Research on
Accuracy Assurance and Auto‐tuning
Takahiro Katagiri (Information Technology Center, Nagoya University, E‐mail: katagiri@cc.nagoya‐u.ac.jp)
Takeshi Ogita (Tokyo Woman’s Christian University, E‐mail: ogita@lab.twcu.ac.jp)
Katsuhisa Ozaki (Shibaura Intitule of Technology, E‐mail:ozaki@sic.shibaura‐it.ac.jp)

Verified Numerical Computations for Linear Systems
Introduction
The target problem is to solve a large-scale linear system, , ∈,, ∈where is a dense matrix.
We produce an approximate solution and its error bound using a supercomputer.

Results
Error Bounds:Dim.
1*105
2*105
3*105
4*105
5*105
E‐bound 3.38e‐07 8.01e‐06 3.29e‐06 1.52e‐05 6.35e‐06
Computing Times:

Verification Methods

Ratio of Computing Times

Ratio of Computing Times: an approximate inverse of: an approximate solution of: the identity matrix
1 is satisfied, thenIf

6.5

5

6

4.5

5.5

45

3.5

4.5

.

34100000

200000300000400000A

We adopt the maximum norm for the above inequalities.
Ref. S. Oishi, S. M. Rump: Fast verification of solutions of matrix
equations, Numer. Math., 90:4 (2002), pp. 755-773.

Experimental Settings
 FUJITSU Supercomputer PRIMEHPC FX100 in Nagoya University
 Total Nodes: 2,880
 Cores per node: 32
 Peak per node: 1.126TFLOPS (Binary64)
 Memory per node : 32GB
 Set as a random matrix.
1,1, … , 1 .
 Set such that

500000600000

2.5
200

400

Dimension

B

600
A800

Nodes

BMethod A: #MPI procs. == #nodes
Method B: #MPI procs. == 2*#nodes
Ratio: Time for approximate solutions (PDGESV in ScaLAPCK) isnormalized to 1.
Theoretical Ratio (of operations) is 9.

The actual ratio of computing time is much less than 9!
Large‐scale Problem: Up to 1,750,000 dimensional problem, we have produced an approximate
solution and its error bound using 2,500 nodes on FX100.

Dynamic Selection of Preconditioners by Deep Learning
Introduction

ResultSelection of preconditioners for sparse iterative solver is one of crucial
issues for many numerical simulations. It is difficult to select the best
preconditioners without expert knowledge. In this research, we aredeveloping an auto-tuning (AT) method for the selection by using deep
leaning (DL). To adapt DL for the selection, we utilize feature image forinput sparse matrices.

The Best Preconditioner

Our method is based on conventional AT approach with DL by Yamada,et. al, in 2018. In this method, color image for the input sparse matrix isused. In our example, the best preconditioners is dynamically changed
according to time step on an application. Hence dynamic selection ofpreconditioners is required.
To solve the above new requirement, we propose an approach for thecolor images for learning phase. In the application, elements of right-handside vector for liner equations, are dynamically changed. To utilize thisfeature, we make a new image by dividing each pixel by the element by row.

ILU0
prediction
JACOBI
ILU0

ILU0

real

Experimental Settings
 FFVC (FrontFlow/violet Cartesian): A simulator for three dimensional
Unsteady incompressible heat flow with cartesian mesh is used.
*According to a simulation parameter, number of iterations for linear
equation solver is dramatically increase. → Need of dynamic
preconditioner selection.
 Xabclib is used for sparse iterative solver (Restart GMRES method).
 Tensorflow 2.2.0 is used for DL tool.

140Estimation Accuracy by the Proposal AT Method

JACOBI

Proposed Method

 Accelerating Time (a simulation parameter for FFVC) is set to0.3.
 6 Kinds of preconditioners (None, Jacobi, SSOR, ILU0D, ILU0,andILUT ) on Xabclib are checked.
 Time step on simulation: 0 [s] to 800 [s].
The best preconditioner Number of the best preconditioner
JACOBI
1959

1938

215135This research is a study of FY2020 master thesis by Ryoto Yamamoto
in Graduate School of Informatics, Nagoya University.

Information Technology Center, Nagoya University
http://www.icts.nagoya‐u.ac.jp/en/center/