We describe our software package Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) recently publicly released. BLOPEX is available as a stand-alone serial library, as an external package to PETSc (Portable, Extensible Toolkit for Scientific Computation, a general purpose suite of tools developed by Argonne National Laboratory for the scalable solution of partial differential equations and related problems), and is also built into hypre (High Performance Preconditioners, a scalable linear solvers package developed by Lawrence Livermore National Laboratory). The present BLOPEX release includes only one solver-the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method for symmetric eigenvalue problems. hypre provides users with advanced high-quality parallel multigrid preconditioners for linear systems. With BLOPEX, the same preconditioners can now be efficiently used for symmetric eigenvalue problems. PETSc facilitates the integration of independently developed application modules, with strict attention to component interoperability, and makes BLOPEX extremely easy to compile and use with preconditioners that are available via PETSc. We present the LOBPCG algorithm in BLOPEX for hypre and PETSc. We demonstrate numerically the scalability of BLOPEX by testing it on a number of distributed and shared memory parallel systems, including a Beowulf system, SUN Fire 880, an AMD dual-core Opteron workstation, and IBM BlueGene/L supercomputer, using PETSc domain decomposition and hypre multigrid preconditioning. We test BLOPEX on a model problem, the standard 7-point finite-difference approximation of the 3-D Laplacian, with the problem size in the range of 10 5 -10 8 .
SUMMARYLinear buckling constraints are important in structural topology optimization for obtaining designs that can support the required loads without failure. During the optimization process, the critical buckling eigenmode can change; this poses a challenge to gradient-based optimization and can require the computation of a large number of linear buckling eigenmodes. This is potentially both computationally difficult to achieve and prohibitively expensive. In this paper, we motivate the need for a large number of linear buckling modes and show how several features of the block Jacobi conjugate gradient (BJCG) eigenvalue method, including optimal shift estimates, the reuse of eigenvectors, adaptive eigenvector tolerances and multiple shifts, can be used to efficiently and robustly compute a large number of buckling eigenmodes. This paper also introduces linear buckling constraints for level-set topology optimization. In our approach, the velocity function is defined as a weighted sum of the shape sensitivities for the objective and constraint functions. The weights are found by solving an optimization sub-problem to reduce the mass while maintaining feasibility of the buckling constraints. The effectiveness of this approach in combination with the BJCG method is demonstrated using a 3D optimization problem.
Abstract. Gradient iterations for the Rayleigh quotient are simple and robust solvers to determine a few of the smallest eigenvalues together with the associated eigenvectors of (generalized) matrix eigenvalue problems for symmetric matrices. Sharp convergence estimates for the Ritz values and Ritz vectors are derived for various steepest descent/ascent gradient iterations. The analysis shows that poorest convergence of the eigenvalue approximations is attained in a three-dimensional invariant subspace; explicit convergence estimates are then derived by means of a mini-dimensional analysis.
The combination of positron emission tomography (PET) with magnetic resonance (MR) imaging opens the way to more accurate diagnosis and improved patient management. At present, the data acquired by PET and MR scanners are essentially processed separately, and the search for ways to improve accuracy of the tomographic reconstruction via synergy of the two imaging techniques is an active area of research. The aim of the collaborative computational project on PET and MR (CCP-PETMR), supported by the UK engineering and physical sciences research council (EPSRC), is to accelerate research in synergistic PET-MR image reconstruction by providing an open access software platform for efficient implementation and validation of novel reconstruction algorithms.
The combination of positron emission tomography (PET) with magnetic resonance (MR) imaging opens the way to more accurate diagnosis and improved patient management. At present, the data acquired by PET and MR scanners are essentially processed separately, and the search for ways to improve accuracy of the tomographic reconstruction via synergy of the two imaging techniques is an active area of research. The aim of the collaborative computational project on PET and MR (CCP-PETMR), supported by the UK engineering and physical sciences research council (EPSRC), is to accelerate research in synergistic PET-MR image reconstruction by providing an open access software platform for efficient implementation and validation of novel reconstruction algorithms.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.