Multigrid-augmented deep learning preconditioners for the Helmholtz equation

Understand

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.