Theory and Algorithms for Eigenvector-Dependent Nonlinear Eigenvalue Problems

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The mathematical questions in this project arise in electronic structure calculations in computational materials science, signal processing of brain–computer interfaces in neuroscience and biomedical engineering, among other applications. The questions come in a variety of forms and pose intriguing challenges for mathematical analysis and numerical solutions. This project seeks to advance the state-of-the-art of the analysis and computation. The project's outcome will advance understanding of the mathematical problems and provide tools for researchers and practitioners to perform simulations in less time using advanced models that were previously unavailable. This project will integrate research into teaching and education and will engage students at various levels in computational mathematics and interdisciplinary research.This project will support one graduate per year in each of the three years of the grant. Technically, this project will focus on an important class of Eigenvector-dependent Nonlinear Eigenvalue Problems NEPv, called affine-linear NEPv (al-NEPv). In an al-NEPv, the coefficient matrix of NEPv poses an affine-linear structure. Origins of al-NEPv include trace-related optimizations such as the trace-ratio optimization for dimension reduction and robust Rayleigh-quotient optimization for handling data uncertainties, among others. The PI plans to conduct systematic analysis and algorithmic development for al-NEPv. The main components of the proposed research are threefold: analysis of al-NEPv, such as a novel geometric description and a variational characterization; new geometric interpretation of the self-consistent field (SCF) iteration for solving al-NEPv, and variants of SCF for handling the local optimal issue and for accelerating the convergence of SCF; availability of a public-domain repository for the collection of NEPv from real-life applications.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.