skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Scalable Nonlinear Compact Schemes

Abstract

In this work, we focus on compact schemes resulting in tridiagonal systems of equations, specifically the fifth-order CRWENO scheme. We propose a scalable implementation of the nonlinear compact schemes by implementing a parallel tridiagonal solver based on the partitioning/substructuring approach. We use an iterative solver for the reduced system of equations; however, we solve this system to machine zero accuracy to ensure that no parallelization errors are introduced. It is possible to achieve machine-zero convergence with few iterations because of the diagonal dominance of the system. The number of iterations is specified a priori instead of a norm-based exit criterion, and collective communications are avoided. The overall algorithm thus involves only point-to-point communication between neighboring processors. Our implementation of the tridiagonal solver differs from and avoids the drawbacks of past efforts in the following ways: it introduces no parallelization-related approximations (multiprocessor solutions are exactly identical to uniprocessor ones), it involves minimal communication, the mathematical complexity is similar to that of the Thomas algorithm on a single processor, and it does not require any communication and computation scheduling.

Authors:
 [1];  [2];  [3]
  1. Argonne National Lab. (ANL), Argonne, IL (United States)
  2. Univ. of Chicago, IL (United States)
  3. Univ. of Colorado, Boulder, CO (United States)
Publication Date:
Research Org.:
Argonne National Lab. (ANL), Argonne, IL (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1171189
Report Number(s):
ANL/MCS-TM-340
104241
DOE Contract Number:
AC02-06CH11357
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English

Citation Formats

Ghosh, Debojyoti, Constantinescu, Emil M., and Brown, Jed. Scalable Nonlinear Compact Schemes. United States: N. p., 2014. Web. doi:10.2172/1171189.
Ghosh, Debojyoti, Constantinescu, Emil M., & Brown, Jed. Scalable Nonlinear Compact Schemes. United States. doi:10.2172/1171189.
Ghosh, Debojyoti, Constantinescu, Emil M., and Brown, Jed. 2014. "Scalable Nonlinear Compact Schemes". United States. doi:10.2172/1171189. https://www.osti.gov/servlets/purl/1171189.
@article{osti_1171189,
title = {Scalable Nonlinear Compact Schemes},
author = {Ghosh, Debojyoti and Constantinescu, Emil M. and Brown, Jed},
abstractNote = {In this work, we focus on compact schemes resulting in tridiagonal systems of equations, specifically the fifth-order CRWENO scheme. We propose a scalable implementation of the nonlinear compact schemes by implementing a parallel tridiagonal solver based on the partitioning/substructuring approach. We use an iterative solver for the reduced system of equations; however, we solve this system to machine zero accuracy to ensure that no parallelization errors are introduced. It is possible to achieve machine-zero convergence with few iterations because of the diagonal dominance of the system. The number of iterations is specified a priori instead of a norm-based exit criterion, and collective communications are avoided. The overall algorithm thus involves only point-to-point communication between neighboring processors. Our implementation of the tridiagonal solver differs from and avoids the drawbacks of past efforts in the following ways: it introduces no parallelization-related approximations (multiprocessor solutions are exactly identical to uniprocessor ones), it involves minimal communication, the mathematical complexity is similar to that of the Thomas algorithm on a single processor, and it does not require any communication and computation scheduling.},
doi = {10.2172/1171189},
journal = {},
number = ,
volume = ,
place = {United States},
year = 2014,
month = 4
}

Technical Report:

