Sample complexity for inverse problems in PDE

Introduction

This project will develop a mathematical theory of sample complexity, i.e. of finite measurements, for inverse problems in partial differential equations (PDE). Inverse problems are ubiquitous in science and engineering, and appear when a quantity has to be reconstructed from indirect measurements.

Importance of Physics

Whenever physics plays a crucial role in the description of an inverse problem, the mathematical model is based on a PDE. Many imaging modalities belong to this category, including:

• Ultrasonography
• Electrical impedance tomography
• Photoacoustic tomography

Many different PDEs appear, depending on the physical domain.

Current Challenges

Currently, there is a substantial gap between theory and practice:

1. All theoretical results require infinitely many measurements.
2. In all applied studies and practical implementations, only a finite number of measurements are taken.

We argue that this gap is crucial, since the number of measurements is usually not very large, and has important consequences regarding:

• The choice of measurements
• The priors on the unknown
• The reconstruction algorithms

Many safe and effective modalities have had very limited use due to low reconstruction quality.

Proposed Solution

Within a multidisciplinary approach, by combining methods from:

• PDE theory
• Numerical analysis
• Signal processing
• Compressed sensing
• Machine learning

We will bridge this gap by developing a theory of sample complexity for inverse problems in PDE. This will allow for the deriving of a new mathematical theory of inverse problems for PDE under realistic assumptions, which will impact the implementation of many modalities, guiding the choice of priors and measurements.

Expected Outcomes

Consequently, emerging imaging modalities will become closer to actual usage. As a by-product, we will also derive new compressed sensing results which are valid for a general class of problems, including nonlinear and ill-posed problems, and sparsity constraints.

Collaboration

Collaborations with experts in the relevant fields will ensure the project’s success.