2011/10/20 by Haim Avron, Avron, Haim, Sivan Toledo +1 · 1 citation
Computer Science · Decision Sciences · #Complexity and Algorithms in Graphs #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.1110.4437
openalex publication_date 2011/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define the notion of effective stiffness and show that it can used to build sparsifiers, algorithms that sparsify linear systems arising from finite-element discretizations of PDEs. In particular, we show that sampling O(nlog n) elements according to probabilities derived from effective stiffnesses yields a high quality preconditioner that can be used to solve the linear system in a small number of iterations. Effective stiffness generalizes the notion of effective resistance, a key ingredient of recent progress in developing nearly linear symmetric diagonally dominant (SDD) linear solvers. Solving finite elements problems is of considerably more interest than the solution of SDD linear systems, since the finite element method is frequently used to numerically solve PDEs arising in scientific and engineering applications. Unlike SDD systems, which are relatively easy to solve, there has been limited success in designing fast solvers for finite element systems, and previous algorithms usually target discretization of limited class of PDEs like scalar elliptic or 2D trusses. Our sparsifier is general; it applies to a wide range of finite-element discretizations. A sparsifier does not constitute a complete linear solver. To construct a solver, one needs additional components (e.g., an efficient elimination or multilevel scheme for the sparsified system). Still, sparsifiers have been a critical tools in efficient SDD solvers, and we believe that our sparsifier will become a key ingredient in future fast finite-element solvers.