Sample complexity for inverse problems in PDE

Project Information

SAMPDE

Grant agreement ID: 101041040

DOI

10.3030/101041040

EC signature date 29 April 2022

Start date 1 November 2022

End date 31 October 2027

Funded under

• European Research Council (ERC)

Total cost

€ 1 153 125,00

EU contribution

€ 1 153 125,00

Coordinated by

UNIVERSITA DEGLI STUDI DI GENOVA
Italy

Periodic Reporting for period 1 - SAMPDE (Sample complexity for inverse problems in PDE)

Reporting period: 2022-11-01 to 2025-04-30

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Compressed sensing is the mathematical theory that allows for the reconstruction of sparse signals with a limited number of measurements, ideally directly proportional to the sparsity. Traditionally, the use of compressed sensing has been limited to specific scenarios, notably in which the reconstruction with full measurements is a stable process. The key focus of this project is addressing this subsampling issue for inverse problems, for which the reconstruction process is, by construction, unstable. We have developed an abstract compressed sensing theory for linear ill-posed inverse problems, by combining well-established statistical methods with functional analysis techniques for the analysis of the operator involved in the modeling.

A key aspect of the investigation is the interplay between the intrinsic infinite dimensionality of the models under consideration, and of the signals to be reconstructed, and the need of finite measurements. This truncation is especially delicate when dealing with an ill-posed problem, and has to be considered with care.

While compressed sensing techniques are based on deterministic prior knowledge about the signals to be reconstructed, machine learning methods are data-driven and have become very popular in recent years. In this project, we have obtained results regarding the theoretical aspects of learning the optimal regularizers for inverse problems, as well as on designing generative models in function spaces based on constructing neural networks as continuous convolution operators, and on manifold learning for manifolds with non-trivial topologies.

These theoretical insights are complemented by several numerical results. The main common outcome is the superiority of the methods that combine a model-based approach with machine learning, as an effective tool for considering priors that are well adapted to the class of signals under consideration.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The work on compressed sensing for linear inverse problems substantially goes beyond the state of the art. As a rather unexpected byproduct, we were able to obtain the first rigorous recovery result regarding the Radon transform, which models computed tomography. We have shown that it is possible to recover a sparse image (with respect to a suitable wavelet basis) from the subsampled Radon transform along a finite number of angles, directly proportional (up to logarithmic factors) to the sparsity.

The work on the continuous generative neural networks provides the first architecture of a generative model in function spaces. The current construction is based on a multi resolution analysis, and is not ideal for capturing and generating discontinuous signals. We are now investigating an alternative approach based on pseudo-differential operators, in order to overcome this issue.