arXiv subjects

Christopher M. Siefert

Publications and source records attributed to Christopher M. Siefert.

3 recordsLinked to original sources

arXiv2026

Smoothed aggregation algebraic multigrid for problems with heterogeneous and anisotropic materials

This paper introduces a material-aware strength-of-connection measure for smoothed aggregation algebraic multigrid methods, aimed at improving robustness for scalar partial differential equations with heterogeneous and anisotropic material properties. Classical strength-of-connection measures typically rely only on matrix entries or geometric distances, which often fail to capture weak couplings across material interfaces or align with anisotropy directions, ultimately leading to poor convergence. The proposed approach directly incorporates material tensor information into the coarsening process, enabling a reliable detection of weak connections and ensuring that coarse levels preserve the true structure of the underlying problem. As a result, smooth error components are represented properly and sharp coefficient jumps or directional anisotropies are handled consistently. A wide range of academic tests and real-world applications, including thermally activated batteries and solar cells, demonstrate that the proposed method maintains robustness across material contrasts, anisotropies, and mesh variations. Scalability and parallel performance of the algebraic multigrid method highlight the suitability for large-scale, high-performance computing environments.

cs.CE↗

arXiv2023

Graph Neural Networks and Applied Linear Algebra

Sparse matrix computations are ubiquitous in scientific computing. With the recent interest in scientific machine learning, it is natural to ask how sparse matrix computations can leverage neural networks (NN). Unfortunately, multi-layer perceptron (MLP) neural networks are typically not natural for either graph or sparse matrix computations. The issue lies with the fact that MLPs require fixed-sized inputs while scientific applications generally generate sparse matrices with arbitrary dimensions and a wide range of nonzero patterns (or matrix graph vertex interconnections). While convolutional NNs could possibly address matrix graphs where all vertices have the same number of nearest neighbors, a more general approach is needed for arbitrary sparse matrices, e.g. arising from discretized partial differential equations on unstructured meshes. Graph neural networks (GNNs) are one approach suitable to sparse matrices. GNNs define aggregation functions (e.g., summations) that operate on variable size input data to produce data of a fixed output size so that MLPs can be applied. The goal of this paper is to provide an introduction to GNNs for a numerical linear algebra audience. Concrete examples are provided to illustrate how many common linear algebra tasks can be accomplished using GNNs. We focus on iterative methods that employ computational kernels such as matrix-vector products, interpolation, relaxation methods, and strength-of-connection measures. Our GNN examples include cases where parameters are determined a-priori as well as cases where parameters must be learned. The intent with this article is to help computational scientists understand how GNNs can be used to adapt machine learning concepts to computational tasks associated with sparse matrices. It is hoped that this understanding will stimulate data-driven extensions of classical sparse linear algebra tasks.

math.NA↗

arXiv2012

Automating embedded analysis capabilities and managing software complexity in multiphysics simulation part II: application to partial differential equations

A template-based generic programming approach was presented in a previous paper that separates the development effort of programming a physical model from that of computing additional quantities, such as derivatives, needed for embedded analysis algorithms. In this paper, we describe the implementation details for using the template-based generic programming approach for simulation and analysis of partial differential equations (PDEs). We detail several of the hurdles that we have encountered, and some of the software infrastructure developed to overcome them. We end with a demonstration where we present shape optimization and uncertainty quantification results for a 3D PDE application.

cs.MS↗