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Erlangga, Y.A., Vuik, C. and Oosterlee, C.W. (2004) On a Class of Preconditioners for Solving the Helmholtz Equation. Applied Numerical Mathematics, 50, 409-425.
https://doi.org/10.1016/j.apnum.2004.01.009

has been cited by the following article:

• TITLE: Preconditioned Neural Network Solvers for the Frequency-Domain Wave Equation with Nearly Analytic Discretization

  AUTHORS: Shili Pang, Chao Lang

  KEYWORDS: Deep Neural Networks, Preconditioned Iterative Methods, BICGSTAB, Frequency-Domain Wave Equation, Nearly Analytic Discretization

  JOURNAL NAME: Advances in Pure Mathematics, Vol.16 No.3, March 23, 2026

  ABSTRACT: The nearly analytic discretization of the frequency-domain wave equation produces large-scale, sparse, and ill-conditioned linear system, which challenge conventional iterative solvers. To mitigate this problem, we employ deep neural networks to construct an approximate preconditioner, which is then embedded within BICGSTAB algorithm. This integration retains the convergence characteristics of classical methods while effectively accelerating the iterative process. To be specific, the proposed neural network-based preconditioned solver achieves faster convergence and improved stability compared with classical preconditioned solvers. Numerical tests on representative models show that the method reduces iterative numbers without sacrificing solution accuracy, suggesting its effectiveness for scalable and high-fidelity simulations of frequency-domain wave propagation.