| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | May 17, 2019 |
| Latest Amendment Date: | July 15, 2021 |
| Award Number: | 1855788 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Tomek Bartoszynski
tbartosz@nsf.gov (703)292-4885 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | July 1, 2019 |
| End Date: | June 30, 2023 (Estimated) |
| Total Intended Award Amount: | $173,899.00 |
| Total Awarded Amount to Date: | $206,822.00 |
| Funds Obligated to Date: |
FY 2021 = $32,923.00 |
| 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: |
LA US 70803-2701 |
| 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): |
PROBABILITY, EPSCoR Co-Funding |
| Primary Program Source: |
01002122DB NSF RESEARCH & RELATED ACTIVIT |
| 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
Ubiquitous presence of randomness in behaviors of various physical and biological systems is universally acknowledged. For example, cellular processes, fluctuations in stock prices, weather patterns, movement of microscopic particles exhibit different types of random behaviors. Furthermore, randomness is a vital ingredient in most modern algorithms that are popular in data-science. Hence, it is of fundamental importance that various types of randomness are properly characterized for detailed understanding of system properties. Mathematical models incorporating such randomness are typically expressed in terms of stochastic equations, which often have complex dynamics. An important component of mathematical studies of these equations includes computer simulations of their temporal evolution, which are necessary for understanding their behavioral patterns over time. Accurate statistical estimation of some key parameters of these stochastic equations from observed data is also crucial, as it leads to the "most appropriate mathematical model" underlying these data points. This in turn leads to better predictive power of such models. The research supported by this award will undertake proper mathematical analyses of various numerical schemes that are used for these purposes. More specifically, research in this direction will involve precise calculations of probabilities of getting accurate results from these schemes and will describe conditions under which these probabilities are very high. There is a critical need for such theoretical results, since they will inevitably lead to design of faster and more efficient algorithms, which in turn will be beneficial to society. The project will involve undergraduate and graduate students and will help them to gain valuable analytical and computational skills. The results of the project will be published in well-known scientific journals and will also be presented at domestic and international conferences.
The research will focus on the limiting behaviors of complex stochastic systems discretized by properly scaled step sizes. Discretization is at the heart of various numerical schemes, but its effect on the desired convergence properties is not always clearly understood. For example, it is well known that approximating stationary distribution of an ergodic stochastic differential equation (SDE) by time averages of sample paths obtained from an Euler-Maruyama type discretization scheme (using fixed step-size) could be problematic. In particular, such an estimator will have a bias, which may or may not be quantifiable. These are infinite-time horizon problems, and there is a deficit of proper asymptotic results in this direction even for regular Ito-diffusions. Exploring proper scaling techniques to get desired convergence along with convergence rates for such discretization-based schemes is the central goal of this project. Stochastic models that will be considered covers switching jump-diffusion models, multiscale systems and systems of interacting SDEs with possible jumps. The last class is particularly important for understanding particle-based methods for approximating nonlinear integro-differential equations including Boltzmann-type equations. These equations are connected to appropriate systems of interacting SDEs with jumps through McKean-Vlasov type limits. The stochastic equations of interest will also include Langevin SDEs, which model dynamics of molecular systems in presence of particle interaction potential, damping and random forces, and which also play a pivotal role in certain Markov Chain Monte Carlo algorithms. The research puts special emphasis on large deviation asymptotics of error probabilities, which, importantly, because of presence of both upper and lower bounds, give the optimal exponential decay rate.
This project is jointly funded by Probability program and the Established Program to Stimulate Competitive Research (EPSCoR).
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.
Stochastic differential equations (SDEs) play a pivotal role in mathematical modeling, particularly in understanding complex systems where inherent randomness significantly impacts the dynamics. By accounting for uncertainties and random fluctuations, SDEs contribute to a more accurate representation of diverse real-world processes, thereby enhancing the prognostic capabilities of mathematical models in various scientific domains. A comprehensive understanding of these models requires various numerical algorithms analyzing their temporal evolution as well as accurate statistical estimation of the models’ parameters from the observed data. The project studied a collection of problems on such algorithms that are based on suitable discretization techniques with a particular emphasis on how the precision of the algorithms is affected by the process of discretization. Specific goals included (i) understanding stability of the discretized system, (ii) asymptotic analysis of Monte-Carlo estimators of equilibrium distributions of the continuous models based on suitable discretization schemes, (iii) development of statistical inference schemes for proper calibration of model with available data and assessing their accuracy through suitable limit theorems. Research in this direction aids in establishing the most suitable mathematical models, understanding their tractable approximations across different scenarios and enhancing their predictive potential.
The outcomes of these studies are as follows. The work on stability of stochastic processes has identified a weak negative drift criterion which along with a state-dependent restriction on the centered conditional moments of the process guarantees uniform bounds on the moments of discrete processes. An important feature of the result is that it does not require Markovian property of the stochastic system. However, introducing the Markovian property as an additional assumption leads to new ergodic theorems ensuring fast convergence to the system’s equilibrium distribution. The research on asymptotic analysis of Monte-Carlo estimators of equilibrium distributions and different statistical estimators for SDEs has produced multiple results concerning central limit theorems and large deviation principles in suitable scaling regimes. These results establish the proper scaling between the discretization steps and the time-horizon and provide valuable insights into decay rate of error probabilities. They also facilitate the quantification of uncertainty of the estimators through construction of effective confidence regions around them. In this context, a novel method of constructing statistically optimal confidence regions having exponential accuracy via large deviation principles has been developed. The research on nonparametric learning has produced important results on infinite-dimensional optimization. This progress led to the formulation of a rigorous mathematical framework and innovative learning algorithms for estimating the driving functions of stochastic dynamical systems without any prior knowledge of their structures. The significance of the work lies in its ability to construct data-driven mathematical models, unencumbered by unnecessary simplifying assumptions that are often imposed to ensure the model's functionality.
The project's findings were communicated through various invited talks at conferences and seminars and multiple research papers, whose preprints are available on open-access repository, arXiv. The project has played a pivotal role in training up-and-coming mathematical researchers, providing them with invaluable insights and fostering a deeper understanding of the field of probability and statistical learning. Collaborative efforts with a Ph.D. student resulted in the publication of one paper, while another is in the final stages before submission. Moreover, the PI has mentored undergraduate students on select components of the project and anticipates submitting the results for publication soon. The project has spurred further research initiatives led by the PI and his collaborators, both at Louisiana State University and beyond. The PI integrated research insights from this project into the advanced graduate courses he taught, enhancing the educational content and fostering a more comprehensive learning experience for the students.
Last Modified: 10/19/2023
Modified by: Arnab Ganguly
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