Compression: Mathematical Foundations and Applications

High-dimensional scientific and engineering data—such as fluid flows, climate fields, videos, and neural recordings—often exhibit rich compressible multilinear and nonlinear structures. Classical matrix-based algorithms and linear approximation methods frequently fail to fully exploit these intrinsic structures, resulting in high computational cost, loss of structural information, and limited ability to capture nonlinear dynamics. This dissertation develops a unified mathematical framework for discovering and exploiting compressible structures in both linear and nonlinear settings. The framework combines tensor algebra with nonlinear algebraic representations to construct scalable and structure-preserving models for high-dimensional dynamical systems and complex data. The first part of this work studies compressive multilinear dynamical systems based on the t-product algebra and related transform-induced tensor products. By exploiting the block-circulant structure of tensor operators, tensor-valued dynamical systems can be diagonalized in the Fourier domain and decomposed into independent matrix-valued subsystems. This transform-domain decoupling enables efficient algorithms for system identification, controllability and stability analysis, and tensor extensions of classical model reduction methods such as balanced truncation, the eigensystem realization algorithm (ERA), and dynamic mode decomposition (DMD). These tensor-based approaches preserve multilinear structure while significantly reducing computational and memory costs. The second part investigates a nonlinear compression framework motivated by algebraic structures in layer and volume potentials arising from integral equation formulations of partial differential equations. By analyzing polynomial-root structures associated with boundary geometry, new representations are developed that express potential functions using low-dimensional algebraic features. These ideas lead to quadratic-formula-based nonlinear approximation models capable of capturing signals with mixed smoothness, oscillatory behavior, and sharp transitions. Overall, this work provides new mathematical foundations and numerical tools for scalable modeling and analysis of high-dimensional and noisy data.