DUDE — Data Uncertainty and Design/Reconstruction Errors

A research project headed by professor Per Christian Hansen, DTU Compute

Funded by Independent Research Foundation Denmark, from August 1, 2026 to July 31, 2030

Mathematical design and reconstruction tasks are everywhere. MR and PET scanners aid brain diagnostics via reconstruction of the blood flow in 3D images with millions of voxels. Datasets of hundreds of images taken inside the mouth are routinely used to compute a digital model of tooth’s surface, which is then used to 3D-print a crown. There many other such tasks, e.g., in antenna design, non-descructive testing, control of tokamak fusion plasma, and astronomical imaging of distant objects.

In all these tasks we use a complex mathematical formulation to compute the desired design parameters and the reconstructed images. Users always need to assess how the computed results are influence by the inevitable errors in the data.
- Are there unwanted artifacts or errors in medical CT or blood-flow images that lead to misdiagnosis?
- Are there errors in the designed shape of the crown that prevents a perfect fit?
- Do we risk instabilities in a tokamak reactor if the reconstruction of the plasma flow is not entirely correct?

We need mathematical tools to help answer these questions, tailored to problems with large datasets where, due to the complexity and computing times, standard “text-book” tools are unsuited.

Iterative regularization

When solving problems with large amounts of data, classical methods are infeasible due to their long computing times and excessive need for storage. Instead, we use iterative reconstruction methods that, from an initial guess, produce a sequence of increasingly better solutions. Kaczmarz’s method is popular in X-ray tomography; CGLS is used in image deblurring.

Noise propagation

We need to know how the noise from the data propagates through the iterations in CGLS, GMRES, and other Krylov subspace methods. A basis for such studies is given in this paper which sets the stage for the present project:

P. C. Hansen, Insight into semi-convergence of iterative regularization methods, Lin. Alg. Appl. (2025), doi 10.1016/j.laa.2025.07.036.

In particilar, the above paper provides insight about the statistics of noise propagation in the CGLS method. A similar analysis of noise propagation in Kaczmarz's method is given in this paper:

P. C. Hansen and M. E. Hochstenbach, On spectral proterties and fast initical convergence of the Kaczmarz method, BIT Numerical Methods, 66 (2026), paper 8, doi 10.1007/s10543-025-01098-1.

Finally, some numerical experiments that illustrate noise propagation in the GMRES method can be found in this paper:

P. C. Hansen, K. Hayami, and K. Morikuni, GMRES methods for tomographic reconstruction with an unmatched back projector, J. Comp. Appl. Math., 413 (2022), 114352, doi 10.1016/j.cam.2022.114352.

Goals and problem formulation

The goal is to give users of design and reconstruction problems a tool to quantify the uncertainties in the computed results, in cases where large data sets must be handled by iterative methods. The project involves these ingredients:

Development of theoretical results, e.g., for the statistics of the growth of the noise error during the iterations.
Develop a formalism for expressing the regularizing properties of CGLS and other Krylov subspace in a Bayesian setting.
Developent of computational methods and software that can efficiently provide rigorous characterization of the propagation of errors in large-scale design and reconstruction problems.
Ensuring that the new methods work for a range of design and reconstruction problems with the same underlying mathematical structure (inverse problems) as well as the need for computer methods suited for problems with large datasets (iterative methods).

Collaborations