Troy D. Butler
$275,000
George Mason University
Virginia
Mathematical and Physical Sciences (MPS)
Optimization problems constrained by physics are ubiquitous. These problems are nonlinear, nonsmooth and contain unknown parameters. The physics describing the constraints are partial differential equations (PDEs) which are multiscale, multiphysics, and geometric in nature. They capture many realistic scenarios: control of pathogen propagation like COVID-19, blood flow in aneurysms, determining weakness in structures, and vortex control in nuclear reactors. This project will study optimization problems constrained by PDEs that can incorporate data to make decisions that are resilient to uncertainty. The proposed methods will provide new insights into nonsmooth nonconvex optimization, and they will be applied to applications such as identifying weakness in structures (e.g., bridges and nuclear plants). The results of this research will be shared with the community via publications and research talks. The outcomes of this research will benefit scientists working in multiple research areas such as numerical analysis, optimization, structural engineering and bioengineering. A PhD student will be fully supported by the project. <br/> <br/>Particular focus of the project is on risk-averse optimization problems where the PDEs contain uncertainty arising from modeling unknown quantities (coefficients, boundary conditions, etc.) as random variables and dynamic optimization problems. The project will develop: (i) Inexact adaptive Semismooth Newton and Trust-region methods to solve these optimization problems; (ii) Primal dual methods for risk-averse optimization problems with general constraints; (iii) Applications to problems where inexactness arise from finite element discretization. Thrusts (i) and (iii) will enable interaction between finite element discretization and optimization solvers leading to structure preserving algorithms. Additionally, Thrust (ii) will lead to different penalty parameters for different constraints and will allow inexact solves at each iteration which is essential for large systems. This will enable a new paradigm for widely used penalty-based methods. Algorithms for high-dimensional nonsmooth risk-averse optimization will help overcome curse of dimensionality for similar problems.<br/><br/>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.