Computational Physics Group

Karel Matous










The image-based multiscale multigrid solver, preconditioner, and reduced order model

D. Yushu and K. Matous

Center for Shock Wave-processing of Advanced Reactive Materials

Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame,
Indiana 46556, USA.


    We present a novel image-based multiscale multigrid solver that can efficiently address the computational complexity associated with highly heterogeneous systems. This solver is developed based on an image-based, multiresolution model that enables reliable data flow between corresponding computational grids and provides large data compression. A set of inter-grid operators is constructed based on the microstructural data which remedies the issue of missing coarse grid information. Moreover, we develop an image-based multiscale preconditioner from the multiscale coarse images which does not traverse through any intermediate grid levels and thus leads to a faster solution process. Finally, an image-based reduced order model is designed by prolongating the coarse-scale solution to approximate the fine-scale one with improved accuracy. The numerical robustness and efficiency of this image-based computational framework is demonstrated on a two-dimensional example with high degrees of data heterogeneity and geometrical complexity.


    In this work, we propose a novel image-based multiscale multigrid solver, preconditioner, and the reduced order model. An image-based inter-grid operator is developed via incorporating the microstructural information from the multiresolution scheme (i.e., data driven SVB model). A two-stage approach is established for computing the inter-grid operator, which preserves the flux along the grid lines. This multigrid solver is robust for extreme coefficient contrasts and exhibits near-optimal convergence rate.
    A new image-based multiscale preconditioner is developed utilizing the coarse SVB image and the image-based inter-grid operator. Thus, relaxations on the intermediate grids are omitted, resulting in a simpler formulation and a lighter computational demand per iteration. This preconditioner shows high efficiency for ill-conditioned systems, and exhibits greatly improved performance compared to traditional preconditioners such as the Jacobi and/or the Incomplete Cholesky.
    The IROM reduces the number of DOFs by converting the fine level problem to a coarse grid one. It is demonstrated that the IROM reduces the error from the geometrical coarse solutions and restores detailed solution characteristics that are filtered due to the direct data compression.
    This work opens a new possibility for solving a system of linear equations associated with data heterogeneity, which is a fundamental problem in a large array of engineering and science disciplines. Moreover, the multiscale image-based approach is applicable to other fields such as uncertainty quantification, data compression, and adaptive multiscale modeling. The development of the 3D image-based multiscale approach with a larger system size and its parallelization are both important future directions.


This work was supported by the Department of Energy, National Nuclear Security Administration, under the award No. DENA0002377 as part of the Predictive Science Academic Alliance Program II. We would like to give special thanks to Dr. Waad Subber for his help in the development of the multiscale preconditioner.

Download paper here