Error Analysis of the Cholesky QR-Based Block Orthogonalization Process for the One-Sided Block Jacobi SVD Algorithm

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

A fast and accurate computation method for reflective diffraction simulations

We present a new computation method for simulating reflection high-energy electron diffraction and the total-reflection high-energy positron diffraction experiments. The two experiments are used commonly for the structural analysis of material surface. The present paper improves the conventional numerical method, the multi-slice method, for faster computation, since the present method avoids the matrix-eigenvalue solver for the computation of matrix exponentials and can adopt higher-order ordinary differential equation solvers. Moreover, we propose a high-performance implementation based on multi-thread parallelization and cache-reusable subroutines. In our tests, this new method performs up to 2,000 times faster than the conventional method

Roundoff error analysis of the double exponential formula-based method for the matrix sign function

In this paper, we perform a roundoff error analysis of an integration-based method for computing the matrix sign function recently proposed by Nakaya and Tanaka. The method expresses the matrix sign function using an integral representation and computes the integral numerically by the double-exponential formula. While the method has large-grain parallelism and works well for well-conditioned matrices, its accuracy deteriorates when the input matrix is ill-conditioned or highly nonnormal. We investigate the reason for this phenomenon by a detailed roundoff error analysis.Comment: 6 pages, 1 figur

Frequent and Challenging Tasks for Learners in English-Medium Instruction Courses: A Questionnaire Survey for English Majors

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Multiscale Universal Interface: A Concurrent Framework for Coupling Heterogeneous Solvers

Concurrently coupled numerical simulations using heterogeneous solvers are powerful tools for modeling multiscale phenomena. However, major modifications to existing codes are often required to enable such simulations, posing significant difficulties in practice. In this paper we present a C++ library, i.e. the Multiscale Universal Interface (MUI), which is capable of facilitating the coupling effort for a wide range of multiscale simulations. The library adopts a header-only form with minimal external dependency and hence can be easily dropped into existing codes. A data sampler concept is introduced, combined with a hybrid dynamic/static typing mechanism, to create an easily customizable framework for solver-independent data interpretation. The library integrates MPI MPMD support and an asynchronous communication protocol to handle inter-solver information exchange irrespective of the solvers' own MPI awareness. Template metaprogramming is heavily employed to simultaneously improve runtime performance and code flexibility. We validated the library by solving three different multiscale problems, which also serve to demonstrate the flexibility of the framework in handling heterogeneous models and solvers. In the first example, a Couette flow was simulated using two concurrently coupled Smoothed Particle Hydrodynamics (SPH) simulations of different spatial resolutions. In the second example, we coupled the deterministic SPH method with the stochastic Dissipative Particle Dynamics (DPD) method to study the effect of surface grafting on the hydrodynamics properties on the surface. In the third example, we consider conjugate heat transfer between a solid domain and a fluid domain by coupling the particle-based energy-conserving DPD (eDPD) method with the Finite Element Method (FEM).Comment: The library source code is freely available under the GPLv3 license at http://www.cfm.brown.edu/repo/release/MUI