Hiroto Tadano

Efficient Parameter Estimation and Implementation of a Contour Integral-Based Eigensolver

We consider an eigensolver for computing eigenvalues in a given domain and the corresponding eigenvectors of large-scale matrix pencils. The Sakurai-Sugiura (SS) method is an eigensolver based on complex moments given by contour integrals of matrix inverses with several shift points. This method has good parallel scalability, and is suitable for massively parallel computing environments. The SS method has several parameters, and the choice of these parameters is crucial for achieving high accuracy and good parallel performance. We discuss some numerical properties of the method, and present efficient parameter estimation techniques. We demonstrate the efficiency of our method with numerical experiments.

Application of block Krylov subspace algorithms to the Wilson-Dirac equation with multiple right-hand sides in lattice QCD

Abstract It is well known that the block Krylov subspace solvers work efficiently for some cases of the solution of differential equations with multiple right-hand sides. In lattice QCD calculation of physical quantities on a given configuration demands us to solve the Dirac equation with multiple sources. We show that a new block Krylov subspace algorithm recently proposed by the authors reduces the computational cost significantly without losing numerical accuracy for the solution of the O ( a ) -improved Wilson–Dirac equation.

Parallel Fock matrix construction with distributed shared memory model for the FMO‐MO method

A parallel Fock matrix construction program for FMO‐MO method has been developed with the distributed shared memory model. To construct a large‐sized Fock matrix during FMO‐MO calculations, a distributed parallel algorithm was designed to make full use of local memory to reduce communication, and was implemented on the Global Array toolkit. A benchmark calculation for a small system indicates that the parallelization efficiency of the matrix construction portion is as high as 93% at 1,024 processors. A large FMO‐MO application on the epidermal growth factor receptor (EGFR) protein (17,246 atoms and 96,234 basis functions) was also carried out at the HF/6‐31G level of theory, with the frontier orbitals being extracted by a Sakurai‐Sugiura eigensolver. It takes 11.3 h for the FMO calculation, 49.1 h for the Fock matrix construction, and 10 min to extract 94 eigen‐components on a PC cluster system using 256 processors.

Massively parallel implementation of 3D‐RISM calculation with volumetric 3D‐FFT

A new three‐dimensional reference interaction site model (3D‐RISM) program for massively parallel machines combined with the volumetric 3D fast Fourier transform (3D‐FFT) was developed, and tested on the RIKEN K supercomputer. The ordinary parallel 3D‐RISM program has a limitation on the number of parallelizations because of the limitations of the slab‐type 3D‐FFT. The volumetric 3D‐FFT relieves this limitation drastically. We tested the 3D‐RISM calculation on the large and fine calculation cell (20483 grid points) on 16,384 nodes, each having eight CPU cores. The new 3D‐RISM program achieved excellent scalability to the parallelization, running on the RIKEN K supercomputer. As a benchmark application, we employed the program, combined with molecular dynamics simulation, to analyze the oligomerization process of chymotrypsin Inhibitor 2 mutant. The results demonstrate that the massive parallel 3D‐RISM program is effective to analyze the hydration properties of the large biomolecular systems.

Performance comparison of parallel eigensolvers based on a contour integral method and a Lanczos method

Abstract We study the performance of a parallel nonlinear eigensolver SSEig which is based on a contour integral method. We focus on symmetric generalized eigenvalue problems (GEPs) of computing interior eigenvalues. We chose to focus on GEPs because we can then compare the performance of SSEig with that of a publicly-available software package TRLan, which is based on a thick restart Lanczos method. To solve this type of problems, SSEig requires the solution of independent linear systems with different shifts, while TRLan solves a sequence of linear systems with a single shift. Therefore, while SSEig typically has a computational cost greater than that of TRLan, it also has greater parallel scalability. To compare the performance of these two solvers, in this paper, we develop performance models and present numerical results of solving large-scale eigenvalue problems arising from simulations of modeling accelerator cavities. In particular, we identify the crossover point, where SSEig becomes faster than TRLan. The parallel performance of SSEig solving nonlinear eigenvalue problems is also studied.

Modified Block BiCGSTAB for Lattice QCD

Abstract We present results for application of block BiCGSTAB algorithm modified by the QR decomposition and the SAP preconditioner to the Wilson–Dirac equation with multiple right-hand sides in lattice QCD on 32 3 × 64 and 64 4 lattices at almost physical quark masses. The QR decomposition improves convergence behaviors in the block BiCGSTAB algorithm suppressing deviation between true residual and recursive one. The SAP preconditioner applied to the domain-decomposed lattice helps us minimize communication overhead. We find remarkable cost reduction thanks to cache tuning and reduction of number of iterations.

Application of preconditioned block BiCGGR to the Wilson–Dirac equation with multiple right-hand sides in lattice QCD

Abstract There exist two major problems in application of the conventional block BiCGSTAB method to the O ( a ) -improved Wilson–Dirac equation with multiple right-hand sides: One is the deviation between the true and the recursive residuals. The other is the convergence failure observed at smaller quark masses for enlarged number of the right-hand sides. The block BiCGGR algorithm which was recently proposed by the authors succeeds in solving the former problem. In this article we show that a preconditioning technique allows us to improve the convergence behavior for increasing number of the right-hand sides.

On Single Precision Preconditioners for Krylov Subspace Iterative Methods

Large sparse linear systems Ax= barise in many scientific applications. Krylov subspace iterative methods are often used for solving such linear systems. Preconditioning techniques are efficient to reduce the number of iterations of Krylov subspace methods. The coefficient matrix of the linear system is transformed into MAor AMin the left or right preconditioning, where Mis a preconditioning matrix. In this paper, we analyze the influence of perturbation in the computation of preconditioning of Krylov subspace methods. We show that the perturbation of preconditioner does not affect the accuracy of the approximate solution when the right preconditioning is used. Some numerical experiments illustrate the influence of preconditioners with single precision arithmetic.

A master-worker type eigensolver for molecular orbital computations

We consider a parallel method for solving generalized eigenvalue problems that arise from molecular orbital computations. We use a moment-based method that finds several eigenvalues and their corresponding eigenvectors in a given domain, which is suitable for masterworker type parallel programming models. The computation of eigenvalues using explicit moments is sometimes numerically unstable. We show that a Rayleigh-Ritz procedure can be used to avoid the use of explicit moments. As a test problem, we use the matrices that arise in the calculation of molecular orbitals. We report the performance of the application of the proposed method with several PC clusters connected through a hybrid MPI and GridRPC system.

A method for finding zeros of polynomial equations using a contour integral based eigensolver

In this paper, we present a method for finding zeros of polynomial equations in a given domain. We apply a numerical eigensolver using contour integral for a polynomial eigenvalue problem that is derived from polynomial equations. The Dixon resultant is used to derive the matrix polynomial of which eigenvalues involve roots of the polynomial equations with respect to one variable. The matrix polynomial obtained by the Dixon resultant is sometimes singular. By applying the singular value decomposition for a matrix which appears in the eigensolver, we can obtain the roots of given polynomial systems. Experimental results demonstrate the efficiency of the proposed method.