jh200041-NAH
Akihiro Fujii (Faculty of Informatics, Kogakuin University, Japan)

Innovative Multigrid Methods II
Co-PI: K. Nakajima2,4, M. Bolten8
M.Kawai4, A.Ida2, G.Wellein9, C.Alappat9, M.Schreiber11,
T.Hoshino2, S.Ohshima6, T.Hanawa2, O.Marques10, K.Ono5,
M.Iizuka5, T.Iwashita1, Y.Takahashi7, R.Speck12, R.Yoda2, Y.Chen2
1: Hokkaido U., 2: U. Tokyo, 3: Kogakuin U., 4: RIKEN R-CCS, 5: Kyushu U.,
6: Nagoya U., 7: Doshisha U. 8: U. Wuppertal*, 9: FAU*, 10: LBNL, USA
11: Technical U. of Munich*, 12: Juelich Supercomputing Centre*, *:GermanyBackground• Multigrid (MG) for linear equations
ü Scalable multilevel method for solving linear equationsü GMG (Geometrical Multigrid) and AMG (Algebraic)ü Generally, number of iterations until convergence for multigrid method
is kept constant as the problem size changes.
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• Robust and efficient algorithms for GMG and AMG methods,and PinST towards the Exascale/Post-Moore Era.

Development: 3-Year Project (FY.2020 to 2022)• GMG and AMG:
ü Research on smoothers• New approaches for PinST
Ono, Iizuka, Fujii, Bolten, Iwashita, Takahashi, Speck, Schreiber,Yoda, Chen
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.

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
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.

Plan for FY.2020
This project is a succession of previous JHPCN project“Innovative Multigrid methods”. The current state of theresearch will be presented as the research results of theprevious project. Here, we list up our plans on FY.2020.
As a first year, we will do fundamental researches including
the following items.
• GMGüüüüüResearches on sparse matrix data structure like SELL-C- sSpecific application optimization including pGW3D-FVM
Communication optimization with Pipelined CG solverFile IO optimization in Mesh Generation by IME
Scalability Evaluation by Adaptive Multilevel hCGA (AM-hCGA)• AMG

ü Researches on sparse matrix data structure like SELL-C- s
ü Researches on smoothers like preconditioned Chebyshev
smoothers and MS-BMC-GS.
ü Communication optimization with Pipelined CG solver
ü Communication optimization on halo communication using
user-level RDMAü Further Improvement of the Convergence by Parallel
Reordering/Aggregation• PinSTüüüüü

MGRIT Preconditioning convergence analysis
Research on Parareal method for hyperbolic PDEs
TSC method application to pHEAT-3D
Research on PinST method to Navier-Stokes applications
Research on TSC method for simulation codes based on
explicit method.