CUQI is a research project at DTU Compute, headed by Prof. Per Chr. Hansen.
It is financed by VILLUM Fonden (the Villum Foundation) through a
Villum Investigator grant of DKK 34,899,255.
The project runs from Sept. 1, 2019 to November 30, 2025.

The CUQI team as of Nov. 2021. Left to right: Babak, Mirza, Aksel, Kim (tall guy), Felip, Dorte, Amal, Per Chr., Kristoffer, Martin, Katrine, Jakob, Silja, Rafael, Nicolai, Jasper, Puyuan, Yiqiu.

What is CUQI

CUQI is a research project where we develop a mathematical, statistical and computational framework for applying uncertainty quantification to inverse problems. We also develop a software package for UQ modeling and computations, see below.

Uncertainty Quantification, or simply UQ, characterizes the sensitivity of a solution taking into account errors and inaccuracies in the data, models, etc.

Inverse Problems determine hidden information from measurements in, e.g., deconvolution, image deblurring, tomographic imaging, source reconstruction, and fault inspection.

As an example, engineers who use X-ray imaging to inspect an object for defects can characterize the reliability of every details in the image - thus aiding the interpretation and decision whether a crack is actually present.

Click here to see slides with a brief introduction to inverse problems and UQ.

Click here to see video: how to handle view-angle errors in X-ray CT by means of UQ.

Funding

Villum Fonden (the Villum Foundation) is one of Denmark's largest research funding foundations with annual grants of approx. DKK 500 million. Their aim is to foster research at the highest international level platformed at Danish universities and research institutions.

The Villum InvestIgator

The project is headed by Prof. Per Christian Hansen who specializes in numerical analysis, matrix computations, iterative methods, and computational methods for inverse problems.

His recent research projects were:
> High-Definition Tomography (HD-Tomo) funded by an ERC Advanced Research grant
> Improved Impedance Tomography with Hybrid Data, together with Prof. Kim Knudsen, funded by the Danish Council for Independent Research.

Per Christian Hansen is the (co-)author of five books, 120 papers and seven software packages. His research focuses on computational methods, e.g., for regularization methods, imaging, tomography, and subspace methods for signal processing.

All his activities comprise a combination of theoretical insights and computational methods.

Per Christian Hansen's homepage

The Goal of the CUQI Project

Our research allows end-users of inverse problems, in science and engineering, to quantify the accuracy in their solutions, and in this way lower the risks and make more correct decisions.

We reach this goal by building general theoretical and computational foundations to quantify how model uncertainties, measurement noise, and other errors influence the solutions to inverse problems.

To do this we build on the solid mathematical and statistical foundation of uncertainty quantification. We also build a software package CUQIpy that allows experts and non-experts to apply UQ to their inverse problems, see below.

Based on our work, our aim is that UQ will become a natural part of solving inverse problems, in science and engineering.

Research

In inverse problems we operate with a very large number of unknowns (e.g., pixels, voxels) and the computational burden can be overwhelming. Also, UQ for inverse problems raises new fundamental scientific questions that we must identify, formulate and answer properly. Hence, we need research in both theory and computational methods.

Outcome

> The theoretical outcome is a mathematical and statistical basis for UQ studies of a range of inverse problems with different priors and noise models.
> The practical outcome is a modeling and computational platform – including an abstraction layer aimed at non-experts.
> The overall outcome is a framework to rigorously quantify the uncertainty of the solution’s details and identify the dominating types of errors.

Snapshots of CUQI research in 2023


CUQI was well represented at the AIP 2023 conference in Göttingen, Sept. 4-8, where we co-organized 7 mini-symposia and gave 12 presentations and one poster.

The presentations, which are available on this page, give a good snapshot of our research activities in 2023.

CUQIpy Software


To enable end-users to make use of our research, we release free open-source software that implements the theoretical and computational foundations developed during the project.

This is all released as part of the Python software package CUQIpy, which implements a computational abstraction layer for UQ studies of inverse problems aimed at non-experts.

In addition to the main software we also host individual software from papers and projects under the CUQI GitHub organization, including a growing collection of plugins for CUQIpy.

Click the logo to learn about our software.

Click this link to see the CUQI GitHub.

The picture shows the CUQIpy software team, as of Jan. 2023. From left to right: postdoc Nicolai A. B. Riis, postdoc Amal M. A. Alghamdi, and senior researcher Jakob S. Jørgensen.

