LEAPS-MPS: Analytic and Numerical Treatment of Nonlinear Maxwell's Equations

Loyola University Of Chicago, Chicago IL

Investigators

Abstract

Electromagnetism plays a central role in almost every field of science and engineering. Areas include energy science, nanotechnology, life science, magnetism confinement fusion, magnetic levitation technology, sensor technology and many others. The physical laws of electromagnetism are mathematically described by Maxwell’s equations, which are commonly used as "building blocks" to derive coupled systems involving electromagnetic phenomena. Problems of optimal control of Maxwell’s equations arise in a variety of applications including, but not limited to, stealth technology, design and control of antennas, diffraction optics, magnetotellurics and related fields. The project will investigate these optimal control problems both theoretically and numerically. More specifically, this project will shed light on the optimal control of non-linear Maxwell’s equations and provide efficient algorithmic tools of optimal complexity to solve a class of important and challenging problems in nonlinear optimization. Moreover, the project will assess the algorithms’ performance by applications of major scientific significance in material and life sciences. The outcomes of this project will advance the greater fields of applied mathematics and computational sciences to meet the growing need for non-invasive technologies in superconductor simulation or particle accelerator design, etc. The project will also involve state-of-the-art research training for students at Loyola University Chicago and promote advanced-degree pursuits and future workforce in STEM, especially from women, minorities, and underrepresented groups. The results from this project, including computer code, will be disseminated to the scientific community and will be made accessible to the public; in particular, to researchers working on related problems and industrial practitioners. The main objective of this project is to study analytical and computational aspects related to the optimization of quasilinear Maxwell’s equations. Most importantly, the optimization framework assumes the form of an optimal control problem for Maxwell’s equations seeking a control strategy via distributed electric currents or boundary steering functions to optimize a prescribed objective functional under suitable constraints on the state and/or control variables. The main challenge lies in the complexity of the full time-dependent Maxwell’s equations which feature a first-order hyperbolic coupled structure of quasilinear type. The investigator will develop analytical and computational tools to theoretically understand, numerically simulate and optimally manipulate electromagnetic phenomena. The project will involve fundamental research in nonlinear control theory, finite elements method, nonlinear model reduction, time integrators, operator splitting, mesh adaptivity, to name a few. The anticipated developments will contribute to advance the greater fields of applied mathematics and computational sciences by involving a thorough investigation of the forward map arising from the underlying direct problem and its regularity properties, Newton-type iteration for solving Pontryagin’s optimality equations and its space-time discretization, development of efficient time-integrators and model reduction techniques. 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.