An energy-minimizing interpolation for robust multigrid methods
SIAM Journal on Scientific Computing
Multigrid
On Generalizing the Algebraic Multigrid Framework
SIAM Journal on Numerical Analysis
Algebraic Multigrid Solvers for Complex-Valued Matrices
SIAM Journal on Scientific Computing
A New Petrov-Galerkin Smoothed Aggregation Preconditioner for Nonsymmetric Linear Systems
SIAM Journal on Scientific Computing
Towards Adaptive Smoothed Aggregation ($\alpha$SA) for Nonsymmetric Problems
SIAM Journal on Scientific Computing
General Constrained Energy Minimization Interpolation Mappings for AMG
SIAM Journal on Scientific Computing
Generalizing smoothed aggregation-based algebraic multigrid
Generalizing smoothed aggregation-based algebraic multigrid
A Quasi-algebraic Multigrid Approach to Fracture Problems Based on Extended Finite Elements
SIAM Journal on Scientific Computing
Design considerations for a flexible multigrid preconditioning library
Scientific Programming
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Algebraic multigrid methods solve sparse linear systems $Ax=b$ by automatic construction of a multilevel hierarchy. This hierarchy is defined by grid transfer operators that must accurately capture algebraically smooth error relative to the relaxation method. We propose a methodology to improve grid transfers through energy minimization. The proposed strategy is applicable to Hermitian, non-Hermitian, definite, and indefinite problems. Each column of the grid transfer operator $P$ is minimized in an energy-based norm while enforcing two types of constraints: a defined sparsity pattern and preservation of specified modes in the range of $P$. A Krylov-based strategy is used to minimize energy, which is equivalent to solving $A P_j = \boldsymbol{0}$ for each column $j$ of $P$, with the constraints ensuring a nontrivial solution. For the Hermitian positive definite case, a conjugate gradient (CG-)based method is utilized to construct grid transfers, while methods based on generalized minimum residual (GMRES) and CG on the normal equations (CGNR) are explored for the general case. The approach is flexible, allowing for arbitrary coarsenings, unrestricted sparsity patterns, straightforward long-distance interpolation, and general use of constraints, either user-defined or auto-generated. We conclude with numerical evidence in support of the proposed framework.