Parallel matching-based AMG preconditioners for elliptic equations discretized by IgA
For computational scientists and engineers solving large-scale IgA problems, this work shows that AMG preconditioners can be effective in high-performance computing environments, addressing a key bottleneck in high-order and 3D simulations.
This paper investigates algebraic multigrid (AMG) preconditioners for large, ill-conditioned linear systems arising from isogeometric analysis (IgA) discretizations, demonstrating robust and scalable performance on distributed-memory and GPU-accelerated architectures using the PSCToolkit.
Isogeometric analysis (IgA) offers enhanced approximation capabilities for the discretization of elliptic boundary-value problems, yet it results in large, sparse, and increasingly ill-conditioned linear systems due to higher interconnectivity among degrees of freedom. In particular, the discretization with tensor-product B-splines or NURBS of degree $p$ on a mesh with $n$ elements per parametric direction leads to symmetric positive-definite systems of the form $K\mathbf{u} = \mathbf{F}$, where the matrix bandwidth and condition number scale unfavorably with both $p$ and spatial dimension $d$. To address the computational challenges posed by such systems, especially in three-dimensional or high-order scenarios, Krylov subspace methods with specialized preconditioners become essential. This paper investigates the efficacy of algebraic multigrid (AMG) preconditioners tailored for IgA-based discretizations, with a focus on performance in modern high-performance computing (HPC) environments. Leveraging the Parallel Sparse Computation Toolkit (PSCToolkit), we explore distributed-memory and GPU-accelerated strategies for solving large-scale problems. The study assesses algorithmic efficiency and scalability across a range of benchmark tests. The results demonstrate that AMG preconditioners can achieve robust and scalable performance, confirming their potential as practical solvers for large IgA systems in engineering and scientific applications.