Miscellaneous Methods

RandomEM()

RandomEM: Random Euler Method (RODE)

Euler method for Random Ordinary Differential Equations (RODEs) with random parameters.

Method Properties

• Problem type: Random ODEs (RODEs)
• Strong Order: 1.0 (for deterministic part)
• Randomness: Handles random parameters, not Brownian motion
• Time stepping: Fixed step size

When to Use

• Random ODEs with random parameters but no Brownian motion
• Uncertainty quantification with parameter randomness
• Problems with random coefficients or initial conditions
• Monte Carlo simulation of deterministic systems with random inputs

RODE vs SDE

• RODE: Random parameters, deterministic evolution
• SDE: Fixed parameters, stochastic (Brownian) evolution

References

• Random ordinary differential equation methods

RandomHeun()

RandomHeun: Random Heun Method (RODE)

Heun method for Random Ordinary Differential Equations with improved accuracy.

Method Properties

• Problem type: Random ODEs (RODEs)
• Strong Order: 2.0 (for deterministic part)
• Randomness: Handles random parameters
• Time stepping: Fixed step size

When to Use

• RODEs requiring higher accuracy than RandomEM
• When computational cost per step is acceptable
• Random parameter problems needing second-order accuracy

References

• Higher-order methods for random ODEs

RandomTamedEM()

RandomTamedEM: Tamed Random Euler Method (RODE)

Tamed Euler method for RODEs with potentially explosive behavior.

Method Properties

• Problem type: Random ODEs with potential blow-up
• Approach: Taming to prevent numerical explosion
• Stability: Enhanced stability for unstable random systems
• Time stepping: Fixed step size with taming

When to Use

• RODEs that may exhibit explosive growth
• When RandomEM gives unstable or explosive solutions
• Random systems with strong nonlinearities
• Problems requiring enhanced numerical stability

Taming Mechanism

Applies taming technique to prevent numerical blow-up while maintaining accuracy for well-behaved solutions.

References

• Tamed methods for random differential equations

BAOAB(;gamma=1.0, scale_noise=true)

BAOAB: Langevin Dynamics Integrator (Specialized)

Specialized integrator for Langevin dynamics in molecular dynamics simulations, particularly effective for configurational sampling.

Method Properties

• Problem type: Langevin dynamics (second-order SDEs)
• Structure: Position-velocity formulation
• Sampling: Designed for equilibrium sampling
• Time stepping: Fixed step size
• Conservation: Preserves equilibrium distributions

Parameters

• gamma::Real = 1.0: Friction coefficient
• scale_noise::Bool = true: Whether to scale noise appropriately

System Structure

Designed for Langevin systems:

\[\begin{align*} du &= v \, dt \\ dv &= f(v,u) \, dt - γv \, dt + g(u) \sqrt{2γ} \, dW \end{align*}\]

where:

• $u$: position coordinates
• $v$: velocity coordinates
• $γ$: friction coefficient
• $f(v,u)$: force function
• $g(u)$: noise scaling function

When to Use

• Molecular dynamics simulations with Langevin thermostat
• Configurational sampling of molecular systems
• Equilibrium sampling from canonical ensemble
• Second-order SDEs with damping and noise

Algorithm Features

• BAOAB splitting: B(kick) - A(drift) - O(Ornstein-Uhlenbeck) - A(drift) - B(kick)
• Preserves correct equilibrium distribution
• Robust and efficient for molecular sampling
• Well-suited for long-time integration

References

• Leimkuhler B., Matthews C., "Robust and efficient configurational molecular sampling via Langevin dynamics", J. Chem. Phys. 138, 174102 (2013)

IIF1M(;nlsolve=NLSOLVEJL_SETUP())

IIF1M: Integrating Factor Method 1 (Semi-Linear)

First-order integrating factor method for semi-linear SDEs with stiff linear parts.

Method Properties

• Strong Order: 1.0
• Weak Order: 1.0
• Time stepping: Fixed or adaptive
• Problem type: Semi-linear SDEs with stiff linear components
• Treatment: Exponential integrator approach

Parameters

• nlsolve: Nonlinear solver configuration

When to Use

• Semi-linear SDEs: du = (L*u + N(u))dt + g(u)dW where L is stiff linear operator
• Problems amenable to integrating factor techniques
• When exponential integrators are appropriate
• Stiff linear parts with nonlinear perturbations

Algorithm Description

Applies integrating factor exp(L*t) to handle stiff linear part exactly while treating nonlinear parts numerically.

References

• Integrating factor methods for stiff SDEs

IIF2M(;nlsolve=NLSOLVEJL_SETUP())

IIF2M: Integrating Factor Method 2 (Semi-Linear)

Second-order integrating factor method for semi-linear SDEs.

Method Properties

• Strong Order: 2.0
• Weak Order: 2.0
• Time stepping: Fixed or adaptive
• Problem type: Semi-linear SDEs with stiff linear components
• Treatment: Higher-order exponential integrator

Parameters

• nlsolve: Nonlinear solver configuration

When to Use

• When higher accuracy than IIF1M is needed
• Semi-linear problems requiring second-order convergence
• More expensive but more accurate than IIF1M

References

• Higher-order integrating factor methods for SDEs

IIF1Mil(;nlsolve=NLSOLVEJL_SETUP())

IIF1Mil: Integrating Factor Milstein Method (Semi-Linear)

Integrating factor method combined with Milstein correction for semi-linear SDEs.

Method Properties

• Strong Order: 1.0 (with Milstein correction)
• Weak Order: 1.0
• Time stepping: Fixed or adaptive
• Problem type: Semi-linear SDEs with stiff linear components
• Treatment: Exponential integrator with Milstein correction

Parameters

• nlsolve: Nonlinear solver configuration

When to Use

• Semi-linear SDEs requiring Milstein-type accuracy
• When both stiff linear treatment and higher-order stochastic accuracy are needed
• Alternative to IIF1M with enhanced stochastic treatment

References

• Integrating factor methods with Milstein correction