# A Technique for Updating Hierarchical Skeletonization-Based Factorizations of Integral Operators

@article{Minden2016ATF, title={A Technique for Updating Hierarchical Skeletonization-Based Factorizations of Integral Operators}, author={Victor Minden and Anil Damle and Kenneth L. Ho and Lexing Ying}, journal={Multiscale Model. Simul.}, year={2016}, volume={14}, pages={42-64} }

We present a method for updating certain hierarchical factorizations for solving linear integral equations with elliptic kernels. In particular, given a factorization corresponding to some initial geometry or material parameters, we can locally perturb the geometry or coefficients and update the initial factorization to reflect this change with asymptotic complexity that is poly-logarithmic in the total number of unknowns and linear in the number of perturbed unknowns. We apply our method to…

## Figures, Tables, and Topics from this paper

## 9 Citations

A Recursive Skeletonization Factorization Based on Strong Admissibility

- Mathematics, Computer ScienceMultiscale Model. Simul.
- 2017

We introduce the strong recursive skeletonization factorization (RS-S), a new approximate matrix factorization based on recursive skeletonization for solving discretizations of linear integral…

FAST FACTORIZATION UPDATE FOR GENERAL ELLIPTIC

- 2020

For discretized elliptic equations, we develop a new factorization update algorithm 4 that is suitable for incorporating coefficient updates with large support and large magnitude in 5 subdomains.…

A fast direct solver for boundary value problems on locally perturbed geometries

- Computer Science, MathematicsJ. Comput. Phys.
- 2018

A fast direct solver for boundary value problems that are recast as boundary integral equations is presented and for problems where perturbation is localized the fast directsolver is three times faster than building a new solver from scratch.

Fast Factorization Update for General Elliptic Equations Under Multiple Coefficient Updates

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 2020

A new factorization update algorithm is developed that is suitable for incorporating coefficient updates with large support and large magnitude in subdomains for discretized elliptic equations.

FLAM: Fast Linear Algebra in MATLAB - Algorithms for Hierarchical Matrices

- Computer ScienceJ. Open Source Softw.
- 2020

Many large matrices in science and engineering possess a special hierarchical low-rank structure that enables fast multiplication and inversion, among other fundamental operations. Such matrices…

Flexibly imposing periodicity in kernel independent FMM: A multipole-to-local operator approach

- Mathematics, Computer ScienceJ. Comput. Phys.
- 2018

Parallel Skeletonization for Integral Equations in Evolving Multiply-Connected Domains

- Computer Science, MathematicsSIAM J. Sci. Comput.
- 2021

This paper presents an application that leverages a parallel implementation of skeletonization with updates in a shape optimization regime and retains the locality property afforded by the "proxy point method," and allows for a parallelized implementation where different processors work on different parts of the boundary simultaneously.

FMM-LU: A fast direct solver for multiscale boundary integral equations in three dimensions

- Computer Science, MathematicsArXiv
- 2022

This work presents a fast direct solver for boundary integral equations on complex surfaces in three dimensions, using an extension of the recently introduced strong recursive skeletonization scheme, and studies its performance in acoustic scattering at low to moderate frequencies.

A fast direct solver for integral equations on locally refined boundary discretizations and its application to multiphase flow simulations

- Computer Science, MathematicsArXiv
- 2021

A new computational approach is presented that avoids this issue by pre-constructing a fast direct solver for the wall ahead of time, computing a low-rank factorization to capture the changes due to the refinement, and solving the problem on the refined discretization via a Woodbury formula.

## References

SHOWING 1-10 OF 40 REFERENCES

Hierarchical Interpolative Factorization for Elliptic Operators : Differential Equations

- 2015

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…

A Fast Direct Solver for Structured Linear Systems by Recursive Skeletonization

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 2012

Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, an approximation of the original matrix is embedded into a larger but highly structured sparse one that allows fast factorization and application of the inverse.

A fast direct solver for boundary integral equations in two dimensions

- Mathematics
- 2003

We describe an algorithm for the direct solution of systems of linear algebraic equations associated with the discretization of boundary integral equations with non-oscillatory kernels in two…

Fast direct solvers for integral equations in complex three-dimensional domains

- Computer ScienceActa Numerica
- 2009

Methods that are currently under development for the fast, direct solution of boundary integral equations in three dimensions are discussed, based on coupling fast matrix-vector multiplication routines with conjugate-gradient-type schemes.

Data-sparse Approximation by Adaptive ℋ2-Matrices

- Mathematics, Computer ScienceComputing
- 2002

The basic ideas of ℋ- andℋ2-matrices are introduced and an algorithm that adaptively computes approximations of general matrices in the latter format is presented.

Efficient Structured Multifrontal Factorization for General Large Sparse Matrices

- Computer Science, MathematicsSIAM J. Sci. Comput.
- 2013

This work presents a framework of structured direct factorizations for general sparse matrices, including discretized PDEs on general meshes, based on the multifrontal method and hierarchically semiseparable (HSS) matrices.

A Fast Solver for HSS Representations via Sparse Matrices

- Mathematics, Computer ScienceSIAM J. Matrix Anal. Appl.
- 2006

A fast direct solver for certain classes of dense structured linear systems that works by first converting the given dense system to a larger system of block sparse equations and then uses standard sparse direct solvers.

A Fast ULV Decomposition Solver for Hierarchically Semiseparable Representations

- Mathematics, Computer ScienceSIAM J. Matrix Anal. Appl.
- 2006

We consider an algebraic representation that is useful for matrices with off-diagonal blocks of low numerical rank. A fast and stable solver for linear systems of equations in which the coefficient…

A direct solver with O(N) complexity for integral equations on one-dimensional domains

- Mathematics
- 2011

An algorithm for the direct inversion of the linear systems arising from Nyström discretization of integral equations on one-dimensional domains is described. The method typically has O(N) complexity…

Maintaining LU factors of a general sparse matrix

- Mathematics
- 1987

Abstract : The authors describe a set of procedures for computing and updating an LU factorization of a sparse matrix A, where A may be square (possibly singular) or rectangular. The procedures…