Akira Imakura

A block Arnoldi-type contour integral spectral projection method for solving generalized eigenvalue problems

Abstract For generalized eigenvalue problems, we consider computing all eigenvalues located in a certain region and their corresponding eigenvectors. Recently, contour integral spectral projection methods have been proposed for solving such problems. In this study, from the analysis of the relationship between the contour integral spectral projection and the Krylov subspace, we conclude that the Rayleigh–Ritz-type of the contour integral spectral projection method is mathematically equivalent to the Arnoldi method with the projected vectors obtained from the contour integration. By this Arnoldi-based interpretation, we then propose a block Arnoldi-type contour integral spectral projection method for solving the eigenvalue problem.

Error bounds of Rayleigh---Ritz type contour integral-based eigensolver for solving generalized eigenvalue problems

We investigate contour integral-based eigensolvers for computing all eigenvalues located in a certain region and their corresponding eigenvectors. In this paper, we focus on a Rayleigh–Ritz type method and analyze its error bounds. From the results of our analysis, we conclude that the Rayleigh–Ritz type contour integral-based eigensolver with sufficient subspace size can achieve high accuracy for target eigenpairs even if some eigenvalues exist outside but near the region.

Simulations of Core-collapse Supernovae in Spatial Axisymmetry with Full Boltzmann Neutrino Transport

We present the first results of our spatially axisymmetric core-collapse supernova simulations with full Boltzmann neutrino transport, which amount to a time-dependent 5-dimensional (2 in space and 3 in momentum space) problem in fact. Special relativistic effects are fully taken into account with a two-energy-grid technique. We performed two simulations for a progenitor of 11.2M, employing different nuclear equations-of-state (EOSs): Lattimer and Swestys EOS with the incompressibility of K = 220MeV (LS EOS) and Furusawas EOS based on the relativistic mean field theory with the TM1 parameter set (FS EOS). In the LS EOS the shock wave reaches ~700km at 300ms after bounce and is still expanding whereas in the FS EOS it stalled at ~200km and has started to recede by the same time. This seems to be due to more vigorous turbulent motions in the former during the entire post-bounce phase, which leads to higher neutrino-heating efficiency in the neutrino-driven convection. We also look into the neutrino distributions in momentum space, which is the advantage of the Boltzmann transport over other approximate methods. We find non-axisymmetric angular distributions with respect to the local radial direction, which also generate off-diagonal components of the Eddington tensor. We find that the r {\theta}-component reaches ~10% of the dominant rr-component and, more importantly, it dictates the evolution of lateral neutrino fluxes, dominating over the {\theta}{\theta}-component, in the semi-transparent region. These data will be useful to further test and possibly improve the prescriptions used in the approximate methods.

Block Krylov-type complex moment-based eigensolvers for solving generalized eigenvalue problems

Complex moment-based eigensolvers for solving interior eigenvalue problems have been studied because of their high parallel efficiency. Recently, we proposed the block Arnoldi-type complex moment-based eigensolver without a low-rank approximation. A low-rank approximation plays a very important role in reducing computational cost and stabilizing accuracy in complex moment-based eigensolvers. In this paper, we develop the method and propose block Krylov-type complex moment-based eigensolvers with a low-rank approximation. Numerical experiments indicate that the proposed methods have higher performance than the block SS–RR method, which is one of the most typical complex moment-based eigensolvers.

Three-dimensional Boltzmann-Hydro code for core-collapse in massive stars II. The Implementation of moving-mesh for neutron star kicks

We present a newly developed moving-mesh technique for the multi-dimensional Boltzmann-Hydro code for the simulation of core-collapse supernovae (CCSNe). What makes this technique different from others is the fact that it treats not only hydrodynamics but also neutrino transfer in the language of the 3+1 formalism of general relativity (GR), making use of the shift vector to specify the time evolution of the coordinate system. This means that the transport part of our code is essentially general relativistic although in this paper it is applied only to the moving curvilinear coordinates in the flat Minknowski spacetime, since the gravity part is still Newtonian. The numerical aspect of the implementation is also described in detail. Employing the axisymmetric two-dimensional version of the code, we conduct two test computations: oscillations and runaways of proto-neutron star (PNS). We show that our new method works fine, tracking the motions of PNS correctly. We believe that this is a major advancement toward the realistic simulation of CCSNe.

