Algorithms and Numerical Methods for Optimization with Partial Differential Equation Constraints

Optimization problems with constraints are ubiquitous in science and engineering. Some examples include designing a drug delivery mechanism to maximize the impact on cancerous cells, maximizing oil recovery from the wells, designing new materials that can be manufactured using additive manufacturing, and machine learning for solving inverse problems.

Many of these problems are inherently non-linear and non-smooth, which makes the development of algorithms and their analysis extremely challenging. The project aims to create new optimization algorithms that will overcome these challenges and that are widely applicable. The precise target applications include, magnetic drug targeting, quantum spin chains, harmonic maps, structure design, and solving inverse problems using machine learning.

Open-source software will be created and collaborations with practitioners will be carried out to maximize the impact of the work.

The applications described above can be modeled using partial differential equations (PDEs). These PDEs are geometric (harmonic maps), nonlocal (fractional), multiphysics (magnetic drug delivery), multiscale with an unknown domain, i.e., free boundary problems (FBPs). The goal of this project is to study optimization problems with PDE constraints, i.e., PDE constrained optimization.

Specifically, it aims to create new optimization methods based on Deep Learning and Augmented Lagrangian frameworks to solve several currently intractable optimization problems, for instance, problems constrained by advection dominated (also limiting transport equations) arising in magnetic targeted drug delivery. All these problems are nonlinear, nonconvex, and non-smooth in nature.

Novel optimization algorithms will provide new insights into nonconvex non-smooth problems. In particular for optimization problems with state or gradient constraints, where concepts from set-valued analysis are typically needed. The deep learning work will help create new research directions.

Cancerous cells absorb only a small amount of medicine, magnetic drug targeting has shown to increase this absorption rate without harming vital organs. This research will also improve our understanding of magnetic fluids and will create mathematical understanding of control of conservation laws. Nonlocal problems are increasingly important in science and engineering.

They lead, for instance, to better models for quantum spin chains, cardiac electrical response, additive manufacturing (materials science), image denoising and phase separation. Open-source software will be created. This will not only benefit scientists in optimization, FBPs, and nonlocal problems but also scientists in nonlinear PDEs and data science.

The results will be disseminated via a special topics course, research publications, and talks. Two PhD students will get PhDs. Reading seminars for students will be created.

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.