
NSF Org: |
DMS Division Of Mathematical Sciences |
Recipient: |
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Initial Amendment Date: | June 30, 2021 |
Latest Amendment Date: | June 30, 2021 |
Award Number: | 2110917 |
Award Instrument: | Standard Grant |
Program Manager: |
Yuliya Gorb
ygorb@nsf.gov (703)292-2113 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
Start Date: | July 1, 2021 |
End Date: | May 31, 2025 (Estimated) |
Total Intended Award Amount: | $152,476.00 |
Total Awarded Amount to Date: | $152,476.00 |
Funds Obligated to Date: |
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History of Investigator: |
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Recipient Sponsored Research Office: |
1 UNIVERSITY OF NEW MEXICO ALBUQUERQUE NM US 87131-0001 (505)277-4186 |
Sponsor Congressional District: |
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Primary Place of Performance: |
NM US 87131-0001 |
Primary Place of
Performance Congressional District: |
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Unique Entity Identifier (UEI): |
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Parent UEI: |
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NSF Program(s): | COMPUTATIONAL MATHEMATICS |
Primary Program Source: |
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Program Reference Code(s): |
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Program Element Code(s): |
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Award Agency Code: | 4900 |
Fund Agency Code: | 4900 |
Assistance Listing Number(s): | 47.049 |
ABSTRACT
Computer simulations and the mathematical methods supporting these are central to the modern study of engineering, biology, chemistry, physics, and other fields. Many simulations are computationally costly and require the large resources of modern supercomputers. New mathematical methods are urgently needed to efficiently utilize next generation supercomputers with millions to billions of processors. This project will develop new parallel-in-time algebraic multigrid methods for complex physical systems specifically designed for next generation computers. These new methods will add a new dimension of parallel scalability (time) and promise dramatically faster simulations in many important application areas, such as the gas and fluid dynamics problems considered (e.g., with relevance to wind turbines and viscoelastic flow). Graduate students will be involved and trained, and open source code will be developed.
This project will develop fast, parallel, and flexible space-time solvers for systems of partial differential equations (PDEs). The project will focus on algebraic multigrid (AMG) within block preconditioning traditionally appropriate for large adaptively refined spatial systems. These techniques will be extended to general space-time systems with a flexible approach that allows for adaptive space-time refinement. This adaptivity helps to accurately resolve lower dimensional features such as shocks at a fraction of the cost and storage of uniform refinement. Furthermore, the project will produce new practical AMG theory for non-SPD (symmetric positive definite) problems as well as solvers for adaptively refined space-time discretizations for a variety of parabolic and hyperbolic PDEs including the Euler and Navier-Stokes equations and Cahn-Hilliard system. The project will design, analyze, and tune parallel AMG solvers that are robust, efficient, and fast over a wide range of PDEs and parameters and will contribute to the widely used packages MFEM and hypre. The solvers will be developed and tested for applications in wind turbines, as well the high Weissenberg number problem in viscoelastic flows.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
This project researched new linear solvers (specifically, algebraic multigrid methods). Linear solvers are a core part of computer simulations for many physical phenomena (e.g., air flow, water flow, deformation and elasticity, and combustion, to name a few). However, linear solvers are often the most expensive part of such computer simulations, and as such, the search for more efficient solver methods is of ongoing interest to the broader community. Algebraic multigrid (AMG) is among the most efficient solver techniques because it’s hierarchical (multilevel) representation of the problem can allow for optimal solution time, relative to the number of variables in the linear equations. However, AMG is not yet applicable to all types of linear equations, and in particular to many nonsymmetric sparse linear systems, including advective problems and problems in space-time. Developing AMG solvers for these problem areas is the primary goal of this project.
During this project, the team developed CLAIR, which is a new type of reduction-based AMG that is suitable for solving nonsymmetric linear equations coming from the discretization of advection-diffusion problems and from space-time problems. By combining techniques from AIR (approximate ideal restriction) that have been effective for advective problems, with energy minimization and root node constraint techniques that are well-suited for diffusion problems, we have developed an efficient method for solving nonsymmetric problems that is insensitive to the varying contribution from the diffusive part of the governing equations. The new method is a constrained local AIR (CLAIR) approach. To our knowledge this is the first such AMG method for nonsymmetric problems. In addition, it is able to achieve standard low complexity hierarchies in the diffusive regime through aggressive coarsening, something that has been previously difficult for reduction-based methods. Please see https://arxiv.org/abs/2307.00229.
The team also developed an optimal interpolation theory for AMG and general nonsymmetric matrices, which is a first for AMG methods. Our new theory was inspired by a specific block 2x2 system of linear equations based on the original general nonsymmetric matrix and by earlier achievements with optimal interpolation theory for symmetric positive definite (SPD) matrices. To our knowledge, this is the first theoretical result for AMG and general nonsymmetric problems that predicts a sharp convergence rate and provides insight into the optimal or best convergence rate for AMG and a nonsymmetric problem. This theoretical work, while important in its own regard, also provides valuable insights into choosing relaxation schemes in the nonsymmetric setting and how one can choose interpolation constraints in the nonsymmetric CLAIR setting. Please see https://arxiv.org/abs/2401.11146.
The team also developed a space-time block preconditioning approach for incompressible resistive magnetohydrodynamics (MHD). Here a block preconditioning framework was applied, similar to how block preconditioners are used for sequential implicit time-stepping and fluid dynamics, only now space-time matrix blocks are considered, instead of purely spatial matrix blocks. To our knowledge, this is the first known space-time AMG solver for MHD. Please see https://arxiv.org/abs/2309.00768.
Last Modified: 06/24/2025
Modified by: Jacob B Schroder
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