Miscellaneous Methods

StochasticCompositeAlgorithm(algs, choice_function)

StochasticCompositeAlgorithm: Multi-Method Composite Algorithm

Composite algorithm that automatically switches between multiple SDE solvers based on problem characteristics.

Method Properties

• Approach: Multi-method solving with automatic switching
• Adaptivity: Changes methods during integration
• Flexibility: Combines strengths of different algorithms

Parameters

• algs::Tuple: Tuple of algorithms to switch between
• choice_function::Function: Function determining which algorithm to use

When to Use

• Problems with changing characteristics during integration
• When different regions require different solution approaches
• Combining methods for different regimes (e.g., stiff/nonstiff)
• When no single method is optimal for entire domain

Choice Function

The choice_function(integrator) should return an integer indicating which algorithm from algs to use:

function choice_function(integrator)
    if stiff_region(integrator.u, integrator.t)
        return 1  # Use first algorithm (e.g., implicit)
    else
        return 2  # Use second algorithm (e.g., explicit)
    end
end

Algorithm Features

• Automatic method switching during integration
• Maintains continuity across method transitions
• Combines computational efficiency with robustness
• Can handle complex multi-scale problems

References

• Composite algorithm methodology for SDEs

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:

du = v dt
dv = f(v,u) dt - γv dt + g(u)√(2γ) dW

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)

PCEuler(ggprime; theta=1/2, eta=1/2)

PCEuler: Predictor-Corrector Euler Method (Nonstiff)

A predictor-corrector variant of the Euler-Maruyama method requiring analytic derivatives of the diffusion term, with adjustable implicitness parameters for drift-diffusion coupling.

Method Properties

• Strong Order: 0.5 (in the Itô sense)
• Weak Order: 1.0
• Time stepping: Fixed time step only
• Noise types: General noise with available derivative information
• SDE interpretation: Itô only

Parameters

• ggprime::Function: The required derivative of the diffusion term
  • For scalar problems: ggprime = g * ∂g/∂x
  • For multi-dimensional problems: ggprime_k = Σ_{j=1...M, i=1...D} g^(j)_i * ∂g^(j)_k/∂x_i
  • where g^(j) corresponds to the noise vector due to the j-th noise channel
  • Must match the in-place/out-of-place specification of the problem
• theta::Real = 0.5: Degree of implicitness in the drift term (default: 0.5)
• eta::Real = 0.5: Degree of implicitness in the diffusion term (default: 0.5)

When to Use

• Problems requiring specific drift-diffusion coupling
• When analytical ggprime function is available
• Specialized predictor-corrector applications
• When the derivative g*g' provides stability or accuracy benefits

Algorithm Description

The method uses a predictor-corrector approach with the specific requirement of computing the derivative of the diffusion coefficient. This additional information allows for improved handling of drift-diffusion interactions through the adjustable parameters θ and η.

Limitations

• Requires analytical computation of ggprime (cannot be approximated)
• Fixed time step only (no adaptive versions available)
• Limited to Itô interpretation

References

• Jentzen, A., Kloeden, P.E., "The numerical approximation of stochastic partial differential equations", Milan J. Math. 77, 205–244 (2009). https://doi.org/10.1007/s00032-009-0100-0
• Originally introduced in PR #88 (commit 42e2510) by Tatsuhiro Onodera (2018)

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

Kloeden, P.E., Platen, E., Numerical Solution of Stochastic Differential Equations. Springer. Berlin Heidelberg (2011)

SimplifiedEM()

SimplifiedEM: Simplified Euler-Maruyama Method (High Weak Order)

Simplified version of the Euler-Maruyama method optimized for weak convergence with reduced computational cost.

Method Properties

• Strong Order: Not optimized for strong convergence
• Weak Order: 1.0
• Time stepping: Fixed step size
• Noise types: All forms (diagonal, non-diagonal, scalar, and colored noise)
• SDE interpretation: Itô
• Computational cost: Reduced compared to standard EM

When to Use

• Monte Carlo simulations where weak convergence is sufficient
• Problems where computational efficiency is more important than strong accuracy
• Large ensemble simulations
• When only statistical properties (expectations, moments) are needed

Algorithm Features

• Simplified implementation reducing computational overhead
• Maintains weak order 1.0 convergence
• More efficient than standard EM for weak convergence applications
• Handles general noise structures

Weak vs Strong Convergence

• Optimized for E[f(XT)] convergence, not pathwise |XT - X_h| convergence
• Ideal for computing expectations and statistical properties
• Less suitable when individual trajectory accuracy is important

References

• Kloeden, P.E., Platen, E., "Numerical Solution of Stochastic Differential Equations", Springer. Berlin Heidelberg (2011)