Hierarchical Interpolative Factorization Preconditioner for Parabolic Equations

Hierarchical Interpolative Factorization for Elliptic Operators: Integral Equations

This paper introduces the hierarchical interpolative factorization for integral equations (HIF-IE) associated with elliptic problems in two and three dimensions. This factorization takes the form of an approximate generalized LU decomposition that permits the efficient application of the discretized operator and its inverse. HIF-IE is based on the recursive skeletonization algorithm but incorpo...

Hierarchical Interpolative Factorization for Elliptic Operators: Differential Equations

This paper introduces the hierarchical interpolative factorization for elliptic partial differential equations (HIF-DE) in two (2D) and three dimensions (3D). This factorization takes the form of an approximate generalized LU/LDL decomposition that facilitates the efficient inversion of the discretized operator. HIF-DE is based on the nested dissection multifrontal method but uses skeletonizati...

Distributed-memory Hierarchical Interpolative Factorization

The hierarchical interpolative factorization (HIF) offers an efficient way for solving or preconditioning elliptic partial differential equations. By exploiting locality and low-rank properties of the operators, the HIF achieves quasi-linear complexity for factorizing the discrete positive definite elliptic operator and linear complexity for solving the associated linear system. In this paper, ...

Interpolative Butterfly Factorization

This paper introduces the interpolative butterfly factorization for nearly optimal implementation of several transforms in harmonic analysis, when their explicit formulas satisfy certain analytic properties and the matrix representations of these transforms satisfy a complementary low-rank property. A preliminary interpolative butterfly factorization is constructed based on interpolative low-ra...

AILU for Helmholtz problems: A new Preconditioner Based on the Analytic Parabolic Factorization.∗