Save / Share:
  • The Photocopia photomultiplier tube (PMT) takes advantage of two of the many unique properties of the hydrogenated amorphous silicon-germanium (a-SiGe) photoemitter material: its mechanical flexibility and mostly substrate-independent properties. The a-SiGe photoemitter has high secondary electron (SE) yield. It can be used both as the photocathode and as the gain medium. The active material can be grown on a flat, thin unibody substrate, formed and then “rolled up” ex situ. The completed structure would then be activated and sealed within a tube. The Ge component can be increased to enhance red-sensitivity. Compact sizes are possible, minimizing magnetic field effects. Themore » Photocopia PMT will be a low cost alternative to MCPs for TOF detectors and provide better timing discrimination for Cherenkov detectors. Retention of the ability to activate to a normal photoyield state upon flexing (bending) the substrate of the a-SiGe material after growth, but prior to activation has been shown. The SE coefficient of the activated material has been characterized over the voltage range suitable for utilization as the gain material. The time response of the material is suited to PMT use.« less
  • Conventional warm-white LED component fabrication consists of a large number of sequential steps which are required to incorporate electrical, mechanical, and optical functionality into the component. Each of these steps presents cost and yield challenges which multiply throughout the entire process. Although there has been significant progress in LED fabrication over the last decade, significant advances are needed to enable further reductions in cost per lumen while not sacrificing efficacy or color quality. Cree conducted a focused 18-month program to develop a new low-cost, high-efficiency light emitting diode (LED) architecture enabled by novel large-area parallel processing technologies, reduced number ofmore » fabrication steps, and minimized raw materials use. This new scheme is expected to enable ultra-compact LED components exhibiting simultaneously high efficacy and high color quality. By the end of the program, Cree fabricated warm-white LEDs with a room-temperature “instant on” efficacy of >135 lm/W at ~3500K and 90 CRI (when driven at the DOE baseline current density of 35 A/cm2). Cree modified the conventional LED fabrication process flow in a manner that is expected to translate into simultaneously high throughput and yield for ultra-compact packages. Building on its deep expertise in LED wafer fabrication, Cree developed these ultra-compact LEDs to have no compromises in color quality or efficacy compared to their conventional counterparts. Despite their very small size, the LEDs will also be robustly electrically integrated into luminaire systems with the same attach yield as conventional packages. The versatility of the prototype high-efficacy LED architecture will likely benefit solid-state lighting (SSL) luminaire platforms ranging from bulbs to troffers. We anticipate that the prototype LEDs will particularly benefit luminaires with large numbers of distributed compact packages, such as linear and area luminaires (e.g. troffers). The fraction of total SSL luminaire cost made up by the LEDs themselves has steadily fallen over the past several years, but can still make up 30% or more of the bill of materials; the new LED design will radically lower this proportion. Ultra-compact, highly efficient LEDs with optimal distribution in the system will further benefit luminaire materials and assembly costs by reducing the complexity and volume of thermal management and optical subsystems.« less
  • SNES (Scalable Nonlinear Equations Solvers) is a software package for the numerical solution of large-scale systems of nonlinear equations on both uniprocessors and parallel architectures. SNES also contains a component for the solution of unconstrained minimization problems, called SUMS (Scalable Unconstrained Minimization Solvers). Newton-like methods, which are known for their efficiency and robustness, constitute the core of the package. As part of the multilevel PETSc library, SNES incorporates many features and options from other parts of PETSc. In keeping with the spirit of the PETSc library, the nonlinear solution routines are data-structure-neutral, making them flexible and easily extensible. This usersmore » guide contains a detailed description of uniprocessor usage of SNES, with some added comments regarding multiprocessor usage. At this time the parallel version is undergoing refinement and extension, as we work toward a common interface for the uniprocessor and parallel cases. Thus, forthcoming versions of the software will contain additional features, and changes to parallel interface may result at any time. The new parallel version will employ the MPI (Message Passing Interface) standard for interprocessor communication. Since most of these details will be hidden, users will need to perform only minimal message-passing programming.« less
  • The focus of the project is on the development and customization of some highly scalable domain decomposition based preconditioning techniques for the numerical solution of nonlinear, coupled systems of partial differential equations (PDEs) arising from nuclear fuel simulations. These high-order PDEs represent multiple interacting physical fields (for example, heat conduction, oxygen transport, solid deformation), each is modeled by a certain type of Cahn-Hilliard and/or Allen-Cahn equations. Most existing approaches involve a careful splitting of the fields and the use of field-by-field iterations to obtain a solution of the coupled problem. Such approaches have many advantages such as ease of implementationmore » since only single field solvers are needed, but also exhibit disadvantages. For example, certain nonlinear interactions between the fields may not be fully captured, and for unsteady problems, stable time integration schemes are difficult to design. In addition, when implemented on large scale parallel computers, the sequential nature of the field-by-field iterations substantially reduces the parallel efficiency. To overcome the disadvantages, fully coupled approaches have been investigated in order to obtain full physics simulations.« less
  • We conducted a six-month investigation of the design, analysis, and software implementation of a class of singularity-insensitive, scalable, parallel nonlinear iterative methods for the numerical solution of nonlinear partial differential equations. The solutions of nonlinear PDEs are often nonsmooth and have local singularities, such as sharp fronts. Traditional nonlinear iterative methods, such as Newton-like methods, are capable of reducing the global smooth nonlinearities at a nearly quadratic convergence rate but may become very slow once the local singularities appear somewhere in the computational domain. Even with global strategies such as line search or trust region the methods often stagnate atmore » local minima of {parallel}F{parallel}, especially for problems with unbalanced nonlinearities, because the methods do not have built-in machinery to deal with the unbalanced nonlinearities. To find the same solution u* of F(u) = 0, we solve, instead, an equivalent nonlinearly preconditioned system G(F(u*)) = 0 whose nonlinearities are more balanced. In this project, we proposed and studied a nonlinear additive Schwarz based parallel nonlinear preconditioner and showed numerically that the new method converges well even for some difficult problems, such as high Reynolds number flows, when a traditional inexact Newton method fails.« less