Name
Abstract | ||
|---|---|---|
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. |
Year | DOI | Venue |
|---|---|---|
2011 | 10.1137/100803031 | SIAM J. Scientific Computing |
Keywords | Field | DocType |
algebraic multigrid method,grid transfer operator,proposed framework,energy minimization,multilevel hierarchy,krylov-based strategy,hermitian positive definite case,algebraic multigrid,general use,grid transfer,general case,general interpolation strategy,interpolation | Conjugate gradient method,Mathematical optimization,Algebra,Linear system,Generalized minimal residual method,Mathematical analysis,Positive-definite matrix,Interpolation,Relaxation (iterative method),Hermitian matrix,Mathematics,Multigrid method | Journal |
Volume | Issue | ISSN |
33 | 2 | 1064-8275 |
Citations | PageRank | References |
16 | 0.98 | 10 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Luke Olson | 1 | 235 | 21.93 |
| Jacob B. Schroder | 2 | 60 | 7.93 |
| Raymond S. Tuminaro | 3 | 145 | 15.07 |