Read more about the software in the CUQIpy preprints Part 1 and Part 2, and on this link: Free software quantifies uncertainties in scans.

And here is a YouTube video with a presentation of CUQIpy, given by Amal M. A. Alghamdi at the Bayes@Lund workshop on Jan. 23, 2023.

Examples of our Research in CUQI

UQ provides a tool to assess the "quality" of a reconstruction - a solution to an inverse problem - with respect to the influence of errors. It tells us how much we can trust the reconstruction and its details. UQ is useful in science, engineering, medical imaging and many other applications where we solve an inverse problem to reveal hidden information. For example, we can apply it in industrial inspection when looking for anomalies or defects in an object, in medical imaging when looking for malignant tissue, and in acoustics when identifying the location of unwanted sound sources.

Below we illustrate our work on computational UQ for inverse problems with a few examples from ongoing research projects in CUQI. These examples illustrate various methodologies for implementing, applying and using UQ. For more details see the references in the list of publications below.

Many of these examples use our CUQIpy software mentioned above.

Bayesian Inference with Constraints Through Projection

In many applications, the signal that is being reconstructed must satisfy some constraints, e.g., material densities being positive, pixels of an image lying between 0 and 1, or the energy of a signal being bounded. Therefore, adding such prior information to a Bayesian model improves the meaningfulness of the resulting posterior distribution.

A common method to model constraints in a prior distribution is truncation, i.e., choosing an unconstrained prior distribution and remove all probability outside of the constraint set. Another method is to reparametrize the signal, e.g., if the signal x is assumed to be positive, then it can be written as x = exp(z) and one can choose an unconstrained prior distribution for z. These two methods focus on the interior of the constraint set, but very often, signals of interest lie on the boundary of the constraint set.

As a natural alternative, one can first sample from an unconstrained posterior distribution and then apply a suitable projection onto a convex constraint set C. The resulting implicit posterior will consist of the original density in the interior of the constraint set, and a “lower dimensional” distribution on the boundary, see the illustration below.

If the forward model is linear and both the likelihood and prior are Gaussian distributions, then the unconstrained sampling and projection can be done in a single step by repeatedly solving randomized constrained linear least square problems of the form

minx∈C || A x - bs ||22 + α || x - cs ||22 ,

where bs and cs are samples from suitable random distributions. The solutions of these optimization problems will be samples of a projected posterior.

Consider the problem of deblurring a noisy 1D signal with prior knowledge that the signal lies in the interval [0,1]. The figure above shows the pointwise median and 99% credibility interval for such a problem using no constraints (left), only the lower bound (middle) and both upper and lower bounds (right). The projection framework significantly improves the quality of the median near the boundary.

This example was created by PhD student Jasper Everink.

The Horseshoe Prior for Edge-Preserving Reconstruction

If we want to reconstruct an image with sharp edges, we can use a Cauchy or Laplace distribution of the difference between neighbor pixels as the prior (this is related to total variation regularization). Unfortunately, these priors are computationally demanding. As a computationally attractive alternative, we can use a so-called horseshoe prior that resembles the Cauchy and Laplace priors - see the figure below where the horseshoe prior's density is guaranteed to lie in the shaded band.

The main advantage of the horseshoe prior is that it imposes a conditionally Gaussian distribution on the differences, which allows more efficient computations because Gaussians can be handled with efficient least-squares methods. The difficulty is that the horseshoe prior has hyperpriors with heavy tails, and to alleviate this issue we propose an extended horseshoe prior that uses a scale-mixture representation of the heavy-tailed hyperpriors.

For the posterior, we exploit the extended horseshoe prior to compute conditional distributions for the associated parameters that can be sampled in closed form due to conjugacy. This allows the application of a Gibbs sampler.

Here we illustrate our method with a 1D deconvolution problem for two different noise levels, and we compare with posterior statistics obtained with a Laplace-difference prior proposed by Uribe, Bardsley, Dong, Hansen & Riis (2022).  Compared to the Laplace-difference prior, we see that the horseshoe prior gives a sharper posterior (dotted lines) and a lower posterior uncertainty (the shaded area shows the 95% credibility interval). The solid line shows the ground truth.

The horseshoe prior originated in the statistical community, and here we introduce it in the setting of Bayesian inverse problem - see the paper by Uribe, Dong & Hansen (2023).

