Mohammad Farazmand
Publications
- Stochastic compartmental models of the COVID-19 pandemic must have temporally correlated uncertainties , PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES (2023)
- EVOLUTION OF NONLINEAR REDUCED-ORDER SOLUTIONS FOR PDEs WITH CONSERVED QUANTITIES , SIAM JOURNAL ON SCIENTIFIC COMPUTING (2022)
- Model-assisted deep learning of rare extreme events from partial observations , CHAOS (2022)
- Quantifying rare events in spotting: How far do wildfires spread? , FIRE SAFETY JOURNAL (2022)
- Shape-morphing reduced-order models for nonlinear Schrodinger equations , NONLINEAR DYNAMICS (2022)
- Data-driven prediction of multistable systems from sparse measurements , CHAOS (2021)
- Evolution of nonlinear reduced-order solutions for PDEs with conserved quantities , SIAM J. on Scientific Computing, In press (2021)
- Investigating climate tipping points under various emission reduction and carbon capture scenarios with a stochastic climate model , PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES (2021)
- Mitigation of rare events in multistable systems driven by correlated noise , PHYSICAL REVIEW E (2021)
- MULTISCALE ANALYSIS OF ACCELERATED GRADIENT METHODS , SIAM JOURNAL ON OPTIMIZATION (2020)
Grants
The dynamics of spatiotemporal systems are routinely described by time-dependent partial differential equations (PDEs). The solutions of these PDEs often exhibit time-varying localized structures, with sharp gradients, surrounded by regions of relative quiescence. Efficiently resolving these multiscale structures has been a long-standing challenge in scientific computing. Currently, there are two broad classes of methods for addressing this challenge: 1. Adaptive methods which dynamically evolve the spatial discretization so that the computational grid is refined around the localized structure and less so in the quiescent regions. 2. Multiresolution methods, such as wavelets, which encode various scales in the basis instead of the discretization. The proposed program will develop a new and computationally efficient method called shape-morphing modes. The main idea behind this method is to use a time-dependent basis of functions that automatically morph their shapes over time and space in order to efficiently resolve all scales. Being mesh-free, the proposed method substantially reduces the computational cost as compared to existing adaptive methods. Furthermore, since the modes adapt themselves to the solution of the PDE, far fewer modes are needed to resolve all scales. This significantly reduces the memory requirements, thus outperforming the existing multiresolution methods.