Symposium1

Academic Center for Computing and Media Studies, Kyoto University

Natural systems are often complex and dynamic. Reductionism is not universally applicable for natural complex systems found in biology and elsewhere where behavior is driven by complex interactions between many components acting together in time nonlinear dynamic systems. In this talk, I will introduce a minimalist paradigm, empirical dynamic modeling (EDM), for studying non-linear systems. It is a data-driven approach that uses time-series information to study a system by reconstructing its attractor (a geometric object) that embodies the rules of a full set of equations for the system. Particularly, I will introduce a few recent works of visual analytics to support the identification and mechanistic interpretation of system states using integration EDM with visualization techniques, leads to our understanding of complex dynamics.

Department of Applied Physics, Nagoya University

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.

Faculty of Economics and Information, Gifu Shotoku Gakuen University

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.