Efficient and scalable calculation of complex band structure using Sakurai-Sugiura method

Complex band structures (CBSs) are useful to characterize the static and dynamical electronic properties of materials. Despite the intensive developments, the first-principles calculation of CBS for over several hundred atoms are still computationally demanding. We here propose an efficient and scalable computational method to calculate CBSs. The basic idea is to express the Kohn-Sham equation of the real-space grid scheme as a quadratic eigenvalue problem and compute only the solutions which are necessary to construct the CBS by Sakurai-Sugiura method. The serial performance of the proposed method shows a significant advantage in both run-time and memory usage compared to the conventional method. Furthermore, owing to the hierarchical parallelism in Sakurai-Sugiura method and the domain-decomposition technique for real-space grids, we can achieve an excellent scalability in the CBS calculation of a boron and nitrogen doped carbon nanotube consisting of more than 10,000 atoms using 2,048 nodes (139,264 cores) of Oakforest-PACS.

Alternating Optimization Method Based on Nonnegative Matrix Factorizations for Deep Neural Networks

The backpropagation algorithm for calculating gradients has been widely used in computation of weights for deep neural networks (DNNs). This method requires derivatives of objective functions and has some difficulties finding appropriate parameters such as learning rate. In this paper, we propose a novel approach for computing weight matrices of fully-connected DNNs by using two types of semi-nonnegative matrix factorizations (semi-NMFs). In this method, optimization processes are performed by calculating weight matrices alternately, and backpropagation (BP) is not used. We also present a method to calculate stacked autoencoder using a NMF. The output results of the autoencoder are used as pre-training data for DNNs. The experimental results show that our method using three types of NMFs attains similar error rates to the conventional DNNs with BP.

An Auto-Tuning Technique of the Weighted Jacobi-Type Iteration Used for Preconditioners of Krylov Subspace Methods

The Jacobi iteration is often used for preconditioners with high parallel efficiency of Krylov subspace methods to solve very large linear systems. However, these preconditioners do not always show great improvement of the convergence rate, because of the strict convergence condition and the poor convergence property of the Jacobi iteration. In order to resolve this difficulty, we recently introduced the weighted Jacobi-type iteration which has a weight parameter and a scaling diagonal matrix, and proposed the optimization technique for its weight parameter. As its efficient development, in this paper, we propose an auto-tuning technique not only for the weight parameter but also for the scaling diagonal matrix of the weighted Jacobi-type iteration used for preconditioners. The numerical experiments indicate that our auto-tuning technique is well played to solve very large linear systems.

Parallel Implementation of the Nonlinear Semi-NMF Based Alternating Optimization Method for Deep Neural Networks

For computing weights of deep neural networks (DNNs), the backpropagation (BP) method has been widely used as a de-facto standard algorithm. Since the BP method is based on a stochastic gradient descent method using derivatives of objective functions, the BP method has some difficulties finding appropriate parameters such as learning rate. As another approach for computing weight matrices, we recently proposed an alternating optimization method using linear and nonlinear semi-nonnegative matrix factorizations (semi-NMFs). In this paper, we propose a parallel implementation of the nonlinear semi-NMF based method. The experimental results show that our nonlinear semi-NMF based method and its parallel implementation have competitive advantages to the conventional DNNs with the BP method.

Block SS–CAA: A complex moment-based parallel nonlinear eigensolver using the block communication-avoiding Arnoldi procedure

Abstract Complex moment-based parallel eigensolvers have been actively studied owing to their high parallel efficiency. In this paper, we propose a block SS–CAA method, which is a complex moment-based parallel nonlinear eigensolver that makes use of the block communication-avoiding Arnoldi procedure. Numerical experiments indicate that the proposed method has higher performance compared with traditional complex moment-based nonlinear eigensolvers, i.e., the block SS–Hankel and Beyn methods.