langevin equations with slow noise and friction

  • For slow-noise Langevin dynamics, FLAVOR and GLA moments agreed in the deterministic-dominated regime, while slow autocorrelations agreed and fast autocorrelations agreed only in the sense of measures.The agreement indicates the same rate of convergence toward the Boltzmann–Gibbs distribution; fluctuations at large times were attributed to using 100 sample trajectories.Source: §6.8 · p. 47, Figure 19 · p. 47, Figure 20 · p. 48
  • When noise acts on slow variables, FLAVOR does not converge strongly, but only in the sense of distributions.For the tested slow-noise Langevin system, fast-variable autocorrelations agreed with GLA only after time averaging over a mesoscopic time span.Source: §6.8 · p. 47

6.9 Langevin equations with fast noise and friction

  • 100x computational acceleration was achieved for the fast-noise Langevin system, whose slow-variable moments concurred with GLA, numerically suggesting weak convergence and Boltzmann-Gibbs preservation.The plotted quantities include the first and second moments of the dominantly slow variable and the first moment of the dominantly fast variable.Source: §6.9 · p. 48, Figure 21 · p. 49

7 Appendix

The appendix proves the deterministic and stochastic convergence results by reducing coordinate transforms and controlling the resulting dynamics. The stochastic proof then uses distributional convergence to establish F-convergence.

7.1 Proof of theorems 1.1 and 1.2

  • The ODE proof reduces the coordinate transform to the identity and establishes the needed inequalities using bounded trajectories and regularity of the vector fields.The proof combines intermediate lemmas to obtain the inequalities required for Theorems 1.1 and 1.2.Source: §7.1 · pp. 50, 51, 55

7.2 Proof of Theorem 3.1

  • The SDE proof reduces the coordinate transform to the identity, derives estimates using Itô expansions, and uses convergence in distribution to obtain F-convergence.The proof uses independent Gaussian increments in its estimates; convergence in distribution of the slow component leads to F-convergence of the pair.Source: §7.2 · pp. 57, 58, 60
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