### Abstract

We present a framework using the Quantized Tensor Train (QTT) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the QTT decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations. For a broad range of problems, computational and storage costs of the inversion scheme are extremely modest O(logN) and once the inverse is computed, it can be applied in O(NlogN). We analyze the QTT ranks for hierarchically low rank matrices and discuss its relationship to commonly used hierarchical compression techniques such as FMM and HSS. We prove that the QTT ranks are bounded for translation-invariant systems and argue that this behavior extends to non-translation invariant volume and boundary integrals. For volume integrals, the QTT decomposition provides an efficient direct solver requiring significantly less memory compared to other fast direct solvers. We present results demonstrating the remarkable performance of the QTT-based solver when applied to both translation and non-translation invariant volume integrals in 3D. For boundary integral equations, we demonstrate that using a QTT decomposition to construct preconditioners for a Krylov subspace method leads to an efficient and robust solver with a small memory footprint. We test the QTT preconditioners in the iterative solution of an exterior elliptic boundary value problem (Laplace) formulated as a boundary integral equation in complex, multiply connected geometries.

Original language | English (US) |
---|---|

Pages (from-to) | 145-169 |

Number of pages | 25 |

Journal | Journal of Computational Physics |

Volume | 334 |

DOIs | |

State | Published - Apr 1 2017 |

### Fingerprint

### Keywords

- Complex geometries
- Fast multipole methods
- Hierarchical matrix compression and inversion
- Integral equations
- Preconditioned iterative solver
- Tensor Train decomposition

### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)
- Computer Science Applications

### Cite this

*Journal of Computational Physics*,

*334*, 145-169. https://doi.org/10.1016/j.jcp.2016.12.051

**A Tensor-Train accelerated solver for integral equations in complex geometries.** / Corona, Eduardo; Rahimian, Abtin; Zorin, Denis.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 334, pp. 145-169. https://doi.org/10.1016/j.jcp.2016.12.051

}

TY - JOUR

T1 - A Tensor-Train accelerated solver for integral equations in complex geometries

AU - Corona, Eduardo

AU - Rahimian, Abtin

AU - Zorin, Denis

PY - 2017/4/1

Y1 - 2017/4/1

N2 - We present a framework using the Quantized Tensor Train (QTT) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the QTT decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations. For a broad range of problems, computational and storage costs of the inversion scheme are extremely modest O(logN) and once the inverse is computed, it can be applied in O(NlogN). We analyze the QTT ranks for hierarchically low rank matrices and discuss its relationship to commonly used hierarchical compression techniques such as FMM and HSS. We prove that the QTT ranks are bounded for translation-invariant systems and argue that this behavior extends to non-translation invariant volume and boundary integrals. For volume integrals, the QTT decomposition provides an efficient direct solver requiring significantly less memory compared to other fast direct solvers. We present results demonstrating the remarkable performance of the QTT-based solver when applied to both translation and non-translation invariant volume integrals in 3D. For boundary integral equations, we demonstrate that using a QTT decomposition to construct preconditioners for a Krylov subspace method leads to an efficient and robust solver with a small memory footprint. We test the QTT preconditioners in the iterative solution of an exterior elliptic boundary value problem (Laplace) formulated as a boundary integral equation in complex, multiply connected geometries.

AB - We present a framework using the Quantized Tensor Train (QTT) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the QTT decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations. For a broad range of problems, computational and storage costs of the inversion scheme are extremely modest O(logN) and once the inverse is computed, it can be applied in O(NlogN). We analyze the QTT ranks for hierarchically low rank matrices and discuss its relationship to commonly used hierarchical compression techniques such as FMM and HSS. We prove that the QTT ranks are bounded for translation-invariant systems and argue that this behavior extends to non-translation invariant volume and boundary integrals. For volume integrals, the QTT decomposition provides an efficient direct solver requiring significantly less memory compared to other fast direct solvers. We present results demonstrating the remarkable performance of the QTT-based solver when applied to both translation and non-translation invariant volume integrals in 3D. For boundary integral equations, we demonstrate that using a QTT decomposition to construct preconditioners for a Krylov subspace method leads to an efficient and robust solver with a small memory footprint. We test the QTT preconditioners in the iterative solution of an exterior elliptic boundary value problem (Laplace) formulated as a boundary integral equation in complex, multiply connected geometries.

KW - Complex geometries

KW - Fast multipole methods

KW - Hierarchical matrix compression and inversion

KW - Integral equations

KW - Preconditioned iterative solver

KW - Tensor Train decomposition

UR - http://www.scopus.com/inward/record.url?scp=85009288360&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85009288360&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2016.12.051

DO - 10.1016/j.jcp.2016.12.051

M3 - Article

AN - SCOPUS:85009288360

VL - 334

SP - 145

EP - 169

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -