**Plenary Talk**

**Presenter:**

### Prof. Hiroaki Natsukawa

Academic Center for Computing and Media Studies, Kyoto University

**Title:**

—

**Abstract:**

**Biography:**

Hiroaki Natsukawa is a junior associate professor / senior lecturer in Academic Center for Computing and Media Studies, Kyoto University. He received a Ph.D. in engineering from Kyoto University in 2013. Although formally trained as a researcher in the field of biomedical engineering at Kyoto University, He has successfully crossed fields into other areas such as information visualization, neuroscience, and more recently into nonlinear dynamics. Currently, he has worked in the field of information visualization and his work focuses on developing visual analytics systems enabling data-driven analytical reasoning by empirical dynamic modeling in collaboration with UC San Diego. He is currently an associate editor of Journal of Visualization, was a The 36th Annual Meeting of Japan Biomagnetism and Bioelectromagneics Society Local Committe, IEEE PacificVis 2019 Poster Co-chairs, IEEE PacificVis 2018 Publication Chair, The 47th Symposium of Visualization Society of Japan Organizer, NICOGRAPH International 2017 Local Committe Chair, JSST2016 Publication Chair, JHBM18 Local Committee.

**Invited Talk**

**Presenter:**

### Prof. Tomohiro Sogabe

Department of Applied Physics, Nagoya University

**Title:**

Computation of Singular Values for Generalized Tensor Sum

(Joint work with Asuka Ohashi (National Institute of Technology, Kagawa)

**Abstract:**

We consider computing singular values of the generalized tensor sum of the form

$$ T:=I_n \otimes I_m \otimes A + I_n \otimes B \otimes I_\ell + C \otimes I_m \otimes I_\ell, \tag{1}$$ where \(I\) is the \(n \times n\) identity matrix and \(A \in {\bf \textit{R }}^{\ell \times \ell}, B \in {\bf \textit{R }}^{m \times m}, C \in {\bf \textit{R }}^{n \times n}.\) The mathematical symbol \(\otimes\) denotes tensor product (or Kronecker’s product). A simple example of the tensor product is given below. $$ A \otimes B := \begin{bmatrix} a_{11}B & a_{12}B \\ a_{21}B & a_{22}B \end{bmatrix} = \begin{bmatrix} a_{11}b_{11} & a_{12}b_{11} & a_{11}b_{12} & a_{12}b_{12} \\ a_{21}b_{11} & a_{22}b_{11} & a_{21}b_{12} & a_{22}b_{12} \\ a_{11}b_{21} & a_{12}b_{21} & a_{11}b_{22} & a_{12}b_{22} \\ a_{21}b_{21} & a_{22}b_{21} & a_{21}b_{22} & a_{22}b_{22} \end{bmatrix}$$ where \( A := \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \). Similarly, \( A \otimes B \otimes C \) is calculated recursively by \( A \otimes (B \otimes C) \) or \( (A \otimes B) \otimes C \).

The size of the generalized tensor sum \( T \) in (1) is of \( n^3 \times n^3 \), which can be extremely large even if matrices \( A, B, C \) are small. Indeed, for \( A, B, C \) being \(1,000 \times 1,000\) matrices, the generalized tensor sum \( T \) can be a \(1,000,000,000 \times 1,000,000,000 \) matrix.

Though the generalized tensor sum \( T \) can be very large, it is easy to compute some fundamental quantities of linear algebra, such as determinant and eigenvalues. In fact, the eigenvalues of \( T \) can be written as the sum of eigenvalues of relatively much smaller matrices \( A, B, \) and \( C \) than \( T \). On the other hand, unlike eigenvalues, it is difficult to compute the singular values of \( T \) since there is no such a simple relation between the singular values of \( T \) and the singular values of \( A, B, \) and \( C \).

In this talk, we present recent progress on efficient computational algorithms [1,2,3] for some specific singular values of the generalized tensor sum \( T \), which are based on the use of the notion of numerical multilinear algebra.

References

[1] A. Ohashi, T. Sogabe, On computing maximum/minimum singular values of a generalized tensor sum, Electronic Transaction on Numerical Analysis, 43 (2015), pp. 244-254.

[2] A. Ohashi, T. Sogabe, On computing the minimum singular value of a tensor sum, Special Matrices, 7 (2019), pp. 95-106.

[3] A. Ohashi, T. Sogabe, On computing an arbitrary singular value of a tensor sum, in preparation.

**Biography:**

Tomohiro Sogabe was graduated from the department of Applied Physics, the University of Tokyo in 2001 and received Ph.D. from the same university in 2006. He worked at Nagoya university as an assistant professor, and Aichi prefectural university as an associate professor. He is currently an associate professor at the department of Applied Physics, Nagoya University, Japan. He served as editors of JSIAM Letters, the Transactions of JSIAM, and currently he serves as an editor of JJIAM journal, Springer. He will be a member of the board of directors of JSIAM from June 2021. His research interests include numerical linear algebra, numerical multilinear algebra, and scientific computing. He published more than 70 international journal papers and 15 domestic journal papers include applied mathematics journals, computational physics journals and quantum computing journal, such as AMC, AML, ETNA, JCAM, JJIAM, NLAA for applied mathematics; IEEE TMTT, JCP, PRB, PRE for computational physics; QIC for quantum computing journal.

**Tutorial Talk**

**Presenter:**

### Prof. Kuniyoshi Abe

Faculty of Economics and Information, Gifu Shotoku Gakuen University

**Title:**

A Numerical Study of Parallel Variants of GPBiCG Method with Vanilla Strategy for Solving Linear Equations

(Joint work with Soichiro Ikuno and Gerard L.G. Sleijpen)

**Abstract:**

The hybrid Bi-conjugate gradient (Bi-CG) methods such as Bi-CG stabilized (Bi-CGSTAB), Generalized Product-type based on Bi-CG (GPBiCG) are well-known for efficiently solving linear equations, but we have seen the convergence behavior with a long stagnation phase. In such cases, it is important to have Bi-CG coefficients that are as accurate as possible. We introduce the stabilization strategy for improving the accuracy of the Bi-CG coefficients.

On present petascale high-performance computing hardware, the main bottleneck for efficient parallelization is the inner products which require a global reduction. The parallel variants of Bi-CGSTAB reducing the number of global communication phases and hiding the communication latency have been proposed. In this paper, therefore, following the analogy of Cools et al., we design parallel variants of GPBiCG, and examine the convergence speed of the parallel variants with the stabilization strategy.

**Biography:**

Kuniyoshi Abe obtained Ph. D at Nagoya University in 1999. He began his career as Assistant professor at Anna National College of Technology in 1998. He moved to Advanced Computing Center, RIKEN as a Contract researcher in 1999. He was associate professor at Faculty of Economics and Information, Gifu Shotoku University in 2003, and is professor from 2012. His research field is numerical linear algebra, especially, he is interested in fast and iterative solvers for linear equations.