Scalable Nonlinear Compact Schemes
Abstract
In this work, we focus on compact schemes resulting in tridiagonal systems of equations, specifically the fifthorder 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 machinezero convergence with few iterations because of the diagonal dominance of the system. The number of iterations is specified a priori instead of a normbased exit criterion, and collective communications are avoided. The overall algorithm thus involves only pointtopoint 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 parallelizationrelated 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:
 Argonne National Lab. (ANL), Argonne, IL (United States)
 Univ. of Chicago, IL (United States)
 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/MCSTM340
104241
 DOE Contract Number:
 AC0206CH11357
 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 fifthorder 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 machinezero convergence with few iterations because of the diagonal dominance of the system. The number of iterations is specified a priori instead of a normbased exit criterion, and collective communications are avoided. The overall algorithm thus involves only pointtopoint 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 parallelizationrelated 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
}

The Photocopia photomultiplier tube (PMT) takes advantage of two of the many unique properties of the hydrogenated amorphous silicongermanium (aSiGe) photoemitter material: its mechanical flexibility and mostly substrateindependent properties. The aSiGe 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 redsensitivity. Compact sizes are possible, minimizing magnetic field effects. Themore »

Scalable, Economical Fabrication Processes for UltraCompact WarmWhite LEDs
Conventional warmwhite 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 18month program to develop a new lowcost, highefficiency light emitting diode (LED) architecture enabled by novel largearea parallel processing technologies, reduced number ofmore » 
Using the scalable nonlinear equations solvers package
SNES (Scalable Nonlinear Equations Solvers) is a software package for the numerical solution of largescale 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). Newtonlike 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 datastructureneutral, making them flexible and easily extensible. This usersmore » 
Scalable Nonlinear Solvers for Fully Implicit Coupled Nuclear Fuel Modeling. Final Report
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 highorder PDEs represent multiple interacting physical fields (for example, heat conduction, oxygen transport, solid deformation), each is modeled by a certain type of CahnHilliard and/or AllenCahn equations. Most existing approaches involve a careful splitting of the fields and the use of fieldbyfield iterations to obtain a solution of the coupled problem. Such approaches have many advantages such as ease of implementationmore » 
Scalable nonlinear iterative methods for partial differential equations
We conducted a sixmonth investigation of the design, analysis, and software implementation of a class of singularityinsensitive, 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 Newtonlike 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 »