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Project Information

STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS AND THEIR NUMERICAL SOLUTION

Agency:
NSF

National Science Foundation

Project Number:
1816378
Contact PI / Project Leader:
BOU-RABEE, NAWAF
Awardee Organization:
RUTGERS THE STATE UNIVERSITY OF NEW JERSEY CAMDEN

Description

Abstract Text:
Space-time phenomena as varied as turbulence, phase-field dynamics, surface growth, neuronal activity, population dynamics, and interest rate fluctuations are modelled by stochastic partial differential equations (SPDE). In comparison with partial differential equations (PDE), SPDEs are advantageous because they allow modelers to explicitly incorporate space-time randomness and capture the effect of random fluctuations inherent to systems with many degrees of freedom. However, they are trickier to numerically solve than PDEs and require new specialized techniques. Indeed, since SPDE solutions are very rough, straightforward adaptations of numerical PDE methods often lead to approximations that do not accurately represent the behaviors of the actual solutions. To address this issue, the project develops a new numerical method, called Spectrwm [spek-trum] (signifying spectral random walk method), that is tailored specifically to SPDEs. REU students are included in the work of the project.Spectrwm is a family of Markov jump process approximations for SPDEs. As such, these approximations can be viewed as infinite-dimensional analogues of random walk methods for Brownian motion. Aside from their mathematical interest, these methods address in a natural way several issues encountered when simulating SPDEs, including long-time simulation, accurately representing the effect of multiplicative noise, and approximation of their path-dependent expected values. by suitably constructing them, Spectrwm is faithful to the domain of their solution. The investigator and his collaborators aim to use probabilistic techniques to quantitatively understand the finite- and long-time behavior of Spectrwm methods for well-posed SPDEs with reflecting boundary conditions, and to develop accelerated variants of Spectrwm methods for a wider range of SPDE problems, including ones that are classically ill-posed or where the operators of the underlying processes are not well understood. As a major byproduct, the project lends insight into SPDEs themselves: the nature of the Kolmogorov equations associated to SPDEs, the influence of boundaries on SPDE solutions, metastability in SPDE solutions, and renormalizations for classically ill-posed SPDEs. REU students are included in the work of the project.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.
Project Terms:
Address; analog; Award; Behavior; Differential Equation; Dimensions; Equation; Evaluation; Family; Foundations; Freedom; Growth; insight; interest; Lead; Mathematics; Methods; Mission; Modeling; Motion; Names; Nature; Neurons; Noise; Phase; Population Dynamics; Process; Research Personnel; simulation; Students; Surface; System; Techniques; Time; Variant; Walking; Work

Details

Contact PI / Project Leader Information:
Name:  BOU-RABEE, NAWAF
Other PI Information:
Not Applicable
Awardee Organization:
Name:  RUTGERS THE STATE UNIVERSITY OF NEW JERSEY CAMDEN
City:  CAMDEN    
Country:  UNITED STATES
Congressional District:
State Code:  NJ
District:  01
Other Information:
Fiscal Year: 2018
Award Notice Date: 03-Aug-2018
DUNS Number: 625216556
Project Start Date: 15-Aug-2018
Budget Start Date:
CFDA Code: 47.049
Project End Date: 31-Jul-2021
Budget End Date:
Agency: ?

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National Science Foundation
Project Funding Information for 2018:
Year Agency

Agency: The entity responsible for the administering of a research grant, project, or contract. This may represent a federal department, agency, or sub-agency (institute or center). Details on agencies in Federal RePORTER can be found in the FAQ page.

FY Total Cost
2018 NSF

National Science Foundation

$46,955

Results

i

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