Multigrid-augmented deep learning preconditioners for the Helmholtz equation

• URL: http://arxiv.org/abs/2203.11025v1
• Date: Mon, 14 Mar 2022 10:31:11 GMT
• Title: Multigrid-augmented deep learning preconditioners for the Helmholtz equation
• Authors: Yael Azulay and Eran Treister
• Abstract summary: We present a data-driven approach to solve the discrete heterogeneous Helmholtz equation at high wavenumbers. We combine classical iterative solvers with convolutional neural networks (CNNs) to form a preconditioner which is applied within a Krylov solver.
• Score: 4.18804572788063
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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