Shuhei Kudo

We consider the problem of transforming a given symmetric matrix into a nearly block diagonal form by permutation of its rows and columns. Such a transformation is useful as preconditioning to accelerate the convergence of an eigenvalue solver, but the problem of finding an optimal permutation that maximizes the Frobenius norms of the diagonal blocks is NP-complete. We formulate this problem as QUBO (Quadratic Unconstrained Binary Optimization) and solve it using D-Wave Advantage quantum annealing machine. Experimental results on small problems show that the true minimum can be obtained with high probability. We also discuss how to improve the mapping of the problem onto the physical qubit network to increase the size of the problems that can be solved.

Our performance benchmark of HPL-AI on the supercomputer Fugaku was awarded in the 55th top500 at ISC20. The effective performance was 1.42 EFlop/s, and the world's first achievement to exceed the wall of exascale in a floating-point arithmetic benchmark. Due to the novelty of HPL-AI, there are few guidelines for large systems and several drawbacks to the large-scale benchmark. It is not enough to replace FP64 operations solely to those on FP32 or FP16. At the least, we need thoughtful numerical analysis for lower-precision arithmetic and introduction of optimization techniques on extensive computing such as on Fugaku. In the poster, we give some comments on the accuracy, implementation, performance improvement, and report on the Exa-scale benchmark on Fugaku.

The one-sided block Jacobi method (OSBJ) has attracted attention as a fast and accurate algorithm for the singular value decomposition (SVD). The computational kernel of OSBJ is orthogonalization of a column block pair, which amounts to computing the SVD of this block pair. Hari proposes three methods for this partial SVD, and we found through numerical experiments that the variant named "V2", which is based on the Cholesky QR method, is the fastest variant and achieves satisfactory accuracy. While it is a good news from a practical viewpoint, it seems strange considering the well-known instability of the Cholesky QR method. In this paper, we perform a detailed error analysis of the V2 variant and explain why and when it can be used to compute the partial SVD accurately. Thus, our results provide a theoretical support for using the V2 variant safely in the OSBJ method.