jh210026-NAH
Akihiro Fujii (Faculty of Informatics, Kogakuin University, Japan)

Innovative Multigrid Methods II
Co-PI: K. Nakajima2,4, M. Bolten9M. Kawai2, A. Ida8, G. Wellein10, C. Alappat10, M. Schreiber12,
T. Hoshino2, S. Ohshima6, T. Hanawa2, O. Marques11, K. Ono5,
M. Iizuka5, T. Iwashita1, Y. Takahashi7, R. Speck13, G. Caklovic13,
A. Miyagi14, T. Tanaka3, R. Yoda2, Y. Chen2, A. Magro2
1: Hokkaido U., 2: U. Tokyo, 3: Kogakuin U., 4: RIKEN R-CCS, 5: Kyushu U.,
6: Nagoya U., 7: Doshisha U. 8: JAMSTEC 9: U. Wuppertal*, 10: FAU*,
11: LBNL, USA,12: Technical U. of Munich*,
13: Juelich Supercomputing Centre*, 14: Taisei Corporation*:GermanyPurpose and Background
Robust and efficient algorithms for Multigrid(MG) methods
towards the Exascale/Post-Moore Era.
• Multigrid (MG) for linear equations
ü Scalable multilevel method for solving linear equationsü GMG (Geometrical Multigrid) and AMG (Algebraic)ü The parallel multigrid method is expected to be one of the mostpowerful tools on exa-scale systems.
ü Many sophisticated methods for efficiency of MG have beendeveloped for ill-conditioned problems derived from real-world
scientific and engineering applications.

• MG for parallel in Space and Timeü Parallel computation in time direction of simulation
ü It adds new dimension to ordinary parallelization of scientificsimulation calculation

Development: 3-Year Project (FY.2020 to 2022)• GMG and AMG:
ü Research on smoothers
Kawai, Ida, Nakajima, Bolten, Wellein, Alappat, SchreiberØ Multicolor-block GS smoother for AMG is the original smootherdeveloped by Our project members.
Ø SpMV based smoothers such as Chebyshev smoother areknown. We will also consider to accelerate these simple
smoothers by fast sparse matrix data structures

ü Lower precision utilization for performance
Nakajima, Hoshino, Wellein, Alappat, Schreiber, OhshimaØ We will study on how to use the lower precision calculations for
efficient multigrid solvers.
Ø First, we will investigate various lower precision usage for solvers
such as only coarse level calculation or lower precision Krylov
preconditioning.
Ø After we checked effective lower precision usage patterns, thenwe will study selection methods according to the problem matrix.

ü Acceleration technique with file IO optimization andcommunication
Hanawa, Ohshima, Fujii, Nakajima, Yoda
Ø Parallel linear solvers and mesh generation routines often need toread and write distributed matrix files. We will investigate efficient
usage of the burst buffer functionality of supercomputers for meshgeneration or sparse linear solvers.
Ø As for halo communication, we will study fast Halocommunication with user-level direct RDMA on Tofu, that will beavailable in Fugaku, post-Kei computer.

ü Evaluation with weak scaling and large sized problems
Nakajima, Fujii, Marques, Ida, Tanaka
Ø Our project already has spent several years to implement
multigrid solvers such as GMG and AMG. We will analyze theperformance of the solvers, and check the space forimprovement.

• New approaches for PinST
Ono, Iizuka, Fujii, Bolten, Iwashita, Takahashi, Speck, Schreiber,Yoda, Chen, Caklovic, Miyagi
ü Coarse level solver in Parareal method Iizuka and Ono studied theconvergence of the Parareal method for hyperbolic PDEs focusing
on coarse solver.
ü Convergence analysis on MGRIT preconditioning for linear
problems
ü Time Segmented Correction (TSC) method and its enhancement,
which is a new parallel time integration method for non-linear
problems.

Plan for FY.2021
This project is a second year of the 3-year project. The current
state of the research will be presented as the research resultsof the last year. Here, we list up our plans on FY.2021.
As a second year, we will enhance the researches including
solver evaluation on Fugaku.
• GMGüüüü
üResearches on sparse matrix data structure like SELL-C- sLower precision GMG solverSpecific application optimization including pGW3D-FVM
File IO optimization in Mesh Generation by IME
Scalability Evaluation by Adaptive Multilevel hCGA (AM-hCGA)• AMG
ü Researches on AM-hCGA algorithm without ParMETISü Researches on smoothers like preconditioned Chebyshev
smoothers and MS-BMC-GS.
ü Communication optimization on halo communication using
user-level RDMA on Flow type1.

• PinST
üüüü

MGRIT Preconditioning convergence analysis
TSC method application to pHEAT-3DPinST method to Navier-Stokes applications
PinST method for explicit method