 

Boundary Detection in X-Ray CT Applications

When an X-ray passes through an object, its intensity is attenuated depending on the material's attenuation coefficient. The accumulated attenuation is the integral of the attenuation coefficient along the ray. We consider the case of 2D parallel-beam systems for which the measured data is called a sinogram.

Any X-ray going through the image is characterized by its slope θ with the horizontal axis. In practice, a full-angle geometry, i.e. when θ covers the entire interval [0,π), is not always possible. This could be due to obstacles in the CT measurement setup or to avoid unhealthy levels of radiation exposure. In such cases, a limited-angle geometry is needed, i.e., θ belongs to the reduced interval [0,θmax) with θmax ≪ π. In such a limited-angle setup, the quality of reconstructed image is compromised when conventional reconstruction methods, such as filtered back projection (FBP), are applied.

In many X-ray CT applications, the boundaries of objects contain valuable information, e.g., to differentiate between benign vs malignant tumors. A common approach in detecting boundaries is via an image segmentation post-processing step, but the quality of the estimated boundaries is compromised in a limited-angel imaging case.

We provide a novel goal-oriented, Bayesian framework for the limited-angel CT problem, in which we reconstruct the boundaries directly and quantify their uncertainty, and without the need for reconstructing the image. This approach avoids the error propagation and reduces the dimensionality of the problem from finding a 2D image to a 1D boundary of a region. In addition, our method estimates and performs UQ of the roughness/smoothness of the boundaries that carry important information for, e.g., medical imaging applications.

In the above plots, we see the performance of our method in estimating the boundaries of an object in the limited-angle CT setup with different θmax. The arrows indicate the angles of the X-rays and thus region of the limited-angle interval [0, θmax ). The estimated boundaries are shown in dark blue color, and the light blue region indicates the uncertainty in this estimation. The smaller the θmax the larger the uncertainty region - which is consistent with the results from microlocal analysis.

A paper by Afkham, Dong, and Hansen is submitted to SIAM/ASA J. Uncertainty Quantification. A related paper is submitted to IEEE Trans. Image Processing.

Defect Detection in X-Ray CT of Subsea Pipes

X-ray computed tomography (CT) is used to monitor the condition of subsea oil or gas pipes in operation, in order to detect defects that might cause leaks. In this example, we use data obtained in the test facilities at FORCE Technology. Computations are done with our CUQIpy software which draws upon the Core Imaging Library (CIL) for the CT models.

We formulate a Bayesian inverse problem with built-in defect detection. The goal is to detect defects and quantify their uncertainties. We express the CT problem as y = A (x+d) + e, where y is the measured X-ray absorption, e is data noise, and A is the linear forward model representing the physics and geometry of the measurements. We use a novel representation x + d of the unknown image to be reconstructed, expressed as a sum of two images; x contains the pipe structure and d contains potential defects.

In the Bayesian setting we formulate the joint posterior distribution p(x,d | y) ∝ p(y | x,d) p(x) p(d), where p(x,d | y) is the joint posterior distribution representing the solution to the CT problem, p(y | x,d) is the likelihood that represents the data misfit, while p(x) and p(d) are prior distributions representing any knowledge we have about the unknown images.

We impose priors that promote the structures we are looking for. For the pipe structure x, we use the structural Gaussian prior proposed by Christensen, Riis, Uribe & Jørgensen (submitted). The above figure shows a sample from this prior which promotes the known layered pipe structure. We expect small and few defects, and therefore we impose a prior that promotes both sparsity and correlation (a gamma Markov random field) in the defect image d (as described in a paper by Christensen, Riis, Pereyra & Jørgensen). We explore the conditional posterior distributions p(x | y,d) and p(d | y,x) using Gibbs sampling.

The figure above shows the means of the x-samples (left), the d-samples (middle), and their sum with annotation (right). These results indicate that our methodology has successfully separated the defects from the overall pipe structure. In the figure below we zoom-in on the defect reconstruction d for the annotated defects indicated in the figure above, for further investigation of these defects. The figure shows the mean of the posterior samples, as well as related UQ in the form of the standard deviation of the samples. Engineers can use these results to identify critical defects in the subsea pipes.

This is joint work with Assoc. Prof. Marcelo Pereyra from Heriot-Watt University. The example was created by PhD student Silja L. Christensen.

