[Paper Review] Hierarchical interpolative factorization for elliptic operators: integral equations

Kenneth L. Ho, Lexing Ying|arXiv (Cornell University)|Jul 10, 2013

Electromagnetic Scattering and Analysis4 citations

TL;DR

This paper introduces Hierarchical Interpolative Factorization for Integral Equations (HIF-IE), a novel method that enables efficient solution of elliptic PDEs via a recursive, tree-based factorization of dense discretized integral operators. By combining matrix sparsification using interpolative decomposition with recursive dimensional reduction, HIF-IE achieves approximate generalized LU decomposition with estimated quasilinear complexity, enabling fast application of the operator and its inverse as a fast multipole method, direct solver, or preconditioner.

ABSTRACT

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 incorporates a novel combination of two key features: (1) a matrix factorization framework for sparsifying structured dense matrices and (2) a recursive dimensional reduction strategy to decrease the cost. Thus, higher-dimensional problems are effectively mapped to one dimension, and we conjecture that constructing, applying, and inverting the factorization all have linear or quasilinear complexity. Numerical experiments support this claim and further demonstrate the performance of our algorithm as a generalized fast multipole method, direct solver, and preconditioner. HIF-IE is compatible with geometric adaptivity and can handle both boundary and volume problems. MATLAB codes are freely available.

Motivation & Objective

• Address the high computational cost of solving dense linear systems arising from discretized elliptic integral equations in 2D and 3D.
• Overcome the limitations of traditional direct solvers (O(N³)) and iterative methods (slow convergence for ill-conditioned problems) in high-contrast or complex geometries.
• Develop a scalable, robust, and accurate direct solver for both boundary and volume integral equations with geometric adaptivity.
• Achieve linear or quasilinear complexity for factorization, application, and inversion of the operator, enabling use in time-stepping, inverse problems, and multiple right-hand sides.
• Provide a unified framework compatible with fast multipole methods, direct solvers, and preconditioners through a single factorization structure.

Proposed method

• Proposes HIF-IE as a hierarchical, tree-structured interpolative factorization of the dense system matrix from elliptic integral equations.
• Employs the recursive skeletonization algorithm (RS) enhanced with a novel matrix sparsification framework based on interpolative decomposition (ID) to compress off-diagonal blocks.
• Introduces recursive dimensional reduction to map higher-dimensional problems to lower-dimensional subproblems, reducing computational cost.
• Constructs an approximate generalized LU decomposition where the factorization is stored in a hierarchical tree format, enabling efficient matrix-vector multiplication.
• Uses skeletonization operators to identify low-rank structures adaptively, with basis vectors selected via ID to optimize compression and accuracy.
• Supports geometric adaptivity and handles both first- and second-kind Fredholm integral equations, including singular kernels from Laplace, Helmholtz, and other elliptic PDEs.

Experimental results

Research questions

• RQ1 Can a hierarchical interpolative factorization achieve quasilinear complexity for solving dense integral equations from 2D and 3D elliptic PDEs?
• RQ2 How does the combination of matrix sparsification via ID and recursive dimensional reduction improve the efficiency and scalability of direct solvers?
• RQ3 To what extent can HIF-IE serve as a fast multipole method, direct solver, and preconditioner with consistent performance across different problem types?
• RQ4 What is the empirical and theoretical complexity of HIF-IE, and how does it compare to existing methods like H-matrices or FMMs?
• RQ5 Can HIF-IE maintain high accuracy while significantly reducing skeleton sizes in 3D, especially at high precision?

Key findings

• HIF-IE achieves estimated quasilinear complexity for factorization, application, and inversion of the discretized operator, with numerical experiments confirming near-linear scaling.
• The method outperforms standard recursive skeletonization (RS) and rskelf3 in terms of both speed and accuracy, especially at high precision, with hifie3x restoring full accuracy at the cost of larger skeleton sizes.
• Numerical experiments show that HIF-IE scales favorably with problem size, with empirical complexity fitting well with theoretical estimates, though asymptotic convergence may not yet be reached.
• HIF-IE successfully handles both boundary and volume integral equations, including ill-conditioned first-kind problems and well-conditioned second-kind problems.
• The algorithm is compatible with geometric adaptivity and supports multiple right-hand sides and low-rank updates efficiently, making it suitable for time-stepping and inverse problems.
• MATLAB implementations are publicly available, enabling reproducibility and broader adoption in scientific computing applications.