TY - UNPB
T1 - Multigrid-augmented deep learning preconditioners for the Helmholtz equation.
AU - Azulay, Yael
AU - Treister, Eran
N1 - DBLP License: DBLP's bibliographic metadata records provided through http://dblp.org/ are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.
PY - 2022/3/14
Y1 - 2022/3/14
N2 - In this paper, we present a data-driven approach to iteratively solve the discrete heterogeneous Helmholtz equation at high wavenumbers. In our approach, we combine classical iterative solvers with convolutional neural networks (CNNs) to form a preconditioner which is applied within a Krylov solver. For the preconditioner, we use a CNN of type U-Net that operates in conjunction with multigrid ingredients. Two types of preconditioners are proposed 1) U-Net as a coarse grid solver, and 2) U-Net as a deflation operator with shifted Laplacian V-cycles. Following our training scheme and data-augmentation, our CNN preconditioner can generalize over residuals and a relatively general set of wave slowness models. On top of that, we also offer an encoder-solver framework where an "encoder" network generalizes over the medium and sends context vectors to another "solver" network, which generalizes over the right-hand-sides. We show that this option is more robust and efficient than the stand-alone variant. Lastly, we also offer a mini-retraining procedure, to improve the solver after the model is known. This option is beneficial when solving multiple right-hand-sides, like in inverse problems. We demonstrate the efficiency and generalization abilities of our approach on a variety of 2D problems.
AB - In this paper, we present a data-driven approach to iteratively solve the discrete heterogeneous Helmholtz equation at high wavenumbers. In our approach, we combine classical iterative solvers with convolutional neural networks (CNNs) to form a preconditioner which is applied within a Krylov solver. For the preconditioner, we use a CNN of type U-Net that operates in conjunction with multigrid ingredients. Two types of preconditioners are proposed 1) U-Net as a coarse grid solver, and 2) U-Net as a deflation operator with shifted Laplacian V-cycles. Following our training scheme and data-augmentation, our CNN preconditioner can generalize over residuals and a relatively general set of wave slowness models. On top of that, we also offer an encoder-solver framework where an "encoder" network generalizes over the medium and sends context vectors to another "solver" network, which generalizes over the right-hand-sides. We show that this option is more robust and efficient than the stand-alone variant. Lastly, we also offer a mini-retraining procedure, to improve the solver after the model is known. This option is beneficial when solving multiple right-hand-sides, like in inverse problems. We demonstrate the efficiency and generalization abilities of our approach on a variety of 2D problems.
U2 - https://doi.org/10.48550/arXiv.2203.11025
DO - https://doi.org/10.48550/arXiv.2203.11025
M3 - Preprint
BT - Multigrid-augmented deep learning preconditioners for the Helmholtz equation.
ER -