| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | August 15, 2024 |
| Latest Amendment Date: | August 15, 2024 |
| Award Number: | 2348650 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Elizabeth Wilmer
ewilmer@nsf.gov (703)292-7021 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | August 15, 2024 |
| End Date: | July 31, 2027 (Estimated) |
| Total Intended Award Amount: | $293,784.00 |
| Total Awarded Amount to Date: | $293,784.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
202 HIMES HALL BATON ROUGE LA US 70803-0001 (225)578-2760 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
202 HIMES HALL BATON ROUGE LA US 70803-0001 |
| 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): |
OFFICE OF MULTIDISCIPLINARY AC, PROBABILITY |
| Primary Program Source: |
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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 research aims to understand and exploit randomness as it appears in a diverse collection of settings. For example, defects and impurities in materials or microscopic turbulence in fluids are fundamentally random, but they can be described deterministically using models based on a complicated averaging, which provide an accurate description of the original system and are used to develop efficient numerical simulations. In other settings one is interested in modeling the likelihood of rare events, such as mechanical failures in engines or extreme concentrations of heat. The probability of these events can be understood using equations with a random noise, where the structure of the noise is determined by the system's small-scale dynamics. As a final example, in machine learning randomized algorithms are used to explore enormous data sets whose size makes computation impractical if not outright impossible. Neural networks are trained by deliberately introducing randomness into the algorithm, where at each step the system is optimized over a small but random sample of the data. Graduate and undergraduate students will be mentored as part of this project. In addition, seminars and workshops in probability will be held at LSU, and a K12-outreach program with local students will be developed.
The research will focus on the areas of stochastic homogenization, stochastic partial differential equations (SPDEs) and interacting particle systems, and randomized algorithms in machine learning. The first topic, homogenization theory, analyzes the properties of systems with complicated microstructures. The current project will make the first connection between the homogenization of incompressible flows and SPDEs with Brownian transport noise and will enhance our understanding of the longtime and equilibrium behavior of diffusion processes in random environments. For the second topic, this project will make a mathematically rigorous connection between macroscopic fluctuation theory and fluctuating hydrodynamics in the context of certain interacting particle processes and interacting diffusive systems. The PI will develop a robust well-posedness theory for the related SPDEs and characterize their stochastic dynamics. The third area of research will put forth a local analysis of randomized algorithms in machine learning that avoids unrealistic conditions like global convexity or contractivity. The research will establish a quantitative criterion to identify basins of attraction in the loss landscape and sharp estimates for the convergence of stochastic gradient descent in the basin of attraction.
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.
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