Handling Model Errors in Inverse Problems

Using an ideal and complex mathematical model will produce a good reconstruction in an inverse problem, but at a high computational cost. Instead, one can use a simplified approximate model that requires less computational work, but this often results in a poorer reconstruction. We use the Bayesian framework to study the model error between the ideal and approximate models and find ways to reduce its influence when solving a Bayesian inverse problem.

We consider the general model
y = F(x,η) + e ,
where y is the data, x is the reconstruction, F is the exact forward operator with model parameters η, and e is the noise. Given an approximate forward operator f and an initial estimate η0 of η, we obtain the model
y = f(x,η0) + δ + e   with   δ = F(x,η) − f(x,η0) ,
where δ denotes the model error. The corresponding likelihood then takes the form
p(y | x) = pv|x(y − f(x,η0) | x)     with     v = δ + e .

Since p(v | x) is difficult to characterize, we may assume that δ and x are independent. Samples of δ can then be generated by "pushing forward" samples of x from the prior for x through the relation δ = F(x,η) − f(x,η0). Unfortunately, such samples of δ are inferior because they lack information related to the likelihood p. We can obtain better samples of δ by pushing forward samples of x obtained via the posterior — and this process can be repeated in an iterative fashion.

We develop methods based on the above ideas and apply them to reconstruction problems in magnetic resonance electrical impedance tomography (MREIT) where the solution x represents the electric conductivity inside a circular domain.

The plots below show that our method improves on results obtained by either neglecting the model error or representing it approximately. Specifically, we show the exact solution x (top left) and estimates of x obtained without the model error (top right), with the model error obtained by pushing forward samples from the prior (bottom left), and with the model error obtained by pushing forward samples from the posterior, repeated three times (bottom right).

This is joint work with Prof. Bangti Jin from Chinese University of Hong Kong. The example was created by PhD student Puyuan Mi.

UQ with Nonnegativity Constraints

Nonnegativity constraints appear in many applications where the underlying physics dictates that the solution cannot be negative. This is the case, e.g., for absorption coefficients in computed tomography, image intensities in astronomical imaging, and wave velocities in seismic travel-time tomography. Hence, nonnegativity constraints are very important for producing meaningful reconstructions.

We considers computational aspects of UQ with nonnegativity constraints x ≥ 0. If we replace the nonnegativity constraint with a positivity constraint, x > 0, then we can handle this constraint within the "standard" UQ framework with a transformation of variables x = exp(z). In this work we specifically address situations where we expect the existence of zero elements in the solution – in fact, in some imaging problems a substantial fraction of the solution's elements/pixels may be zero. Therefore we use a nonnegative Gaussian distribution.

The above figure shows a simple Gauss distribution (left), a truncated Gauss distribution with zero probability for x = 0 (middle), and a nonnegative Gaussian distribution with nonzero probability for x = 0 (right).

We use a Markov chain Monte Carlo (MCMC) method to sample the posterior with constraints, and we refer to our method as a Nonnegative Hierarchical Gibbs Sampler.

We applied our computational method to positron emission tomography (PET), where a radioactive tracer element is injected into a body and exhibits radioactive decay, resulting in photon emission. The emitted photons that leave the body are recorded by a photon detector, and the resulting mathematical reconstruction problem is equivalent to a classical X-ray CT problem. The above figure shows our simulation results.

  • Top left: the mean of the computed image samples from our nonnegative hierarchical Gibbs sampler.
  • Bottom left: histogram of the regularization parameters produced by the sampler.
  • Top right: the MAP estimate using the mean of the regularization parameters.
  • The pixel-wise standard deviations from the sampler. As expected, image pixels with higher intensity also have higher variance.

For details see the paper by Bardsley & Hansen (2020).

The Team

All members of the CUQI team are associated with the Section for Scientific Computing at DTU Compute.

Follow this LINK to see pictures from our team building event in the spring of 2023.

Permanent Team Members

Martin Skovgaard Andersen

Martin Skovgaard Andersen Associate Professor

Specialist in convex optimization and numerical optimization algorithms.
Yiqiu Dong

Yiqiu Dong Associate Professor

Specialist in noise modeling and computational methods for image processing.

Link to portrait.

Per Christian Hansen

Per Christian Hansen Professor

Specialist in numerical analysis, iterative solvers, and computational methods for inverse problems.

Link to portrait.

