This project seeks to establish a mathematical framework for analyzing sample complexity—specifically, the role of finite measurements—in inverse problems governed by partial differential equations (PDEs). Inverse problems occur frequently in science and engineering when an unknown quantity must be inferred indirectly from available data. The mathematical formulation typically relies on PDEs. Numerous imaging modalities, including ultrasonography, electrical impedance tomography, and photoacoustic tomography, fall within this category, with the specific PDE varying according to the underlying physical context.
Current mathematical theories often assume an infinite number of measurements, which contrasts sharply with real-world scenarios where only a finite—and often small—set of measurements is available. This gap is significant, as the limited number of measurements directly affects decisions about how data is collected, what assumptions (priors) are made about the unknown quantity, and the reconstruction methods used. Many promising imaging techniques remain underutilized because their reconstructions suffer from poor quality.
To address this, the project will integrate methods from PDE theory, numerical analysis, signal processing, compressed sensing, and machine learning. By combining these disciplines, it aims to construct a new theory of sample complexity tailored to PDE-based inverse problems. This work will provide a mathematical foundation for inverse problems that better reflects practical constraints, guiding the selection of measurements, priors, and reconstruction algorithms. Ultimately, this research will enhance the feasibility and effectiveness of emerging imaging technologies, bringing them closer to practical application.
In addition, the project is expected to generate novel results in compressed sensing, extending its applicability to a wider range of problems, including nonlinear and ill-posed scenarios.
