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In this paper, we present an original preconditioner to solve Poisson equation for strongly heterogeneous media. We suggest using the Conjugate Gradient method with the preconditioner based on the solution of the Poisson equation for homogeneous media. Corresponding operator is easy to invert by spectral method, where spectral decomposition is applied in two spatial directions and the Gauss elimination method is applied to solve a series of 1D problems. We illustrate that use of such precondtioner strongly decreases the number of iterations to solve the original Poisson equation, moreover the number of iterations weakly depends on the problem size. Implementation of the suggested approach using modern GPUs allows solving problems of up to the size of \(1000^3\) voxels.

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Implementation of the Spectral Preconditioner to Solve Poisson Equation

  • 会议论文

  • 发表日期：

    2026年

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  • Vadim Lisitsa,
  • Aleksei Manaev,
  • Sergey Solovyev

摘要

In this paper, we present an original preconditioner to solve Poisson equation for strongly heterogeneous media. We suggest using the Conjugate Gradient method with the preconditioner based on the solution of the Poisson equation for homogeneous media. Corresponding operator is easy to invert by spectral method, where spectral decomposition is applied in two spatial directions and the Gauss elimination method is applied to solve a series of 1D problems. We illustrate that use of such precondtioner strongly decreases the number of iterations to solve the original Poisson equation, moreover the number of iterations weakly depends on the problem size. Implementation of the suggested approach using modern GPUs allows solving problems of up to the size of \(1000^3\) voxels.