Jakob Sauer Jørgensen

Jakob Sauer Jørgensen Senior Researcher

Specialist in methods and software for image reconstruction and their application in X-ray CT.
Mirza Karamehmedovic

Mirza Karamehmedovic Associate Professor

Specialist in functional analysis and physical modeling with uncertainty quantification.
Kim Knudsen

Kim Knudsen Professor

Specialist in coupled-physics modeling, PDE-constrained optimization, functional analysis and electrical impedance tomography.

Link to portrait.

Dorte Olesen

Dorte Olesen Consultant

Project coordinator.

Anette Iversen

Anette Iversen Administrativ Coordinator

Project secretary.

Postdocs

Babak Maboudi Afkham

Babak Maboudi Afkham Postdoc

Project: goal-oriented uncertainty quantification.

Amal Mohammed A Alghamdi

Amal Mohammed A Alghamdi Postdoc

Project: computational UQ for PDE problems.

Rémi Laumont

Rémi Laumont Postdoc

Project: machine learning and UQ for inverse problems.

Nicolai Andre Brogaard Riis

Nicolai Andre Brogaard Riis Postdoc

Project: efficient and flexible computational methods.

Former Postdocs

Project: non-Gaussian priors.

Now with LUT University, Finland; homepage.

PhD Students

Lara Baalbaki

Lara Baalbaki PhD student

Project: Bayesian inference for inverse problems with sampling conditioned on functionals.
Silja Westphal Christensen

Silja Westphal Christensen PhD student

Project: UQ for tomographic reconstruction.

(Currently on parental leave.)

Jasper Marijn Everink

Jasper Marijn Everink PhD student

Project: computational UQ for inverse problems with implicit priors.
Rafael Flock

Rafael Flock PhD student

Project: dimensionality challenges in UQ for inverse problems.
Andreas Horst

Andreas Horst PhD student

Project: inverse problems with Besov prior.
Kristoffer Linder-Steinlein

Kristoffer Linder-Steinlein PhD student

Project: UQ for source localization and passive imaging in random media
Puyuan Mi

Puyuan Mi PhD student

Project: model error reduction in inverse problems. 

Former PhD Students

Katrine Ottesen Bangsgaard

Katrine Ottesen Bangsgaard PhD student

Sept. 1, 2019 to June 30, 2023 (incl. parental leave).

Project: prior modeling in computational UQ for inverse problems.

Now a statistician with Novo Nordisk Research & Development.

Aksel Kaastrup Rasmussen

Aksel Kaastrup Rasmussen PhD student

Sept. 1, 2020 to August 31, 2023.

Project: computational UQ of hybrid inverse problems.

Collaborators


Professor Johnathan M. Bardsley, Department of Mathematical Sciences, University of Montana - specialist in computational methods for inverse problems and uncertainty quantification.
Associate Professor Julianne Chung, Department of Mathematics, Emory University - specialist in in computational methods for inverse problems and uncertainty quantification in imaging applications.

Associate Professor Matthias Chung, Department of Mathematics, Emory University - specialist in computational inverse problems, data analytics & learning, uncertainty quantification, and numerical optimization.

Professor Bangti Jin, Department of Mathematics, Chinese University of Hong Kong - specialist in theory and algorithms for inverse and ill-posed problems.

Jakob Lemvig

Jakob Lemvig Associate Professor

DTU Compute, Section for Mathematics - specialist in wavelets, frames, and signal analysis.

Associate Professor Marcelo Pereyra, Maxwell Institute for Mathematical Science and School of Mathematical and Computer Sciences, Heriot-Watts University - specialist in statistical, analytical and machine-learning paradigms.

Professor Tanja Tarvainen, Department of Applied Physics, University of Eastern Finland - specialist in computational methods for inverse problems and uncertainty quantification in imaging applications

Research Professor Faouzi Triki, Laboratoire Jean Kuntzman, Grenoble-Alpes University - specialist in inverse problems and mathematical modeling in optics.
Postdoc Felipe Uribe, LUT University - specialist in applied probability & statistics, and large-scale sampling methods.
Research Scientist Olivier Zahm, INRIA-Grenoble - specialist in model order reduction for uncertainty quantification.

Courses, Workshops, Training and Other Activities

CUQIpy Training Course

December 7-8, 2023

A two-day training course at DTU, aimed at introducing the CUQIpy software to new users. After a brief introduction to UQ for inverse problems, we describe the CUQIpy interface that allows users to easily model and analyze linear and nonlinear inverse problems. This is followed by computer exercises and work on a small project. Users are welcome to bring their own problems and data.

Participants must be familiar with python and inverse problems.More details about the course and registration will follow soon.

