VRPIC: Variance Reduction for Particle-in-Cell method

A CUDA/C++ implementation of the particle-in-cell (PIC) method of solving the Vlasov–Poisson equation equipped with the Variance Reduction (VRPIC). This project provides a high-performance GPU-accelerated framework for simulating plasma dynamics with reduced statistical noise, and enabling more accurate long-time evolution of distribution functions.

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Overview

The Vlasov–Poisson equation describes the evolution of a charged particle system under self-consistent electric fields. Traditional particle-in-cell (PIC) methods suffer from inherent statistical noise when the signal is weak. The proposed method addresses this challenge by taking advantage of the correlation between the non-equilibrium and equilibrium simulations via importance weights. For example, the density and temperature profiles for the Landau Damping test case at the finite time can be estimated with a lower variance compared to standard PIC.

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Features

• Fully GPU-accelerated: Uses CUDA to parallelize moment computation, particle updates, and field (Poisson) solver.
• Variance reduction (VR): Implements control variate methods to reduce noise in moment computations.
• Importance weighting: Dynamically adjusts particle weights using local Maxwellian-Boltzmann distribution as control variate.
• Least biased moment conservation: Deploys MxE formulation to ensure weight conservation during the kick process.
• Self-consistent field solving: Solves the Poisson equation using FFT method.
• Post-processing output: Dumps moment fields for visualization and diagnostics.

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🛠️ Build Instructions

Requirements

• CUDA Toolkit (>= 11.x recommended)
• C++ compiler with C++11 or higher
• CMake

Build

git clone https://github.com/mohsensadr/VRPIC.git
cd VRPIC
mkdir bin && cd bin
cmake ..
make

By default, cmake compile the code for A100 GPU. In case of other architectures, provide cmake with the flag CMAKE_CUDA_ARCHITECTURES, for example

cmake .. -DCMAKE_CUDA_ARCHITECTURES=80

Execution

The compiled executable can be run by

./main N_GRID_X\
       N_GRID_Y\
       N_PARTICLES\
       CFL\
       NSteps\
       Lx\
       Ly\
       threadsPerBlock\
       deposition_mode\
       VRMode\
       RhsMode\
       [pdf_type]\
       [pdf_params...]

where

deposition_mode: brute | tiling | sorting
VRMode: basic | MXE
RhsMode: MC | VR

For example:

./main 100 100 1000000 0.1 200 12.5663706144 12.5663706144 256 sorting mxe vr cosine 0.05 0.5

For the command line of executioning different test cases, see the header in src/main.cpp.