This project develops a mathematical theory of sample complexity – that is, of finite measurements – for inverse problems in partial differential equations (PDEs).

Inverse problems are ubiquitous in science and engineering and arise when a quantity must be reconstructed from indirect measurements. When physics plays a fundamental role in the description of an inverse problem, the mathematical model is based on a PDE. Many imaging techniques fall into this category, including ultrasound, electrical impedance tomography and photoacoustic tomography. Depending on the physical domain under consideration, different types of PDEs may come into play.

Currently, there is a significant gap between theory and practice: all theoretical results require an infinite number of measurements, whilst in applied studies and practical implementations only a finite number of measurements are taken. We argue that this gap is crucial, as the number of available measurements is generally limited, with significant consequences for the choice of measurements, the a priori assumptions regarding the unknown quantities, and the reconstruction algorithms. Many reliable and effective techniques have seen limited adoption precisely because of the poor quality of the reconstruction.

Through a multidisciplinary approach, combining methods from the theory of PDEs, numerical analysis, signal processing, compressed sensing and machine learning, the project aims to bridge this gap by developing a theory of sampling complexity for inverse problems based on PDEs.

This enables the development of a new mathematical theory of inverse problems for PDEs based on realistic assumptions, with a significant impact on the implementation of numerous imaging techniques, guiding the choice of both prior information and the measurements to be taken. Consequently, emerging imaging techniques will come closer to practical and widespread use.

As a side benefit, the project aims to achieve new results in the field of compressed sensing, applicable to a wide class of problems, including non-linear, ill-posed and sparsity-constrained problems. Collaboration with experts in the relevant fields ensures the project’s success.