An O (N) direct solver for integral equations on the plane

Eduardo Corona, Per Gunnar Martinsson, Denis Zorin

Research output: Contribution to journalArticlepeer-review


An efficient direct solver for volume integral equations with O(N) complexity for a broad range of problems is presented. The solver relies on hierarchical compression of the discretized integral operator, and exploits that off-diagonal blocks of certain dense matrices have numerically low rank. Technically, the solver is inspired by previously developed direct solvers for integral equations based on "recursive skeletonization" and "Hierarchically Semi-Separable" (HSS) matrices, but it improves on the asymptotic complexity of existing solvers by incorporating an additional level of compression. The resulting solver has optimal O(N) complexity for all stages of the computation, as demonstrated by both theoretical analysis and numerical examples. The computational examples further display good practical performance in terms of both speed and memory usage. In particular, it is demonstrated that even problems involving 107 unknowns can be solved to precision 10-10 using a simple Matlab implementation of the algorithm executed on a single core.

Original languageEnglish (US)
Pages (from-to)284-317
Number of pages34
JournalApplied and Computational Harmonic Analysis
Issue number2
StatePublished - Mar 1 2015


  • Direct solvers
  • Fast algorithms
  • Fast multipole methods
  • Hierarchical matrix compression
  • Integral equations
  • Interpolative decomposition

ASJC Scopus subject areas

  • Applied Mathematics


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