| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | September 8, 2013 |
| Latest Amendment Date: | September 8, 2013 |
| Award Number: | 1319054 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Leland Jameson
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | September 15, 2013 |
| End Date: | August 31, 2017 (Estimated) |
| Total Intended Award Amount: | $258,196.00 |
| Total Awarded Amount to Date: | $258,196.00 |
| 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: |
MSC01 1247 Albuquerque 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
The Principal Investigator proposes to carry out an interdisciplinary comprehensive research program combining the development, analysis and optimization of a new class of numerical methods, with their application to problems in seismology and electromagnetics. The novel methods hybridizes arbitrary-order Hermite approximations with arbitrary-order discontinuous Galerkin methods. The combination of these two methods will result in a new class of hybrid methods able to handle complex geometries and with unprecedented computational efficiency through large time steps and high-resolution. The methods have very large computation to communication ratio and are well suited for implementation on current and emerging supercomputer systems, enabling the solution of complex, multiple-scale evolutionary systems. The proposed unified analysis of discontinuous Galerkin methods and Hermite methods will require new tools and theories to be developed and will lead to a new theoretical framework for the analysis of hybrid methods. The proposal will consider methods for both first and second order formulations of the governing equations of elasticity and electromagnetics.
The research will have broader impacts in technology and science, as well as in the training of the next generation of computational scientists. As recent events in Japan have shown, earthquakes are a societal problem throughout the world. To better mitigate seismic hazard, effective prevention and prediction is needed. Careful assessment of seismic hazards through accurate computational predictions can lead to appropriate building codes. This can be of enormous impact for human life and societal welfare in the case of a large seismic event in a densely populated area as the greater Los Angeles or the San Francisco bay. The broader impacts of the proposed activities also include education. The project will involve graduate students who will gain experience in state-of-the-art computational science. The research will be performed at the University of New Mexico, a Hispanic serving institution that also serves a large body of native Americans, allowing active recruitment and education of students from underrepresented groups.
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.
Intelectual Merits outcomes includes the development of several numerical techniques for simulating time dependent wave propagation of elastic, acoustic and electro-dynamic type.
A significant result is the development of energy-based discontinuous Galerkin (DG) methods and their basic construction and analysis for the scalar wave equation and the elastic wave equation. The new methods are distinct from other DG methods for second order equations admiting straightforward mesh-independent flux options, and, use a small number of variables. Moreover, they can be directly formulated for any system from the Lagrangian form.
Another relevant result is the development of efficient conservative Hermite methods discussed including a complete convergence analysis in the spatially periodic case, proving the result that the time step can always be taken to be the maximum allowed by domain-of-dependence considerations. In addition an open-source library was developed to illustrate the implementation of Hermite methods. Flux-Conservative Hermite methods for equations describing fluid flow with the ability to handle shocks were also developed. Additionally new multi-order Monte Carlo algorithm for computing the statistics of stochastic quantities of interest described by linear hyperbolic problems with stochastic parameters were developed.
Broader Impacts outcomes include the development of open-source software libraries containing benchmark problems for elastic waves in cylindrical geometries as well as open-source implementations of the energy based discontinuous Galerkin methods.
In addition to the development of open-source libraries, the grant has been a vehicle for training the next generation computational scientists. With the support of the grant the PI served as a research mentor to two PhD students and two Ms students. Out of these four two were from underrepresented groups. One PhD student has graduated and is pursuing a career in academia, currently in the capacity as a postdoc. Both Ms students has graduated, one is now employed in industry and the other is pursuing a PhD at another institution.
Outreach to underrepresented groups and high school students includes the public lecture “From Calculus to music - The mathematics of waves” presented to mathematics students at the tribal college Navajo Tech. University, Crownpoint, NM and providing faculty support for the 2016 UNM-PNM Statewide High School Mathematics Contest.
Last Modified: 11/29/2017
Modified by: Daniel Appelo
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