Application of the Sparse Direct Solver MUMPS to Hybridizable Discontinuous Galerkin Discretization for Wave Modeling

  • Amestoy, Patrick (Mumps Technologies)
  • Buttari, Alfredo (Université de Toulouse, CNRS, IRIT)
  • Faucher, Florian (Inria–TotalEnergies–UPPA)
  • L'Excellent, Jean-Yves (Mumps Technologies)
  • Mary, Theo (Sorbonne Université, CNRS, LIP6)

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We consider the Hybridizable Discontinuous Galerkin (HDG) discretization method [4] for the numerical simulation of time-harmonic waves in acoustic and elastic media. The discretization results in large sparse systems of linear equations of the form AX = B to be solved for different frequencies. Although a significant amount of memory is required, these systems can be efficiently solved using sparse direct solvers, that exploit the block structure of A and that factorize A only once for multiple right-hand sides. In this study, we revisit the potential of the massively-parallel sparse multifrontal solver MUMPS and its recent features to efficiently solve our problem. We analyze in more details the influence of the HDG discretization which, being more appropriate for high-order polynomials, can lead to denser blocks. We also study the consideration of acoustic and elastic wave propagation on the data sparsity of the matrix, and the capacity to use low rank compression possibly with mixed precision [1,2] in the sparse direct solver for efficiency (time and memory footprint). We provide numerical experiments in a parallel setting for the simulation of wave propagation with the code hawen [3]. In particular, we consider a 3D earth domain with an adapted mesh (refined for topography and coarser below), exploiting p-adaptivity. We also illustrate how today challenges related to heterogeinity of computers, mixed-precision arithmetics, new models and applications, motivate research and developments in the field of sparse direct solvers. [1] P. R. Amestoy, A. Buttari, J.-Y. L’Excellent and T. Mary, Performance and scalability of the Block Low-Rank multifrontal factorization on multicore architectures, ACM Transactions on Mathematical Software, 45(1), 2019. [2] P. R. Amestoy, O. Boiteau, A. Buttari, M. Gerest, F. Jézéquel, J.-Y. L’Excellent and T. Mary. Mixed precision low rank approximations and their application to block low rank LU factorization, IMA Journal of Numerical Analysis, 43(4), 2023. [3] F. Faucher. hawen: time-harmonic wave modeling and inversion using hybridizable discontinuous Galerkin discretization, Journal of Open Source Software, 6, 2021. [4] H. Pham, F. Faucher and H. Barucq. On the implementation of Hybridizable Discontinuous Galerkin discretization for linear anisotropic elastic wave equation: Voigt-notation and stabilization, Research report RR-9533, INRIA, 2023.