CAREER: Randomized Multiscale Methods for Heterogeneous Nonlinear Partial Differential Equations

Project: Research project

Project Details

Description

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Heterogeneous systems with salient features at multiple scales are ubiquitous in science and engineering. A direct numerical simulation that aims at capturing relevant phenomena at all scales requires an often prohibitively large amount of computation time. To simulate such systems, multiscale methods include the local behavior of a numerical solution in the approximation process, thus taking into account the various scales. For example, in modeling a wind turbine made from composites, deformations during operation can be simulated for portions of the wind turbine blade. The multiscale approximation for the deformation of the whole wind turbine is then built from these local solutions. Multiscale methods that can guarantee that the error between the multiscale approximation and the global solution is below a given tolerance are of particular interest. The goal of this project is to design and analyze such multiscale methods for the numerical solution of nonlinear partial differential equations that are used in simulating deformations in (realistic) wind turbines. It is anticipated that the new methods will be crucial in building digital twins, that is, mathematical models of physical objects that can be employed in real time to assess, for example, the structural health of a system. Application of the results in digital twins for wind turbines will support the generation of renewable energy for society. The project includes a closely integrated educational plan to increase participation and retention of students from groups underrepresented in STEM by (i) designing and leading courses for high school students, helping them discover via creative and project-based learning techniques how the concepts of mathematics they are learning have important applications; and (ii) establishing a mentoring program for undergraduate mathematics students from underrepresented groups.To develop the desired multiscale methods, in this project, the local ansatz functions will be constructed to (quasi-)optimally approximate the nonlinear set of local solutions of the partial differential equation (PDE). To approximate the latter, randomized versions of model order reduction methods will be developed. While deterministic model reduction algorithms construct provably the optimal space to approximate a set of solutions of a PDE dependent on a parameter (here arbitrary Dirichlet boundary data), they suffer from the curse of dimensionality for high-dimensional parameter sets. Randomizing these methods is expected to break the curse of dimensionality and allow analysis of the error in novel ways suitable for nonlinear systems. The three research objectives of the project are: development and analysis of randomized multiscale methods for (i) elliptic and (ii) parabolic nonlinear PDEs, where the local ansatz functions can be constructed parallel in time, and (iii) application to the simulation of the deformation of wind turbines.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.
StatusActive
Effective start/end date1/06/2231/05/27

Funding

  • National Science Foundation

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