Algebraic Solvers

The Algebraic Solvers research area encompasses fundamental research for scalable linear solvers, eigen solvers, and tensor solvers, targeting multiscale, coupled systems of equations.  These solvers are one of the most common computational kernels in scientific applications of interest to the Department of Energy. Efficient, scalable, and reliable algorithms are crucial for the success of large-scale simulations and data generation for AI methods.

To achieve maximal efficiency, the solvers need to exploit various structures.  Novel applied mathematics research is expanding structure-exploiting methods across two main themes:

• Discretization-aware solvers deliver structure-aware algebraic multigrid (AMG) methods, semi-structured AMG, and high-order methods.  Our solvers team collaborates closely with the discretization team to co-design multilevel and multigrid linear and nonlinear solvers to better exploits the properties of the coupled multi-physics equations.

• Sparsity-aware linear algebra algorithms exploit both structural sparsity and data sparsity in the matrices and operators. These include traditional sparse direct solvers, hierarchical low-rank sparse direct solvers and preconditioners, and sparse linear, nonlinear and tensor eigen solvers that exploit the eigenvector localization properties.

Our solvers introduce new randomized algorithms and reduced and/or mixed precision algorithms to accelerate the computation and improve memory and energy efficiency.  Our solvers capabilities are delivered to SciDAC Partnerships via our packages available through the FASTMath Software Catalog.