PhD Scholarships in Computational Uncertainty Quantification for Inverse Problems

We invite applications for several PhD positions, starting in the fall of 2021, in the Section for Scientific Computing within the Department of Applied Mathematics and Computer Science. The positions are part of the research initiative CUQI: Computational Uncertainty Quantification for Inverse problems funded by the Villum Foundation and headed by Professor Per Christian Hansen.

Image deblurring, tomographic imaging, source reconstruction, and fault inspection are examples of inverse problems. Uncertainty Quantification (UQ) uses methods from Bayesian inference to characterize and study the sensitivity of a reconstruction, taking into account errors and inaccuracies in the data as well as the mathematical and physical models, the regularization terms, etc. Our goal is to create a platform for modeling and computations related to applying UQ to a range of inverse problems in academia and industry.

Responsibilities and qualifications
You will be part of a large team consisting of experts in many areas of inverse problems and scientific computing. Together with the team, you will contribute to the project’s goal by developing theory and computational methods that can handle the challenges we face in developing a versatile platform. 

These PhD positions will focus on four important areas:

1. Dealing with large-scale inverse problems in the CUQI platform, we face a dimensionality challenge that calls for dimension reduction techniques, surrogate modeling, multi-fidelity sampling algorithms, etc. Along this line, we must also study different types of model errors and techniques for handling errors and uncertainties in the reconstruction models. This project requires knowledge of numerical analysis and numerical optimization methods.
2. A different way to handle the dimensionality challenge in the CUQI platform is to utilize recent progress on stochastic optimization methods to construct efficient sampling methods. These techniques allow us to implicitly handle given prior/posterior distributions (e.g., for constrained problems) without the need to tune algorithm parameters. This project requires knowledge of numerical optimization. Familiarity with Bayesian sampling methods is a plus but not a necessity.
3. In Bayesian inference, we often face uncertain parameters in the likelihoods and priors. Therefore, we introduce hyper-parameters with associated hyper-priors, which must be compatible with the data and reconstruction model. The hyper-parameters generalize the regularization parameters from classical methods. We need theory behind the hyper-priors as well as diagnostic tools to check these priors within the CUQI platform. This project requires knowledge of numerical linear algebra and numerical computations.
4. In many inverse problems, we need to go beyond Gaussian priors in order to handle more advanced spatial correlations. In the CUQI platform we will use Besov priors that are suited for producing piecewise smooth reconstructions and for detection of edges and interfaces. This involves the use of linear combinations of wavelets/frames with random coefficients. This project requires knowledge of numerical PDEs, functional analysis and, preferably, harmonic analysis and/or probability theory.

In addition to the above-mentioned requirements, you must have experience with inverse problems or Bayesian inference.

You must have a two-year master's degree (120 ECTS points) or a similar degree with an academic level equivalent to a two-year master's degree. 

Applicable for all scholarships

Approval and Enrolment
The scholarships for the PhD degree are subject to academic approval, and the candidates will be enrolled in one of the general degree programmes at DTU. For information about our enrolment requirements and the general planning of the PhD study programme, please see the DTU PhD Guide. 

Assessment
The assessment of all applicants will be made by Professor Per Christian Hansen, Associate Professor Martin S. Andersen, and Associate Professor Yiqiu Dong.

We offer
DTU is a leading technical university globally recognized for the excellence of its research, education, innovation and scientific advice. We offer a rewarding and challenging job in an international environment. We strive for academic excellence in an environment characterized by collegial respect and academic freedom tempered by responsibility.

Salary and appointment terms
The appointment will be based on the collective agreement with the Danish Confederation of Professional Associations. The allowance will be agreed upon with the relevant union. The period of employment is 3 years.

The workplace is DTU Lyngby Campus, Denmark.

You can read more about career paths at DTU here.  

Further information
Further information about the positions and CUQI may be obtained from Professor Per Christian Hansen pcha@dtu.dk. 

You can read more about the Department of Applied Mathematics and Computer Science at www.compute.dtu.dk/english. 

If you are applying from abroad, you may find useful information on working in Denmark and at DTU at DTU – Moving to Denmark. 

Application procedure
Your complete online application must be submitted no later than 25 May 2021 (Danish time). Applications must be submitted as one PDF file containing all materials to be given consideration. To apply, please open the link "Apply online", fill out the online application form, and attach all your materials in English in one PDF file. The file must include:

• A letter motivating the application (cover letter). In the cover letter, please indicate one or at most two of the above area(s) you would like to work with, and how your background aligns with this choice.
• Curriculum vitae
• Grade transcripts and BSc/MSc diploma
• Excel sheet with translation of grades to the Danish grading system (see guidelines and Excel spreadsheet here – in the right hand column)

In the field “Please indicate which position(s) you would like to apply for”, please indicate which project you are applying for (title from the above list of PhD projects or individual research projects).

Incomplete applications will not be considered. You may apply prior to ob­tai­ning your master's degree but cannot begin before having received it.

All interested candidates irrespective of age, gender, disability, race, religion or ethnic background are encouraged to apply.

DTU Compute
DTU Compute is a unique and internationally recognized academic environment spanning the science disciplines mathematics, statistics, computer science, and engineering. We conduct research, teaching and innovation of high international standard—producing new knowledge and technology-based solutions to societal challenges. We have a long-term involvement in applied and interdisciplinary research, big data and data science, artificial intelligence (AI), internet of things (IoT), smart and secure societies, smart manufacturing, and life science.

The Section for Scientific Computing has a strong track record of research within various branches of applied mathematics, including PDEs, inverse problems, numerical linear algebra, and optimization.

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DTU develops technology for people. With our international elite research and study programmes, we are helping to create a better world and to solve the global challenges formulated in the UN’s 17 Sustainable Development Goals. Hans Christian Ørsted founded DTU in 1829 with a clear vision to develop and create value using science and engineering to benefit society. That vision lives on today. DTU has 12,900 students and 6,000 employees. We work in an international atmosphere and have an inclusive, evolving, and informal working environment. Our main campus is in Kgs. Lyngby north of Copenhagen and we have campuses in Roskilde and Ballerup and in Sisimiut in Greenland.