Solving partial differential equations is crucial to analysing and predicting complex, large-scale physical systems but pushes conventional high-performance computers to their limits. Application specific photonic processors are an exciting computing paradigm for building efficient, ultrafast hardware accelerators. Here, we investigate the synergy between multigrid based partial differential equations solvers and low latency photonic matrix vector multipliers. We propose a mixed-precision photonic multigrid solver, that offloads the computationally demanding smoothening procedure to the optical domain. We test our approach on an integrated photonic accelerator operating at 2 GSPS solving a Poisson and Schrödinger equation. By offloading the smoothening operation to the photonic system, we can reduce the digital operation by more than 80%. Finally, we show that the photonic multigrid solver potentially reduces digital operations by up to 97 % in lattice quantum chromodynamics (LQCD) calculations, enabling an order-of-magnitude gain in computational speed and efficiency.
A Nonlinear Multigrid Steady-State Solver for Microflow
LMAM & School of Mathematical Sciences, Peking University（北京大学数学科学学院）；HEDPS & CAPT, LMAM & School of Mathematical Sciences, Peking University（北京大学数学科学学院）
An implicit $ P $-multigrid flux reconstruction method for simulation of locally preconditioned unsteady Navier-Stokes equations at low Mach numbers
本文发展了一个$ P $ -多重网格求解器，用隐式高阶方法模拟了低马赫数下局部预处理的非定常可压缩纳维-斯托克斯方程。具体而言，高阶通量重建/校正程序通过重建（FR/CPR）方法用于空间离散和高阶时间积分进行的显式第一阶段，单对角隐式龙格库塔（ESDIRK）方法。采用局部预处理来降低可压缩纳维-斯托克斯方程在低马赫数下的刚度，并仅在伪瞬态延拓中进行，以保证ESDIRK方法的高阶精度。我们采用元素雅可比平滑更新解决方案在不同的$ P $ -层次的$ P $ -多重网格求解器。通过自由流马赫数为0.005时的等熵涡传播验证了新求解器在低马赫数流动模拟中的高阶时空精度。通过数值实验研究了多项式阶数对$ P $ -多重网格法收敛速度的影响，包括$\text{Ma} = 0.001$下NACA 0012翼型的二维无粘和粘性流场，以及$\text{Ma} = 0.001$下球体的三维无粘流场。然后应用$ P $ -多重网格求解器对SD 7003机翼在$ 8^\circ $攻角下，雷诺数为60000，马赫数为0.1或0.01时的转捩流场进行了粗分辨率模拟。
Acceleration of an algebraic multigrid pressure solver using graph neural networks
Faculty of Mechanical Engineering, Delft University of Technology, The Netherlands（荷兰代尔夫特理工大学机械工程学院）；Maritime Research Institute Netherlands, The Netherlands（荷兰海事研究院）；Department of Aeronautics, Imperial College London, United Kingdom（英国伦敦帝国理工学院航空系）
Solving the (Navier-)Stokes equations with space and time adaptivity using deal.II
Technical University of Berlin（柏林工业大学）；Charles University（查理大学）；Ruhr University Bochum（波鸿鲁尔大学）；Otto von Guericke University Magdeburg（马格德堡奥托·冯·格里克大学）；Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机性研究所）