Past Activities

 

Per Christian Hansen - Edge-Preserving Computed Tomography (CT) with Uncertain View Angles

Professor Per Christian Hansen gave an invited to talk on Feb. 7, 2023 in the online seminar series MATH4UQ organized by RWTH Aachen University. Here are the slides and a YouTube video.

Workshop: Imaging with Uncertainty Quantification (IUQ22)

September 27-29, 2022

   

This workshop brought together specialists in UQ for imaging, and the talks covered various aspects related to the development of theory, methodology and software, as well as applications of UQ in imaging. The goal was to stimulate networking and collaboration between researchers and students in these areas.

Before the workshop, we arranged a 1-day short course devoted to the Python software CUQIpy that we are currently developing for modeling and computations related to UQ for imaging.

For more details about the workshop, go to the IUQ22 Workshop homepage.

Felipe Uribe - Bayesian Inverse Problems

Postdoc Felipe Uribe was invited to talk about Bayesian inverse problems at the "Summer School on Recent Advancements in Computational and Learning Methods for Inverse Problems" (CLIP22), July 11–15: https://bugs.unica.it/cana/clip22/

Jakob Sauer Jørgensen - CUQIpy

Senior researcher Jakob Sauer Jørgensen was invited to talk about the software package CUQIpy at the "CIMPA Summer School 2022 Mathematical Methods in Data Analysis", July 18–29: https://sites.google.com/view/mathschoolinalbania/

PhD course: Introduction to Uncertainty Quantification for Inverse Problems

January 2022

This course introduces state-of-the-art numerical methods for quantification and reduction of uncertainties in computational models. UQ is paramount to enhance analysis and prediction tasks in multiple applications such as tomography, material science, spatial statistics, reliability, etc. Therefore, the course can be of interest to students from any discipline in applied mathematics and engineering. The course provides the mathematical background for theory and methods of UQ, which are illustrated via Python exercises. We will also cover the use of the software package CUQIpy for performing UQ analysis.
Ects: 5.
Time: January 3–22, 2022 (3 week period at DTU). Next time: June 2024.
Course responsible: Associate Professor Yiqiu Dong, DTU Compute.
More details: Link to DTU's course description, course no. 02975.

PhD course: Bayesian Scientific Computing

December 2019

The lectures focus on basic techniques in Bayesian methods, including probability distributions, Bayes' formula, conditioning, hierarchical models, estimation problems arising in this context, as well as certain numerical techniques for inverse problems, including regularization and iterative methods for solving large systems. The course is based on the book: D.  Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing, Springer, 2007, as well as a new edition of it.
Time: December 9-13, 2019 (one full week).
Teachers: Professor Daniela Calvetti and Professor Erkki Somersalo, both from Case Western Reserve University, Cleveland, Ohio.
Course responsible: Professor Per Christian Hansen, DTU.

Workshop: Uncertainty Quantification for Inverse Problems

December 17 and 18, 2018

The goal of this workshop was to give the participants an introduction to the central ideas and computational methods for uncertainty quantification, with a focus on its application to inverse problems, and with illustrations from applications. The workshop is aimed at newcomers in the field, but more experienced user will also benefit from the presentations.

Workshop homepage with slides etc.

Internships

Our summer internships give master students a unique opportunity to build on their skills, and to obtain experience with research in computational methods and applications of uncertain quantification to inverse problems.

We expect to offer two internships in 2024.

Our interns

2022. Louis Poulain-Auzéau from EPFL, Switzerland: Neural networks for acoustic and electromagnetic field control.

2023. Marco Ratto, Università de Cagliari, Italy: Implementation of fast convolution algorithms in CUQIpy.

PUBLICATIONS

Here we list submitted and published papers produced in the CUQI project.

