Batched Krylov GPU

GPU-resident batched Arnoldi / Krylov subspace methods implemented in Julia using CUDA.

This package is designed for efficient handling of many right-hand sides in parallel, with all core linear algebra operations executed on the GPU. This project focuses on numerical linear algebra for high-performance computing, with applications in time integration, model reduction, and large-scale optimization.

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Key Features

• Batched Arnoldi Method: Implements CGS2 reorthogonalization for numerical stability.
• Fully GPU-Resident: All operations stay on the device (no expensive CPU round-trips).
• High Performance: Utilizes Strided-batched cuBLAS GEMM backend.
• Operator Support: Specialized support for operators of the form: $$A = A_0 + \mathrm{diag}(d)$$
• Robust: Features active trajectory masking and graceful breakdown handling.
• Research Ready: Designed for research, experimentation, and easy extension.

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Installation

You can install the package directly from GitHub:

using Pkg
Pkg.add(url="[https://github.com/Micik24/Batched_Krylov_GPU.jl](https://github.com/Micik24/Batched_Krylov_GPU.jl)")

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Quick Start

Here is a simple example demonstrating how to run the batched Arnoldi process.

using BatchedKrylovGPU
using CUDA

# 1. Define Problem Dimensions
n     = 100      # Matrix size
m     = 20       # Krylov subspace dimension
batch = 8        # Number of right-hand sides

# 2. Setup Operator A = A0 + diag(d)
A0 = CUDA.randn(Float32, n, n)
d  = CUDA.randn(Float32, n)

# 3. Initialize Vectors
X0 = CUDA.randn(Float32, n, batch)

# 4. Run Batched Arnoldi
Vall, Hall, beta0, active = batched_arnoldi(A0, d, X0; m = m)

Outputs

• Vall: Arnoldi basis vectors. Shape: (n, m+1, batch)
• Hall: Upper Hessenberg matrices. Shape: (m+1, m, batch)
• beta0: Initial normalization coefficients for each trajectory.
• active: Boolean mask indicating which trajectories remained active throughout the iteration.

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Requirements

• Julia: ≥ 1.9
• Packages: CUDA.jl
• Hardware: NVIDIA GPU with CUDA support

Note: Continuous Integration (CI) runs CPU-only correctness tests. Full functionality and performance benefits require a CUDA-capable GPU.

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Testing

Basic correctness tests are included and automatically run via GitHub Actions. The test suite verifies:

• API stability
• Output dimensions
• Orthogonality of the Arnoldi basis
• Correct handling of batched trajectories

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Project Status

This is research code. The API may change, and performance tuning is ongoing.

Contributions, discussions, and experimentation are welcome!

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License

MIT License