Implemented

Random Number Generation

[!NOTE] DiffusionX uses the high-quality Xoshiro256++ random number generator as the common entropy source across all distributions.

• Normal distribution
• Uniform distribution
• Exponential distribution
• Poisson distribution
• $\alpha$-stable distribution

Stochastic Processes Simulation

• Brownian motion
• $\alpha$-stable Lévy process
• Cauchy process
• $\alpha$-stable subordinator
• Inverse $\alpha$-stable subordinator
• Poisson process
• Fractional Brownian motion
• Continuous-time random walk
• Ornstein-Uhlenbeck process
• Langevin equation
• Generalized Langevin equation
• Subordinated Langevin equation
• Lévy walk
• Birth-death process
• Random walk
• Brownian excursion
• Brownian meander
• Gamma process
• Geometric Brownian motion
• Brownian yet non-Gaussian process

GPU Acceleration (CUDA/Metal)

The acceleration includes moment calculation and random number generation.

• Brownian motion
• $\alpha$-stable Lévy process
• Ornstein-Uhlenbeck process
• $\alpha$-stable distribution

Usage

Random Number Generation

use diffusionx::random::{normal, uniform, stable};
fn main() -> Result<(), Box<dyn std::error::Error>> {
    // Generate a normal random number with mean 0.0 and std 1.0
    let normal_sample = normal::rand(0.0, 1.0)?;
    // Generate 1000 standard normal random numbers
    let std_normal_samples = normal::standard_rands::<f64>(1000);

    // Generate a uniform random number in range [0, 10)
    let uniform_sample = uniform::range_rand(0..10)?;
    // Generate 1000 uniform random numbers in range [0, 1)
    let std_uniform_samples = uniform::standard_rands(1000);

    // Generate 1000 standard stable random numbers
    let stable_samples = stable::standard_rands(1.5, 0.5, 1000)?;

    Ok(())
}

Stochastic Process Simulation

use diffusionx::simulation::{prelude::*, continuous::Bm};
fn main() -> Result<(), Box<dyn std::error::Error>> {
    // Create standard Brownian motion object
    let bm = Bm::default();
    // Create trajectory with duration 1.0
    let traj = bm.duration(1.0)?;
    // Simulate Brownian motion trajectory with time step 0.01
    let (times, positions) = traj.simulate(0.01)?;
    println!("times: {:?}", times);
    println!("positions: {:?}", positions);

    // Calculate first-order raw moment with 1000 particles and time step 0.01
    let mean = traj.raw_moment(1, 1000, 0.01)?;
    println!("mean: {mean}");
    // Calculate second-order central moment with 1000 particles and time step 0.01
    let msd = traj.central_moment(2, 1000, 0.01)?;
    println!("MSD: {msd}");
    // Calculate EATAMSD with duration 100.0, delta 1.0, 10000 particles, time step 0.1,
    // and Gauss-Legendre quadrature order 10
    let eatamsd = bm.eatamsd(100.0, 1.0, 10000, 0.1, 10)?;
    println!("EATAMSD: {eatamsd}");
    // Calculate first passage time of Brownian motion with boundaries at -1.0 and 1.0
    let fpt = bm.fpt((-1.0, 1.0), 1000, 0.01)?;
    println!("fpt: {fpt}");
    Ok(())
}

Visualization

[!NOTE] The visualization requires the visualize feature to be enabled.

# In your Cargo.toml
[dependencies]
diffusionx = { version = "*", features = ["visualize"] }

use diffusionx::{
    simulation::{continuous::Bm, prelude::*},
};
fn main() -> Result<(), Box<dyn std::error::Error>> {
    // Create Brownian motion trajectory
    let bm = Bm::default();
    let traj = bm.duration(10.0)?;

    // Configure and create visualization
    let config = PlotConfigBuilder::default()
    .time_step(0.01)
    .output_path("brownian_motion.png")
    .caption("Brownian Motion Trajectory")
    .x_label("t")
    .y_label("B")
    .legend("bm")
    .size((800, 600))
    .backend(PlotterBackend::BitMap)
    .build()?;

    // Generate plot
    traj.plot(&config)?;

    Ok(())
}

GPU Acceleration

[!NOTE] This requires the metal or cuda feature to be enabled.

# In your Cargo.toml
[dependencies]
diffusionx = { version = "*", features = ["cuda"] }

use diffusionx::{
    simulation::continuous::Bm,
    gpu::GPUMoment,
};
fn main() -> Result<(), Box<dyn std::error::Error>> {
    let bm = Bm::<f32>::default();

    // GPU-accelerated moment calculations
    let mean = bm.mean_gpu(1.0, 100_000, 0.01)?;
    let msd = bm.msd_gpu(1.0, 100_000, 0.01)?;
    let raw_moment = bm.raw_moment_gpu(1.0, 2, 100_000, 0.01)?;
    let central_moment = bm.central_moment_gpu(1.0, 2, 100_000, 0.01)?;

    // Fractional moments are also supported
    let frac_raw = bm.frac_raw_moment_gpu(1.0, 1.5, 100_000, 0.01)?;
    let frac_central = bm.frac_central_moment_gpu(1.0, 1.5, 100_000, 0.01)?;

    println!("Mean: {mean}, MSD: {msd}");
    Ok(())
}

Architecture and Extensibility

DiffusionX is designed with a trait-based system for high extensibility and performance:

Core Traits

• ContinuousProcess: Base trait for continuous stochastic processes
• PointProcess: Base trait for point processes
• DiscreteProcess: Base trait for discrete stochastic processes
• Moment: Trait for statistical moments calculation, including (fractional) raw and central moments
• Visualize: Trait for plotting process trajectories
• GPUMoment: Trait for simulating the (fractional) moments in CUDA.

The GPUMoment trait provides GPU-accelerated statistical moment calculations. It is implemented for:

• Bm<T> - Brownian Motion
• OrnsteinUhlenbeck<T> - Ornstein-Uhlenbeck Process
• Levy<T> - Lévy Process

Extending with Custom Processes

1. Adding a New Continuous Process:

   #[derive(Debug, Clone)]
   struct MyProcess {
       // Your parameters
       // Should be `Send + Sync` for parallel computation
       // and `Clone`
   }
   
   impl ContinuousProcess for MyProcess {
       fn start(&self) -> f64 {
           0.0  // or any default value
       }
       fn simulate(
            &self,
            duration: f64,
            time_step: f64
        ) -> XResult<(Vec<f64>, Vec<f64>)> {
           // Implement your simulation logic
           todo!()
       }
   }
   

2. Implementing ContinuousProcess trait automatically provides

   • mean mean
   • msd msd
   • (fractional) raw moment raw_moment (frac_raw_moment)
   • (fractional) central moment central_moment (frac_central_moment)
   • first passage time fpt
   • occupation time occupation_time
   • TAMSD tamsd
   • visualization plot

The full example implementing the CIR process is here.

Benchmark

Performance benchmark tests compare the Rust, C++, Julia, and Python implementations, which can be found here.

License

Licensed under either of:

• Apache License, Version 2.0, (LICENSE-APACHE or https://www.apache.org/licenses/LICENSE-2.0)
• MIT license (LICENSE-MIT or https://opensource.org/licenses/MIT)

at your option.

Contribution

Unless you explicitly state otherwise, any contribution intentionally submitted for inclusion in the work by you, as defined in the Apache-2.0 license, shall be dual licensed as above, without any additional terms or conditions.

Dedicated to my brief yet unforgettable years in LZU and to XX.