  1. B. M. Afkham, J. Chung, and M. Chung, Learning regularization parameters of inverse problems via deep neural networks, Inverse Problems, 37 (2021), 105017, doi 10.1088/1361-6420/ac245d.
  2. B. M. Afkham, J. Chung, and M. Chung, Goal-oriented uncertainty quantification for inverse problems via variational encoder-decoder networks, submitted, arxiv.org/abs/2304.08324.
  3. B. M. Afkham, Y. Dong, and P. C. Hansen, Uncertainty quantification of inclusion boundaries in the context of X-ray tomography, SIAM/ASA J. Uncertainty Quantification, 11 (2023), pp. 31-61, doi 10.1137/21M1433782.
  4. B. M Afkham, K. Knudsen, A. K. Rasmussen, and T. Tarvainen, A Bayesian approach for consistent reconstruction of inclusions, submitted, arxiv.2308.13673.
  5. A. M. Afkham, N. A. B. Riis, Y. Dong, and P. C. Hansen, Inferring features with uncertain roughness, arxiv.org/abs/2305.04608.
  6.  A. M. A. Alghamdi, P. Chen, and M. Karamehmedović, Optimal design of photonic nanojets under uncertainty, submitted, arxiv.org/abs/2209.02454.
  7. A. M. Alghamdi, N. A. B. Riis, B. M. Afkham, F. Uribe, S. L. Christensen, P. C. Hansen, and J. S. Jørgensen, CUQIpy – Part II: computational uncertainty quantification for PDE-based inverse problems in Python, submitted to Inverse Problems, arxiv.org/abs/2305.16951.
  8. E. Ametova, G. Burca, S. Chilingaryan, G. Fardell, J. S. Jørgensen, E. Papoutsellis, E. Pasca, R. Warr, M. Turner, W. R. B. Lionheart, and P. J. Withers, Crystalline phase discriminating neutron tomography using advanced reconstruction methods, J. Phys. D: Appl. Phys., 54 (2021), 325502, doi 10.1088/1361-6463/ac02f9 (open access).
  9. E. Ametova, G. Burca, G. Fardell, J. S. Jørgensen, E. Papoutsellis, E. Pasca, R. Warr, M. Turner, W. R. B. Lionheart, and P. J. Withers, Joint reconstruction with a correlative regularisation technique for multi-channel neutron tomography, Proc.16th International Meeting on Fully 3D Image Reconstruction in Radiology and Nuclear Medicine, pp. 431–434 (2021), no doi; arxiv.org/abs/2110.04143.
  10. K. O. Bangsgaard and M. S. Andersen, A statistical reconstruction model for absorption CT with source uncertainty, Inverse Problems, 37 (2021), 085009, doi 10.1088/1361-6420/ac11c7.
  11. K. O. Bangsgaard, M. Andersen, J. G. Heaf, and J. T. Ottesen, Bayesian parameter estimation for phosphate dynamics during hemodialysis, Math, Biosci. Eng., 20 (2022), pp. 4455–4492, doi 10.3934/mbe.2023207 (open access).
  12. K. O. Bangsgaard, G. Burc, E. Ametova, M. S. Andersen, and J. S. Jørgensen, Low-rank flat-field correction for artifact reduction in spectral computed tomography, Appl. Math. in Sci. Eng., 31 (2023), 2176000, doi 10.1080/27690911.2023.2176000 (open access).
  13. J. M. Bardsley and P. C. Hansen, MCMC algorithms for computational UQ of nonnegativity constrained linear inverse problems, SIAM J. Sci. Comput., 41 (2020), pp. A1269–1288, doi 10.1137/18M1234588.
  14. D. Caviedes-Nozal, F. M. Heuchel, J. Brunskog, N. A. B. Riis, and E. Fernandez-Grande, A Bayesian spherical harmonics source radiation model for sound field control, J. Acoust. Soc. Am., 146 (2019), pp. 3425–3435, doi 10.1121/1.5133384.
  15. D. Caviedes-Nozal, N. A. B. Riis, F. M. Heuchel, J. Brunskog, P. Gerstoft, and E. Fernandez-Grande, Gaussian processes for sound field reconstruction, J. Acoust. Soc. Amer., 140, article id 1107 (2021), doi 10.1121/10.0003497.
  16. D. Caviedes-Nozal, N. A. B. Riis, F. M. Heuchel, J. Brunskog, P. Gerstoft, and E. Fernandez-Grande, Acoustic Gaussian processes, J. Acoust. Soc. Am., 149 (2021), pp. 1107–1119, doi 10.1121/10.0003497.
  17. S. L. Christensen, N. A. B. Riis, F. Uribe, and J. S. Jørgensen, Structural Gaussian priors for Bayesian CT reconstruction of subsea pipes, Appl. Math. in Sci. and Eng., 31 (2023), article ID: 2224918, doi 10.1080/27690911.2023.2224918 (open access).
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Bonus Material

Here is a UQ Test (created by Kristoffer) that can reveal how well you are versed in the details and mysteries of UQ: link to pdf file.