- Computational mathematics
- Inverse problems
- Mathematical modeling, analysis, and simulation
- Optimization and control
- Computational uncertainty quantification

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In many areas of engineering and science research in both mathematics and computing plays an essential and increasingly important role. Computational mathematics lies at the interface between research in mathematics and computer science combined with areas of application. With continued improvements in technology leading to increasing computational resources available, the importance and need for advanced computer simulations are becoming more central as a driver for new scientific discoveries, engineering analysis and decision making. Computational mathematics enables advanced simulations based on numerical solution of complex equations; parallel and high-performance computing and are needed for optimisation, control and uncertainty quantification.

*People: Allan Engsig-Karup*

Inverse problems arise, e.g., in tomography when we reconstruct the interior structures of an object from x-ray measurements. In inverse problems it is not possible to observe the sought phenomenon directly, but only an indirect effect of the phenomenon can be measured. In order to solve the inverse problem we need a mathematical model that allows us to determine the effect if we know the phenomenon. Such models are typically given in terms of differential or integral equations.

Inverse problems have a common difficulty known as ill-posedness, i.e., the solution is extremely sensitive to errors in the data, and there may not be a unique solution to the problem. It is necessary to use prior information to compute stable unique solutions through the use of regularization algorithms. Our current research focuses on various kinds of tomography, large-scale iterative algorithms, imaging, and paradigms for the incorporation of all relevant prior information in the reconstruction.

Recent projects include HD-Tomo and Improved Impedance Tomography using Hybrid Data

*People: Per Christian Hansen, Kim Knudsen*

**Mathematical modeling, analysis, and simulation**

**Functional Analysis and Control of PDE’s**
We study general boundary value problems controlled by boundary input, in particular the Hilbert Uniqueness Method. We quantify the regularity of the solution in terms of the regularity of the boundary data. We use variational as well as pseudodifferential methods. We are interested in well-posedness and unique continuation properties and also the behavior with respect to non-smooth data and domains.
**Nonlinear PDE’s and Mathematical Biology**
In the MECOBS network (Modelling, Estimation and Control of Biotechnological Systems) we study various PDE models related to mathematical biology. We have a strong collaboration with the HelmholtzZentrum and TUM in Munich, as well as a number of universities in the US and China.
**Computational Structural Biology**
We develop methods for topological and fast structural comparison of protein structures. We also investigate metric training of knowledge based potentials and its application to protein structure prediction.

*People: Michael Pedersen, Kim Knudsen, Mirza Karamehmedović
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*People: Michael Pedersen, Kim Knudsen*

*People: Peter Røgen
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**Optimization**
Mathematical optimization is an important tool in many science and engineering disciplines. Optimization models and methods are used for a wide range of tasks ranging from parameter estimation and data fitting to design and control of systems where some notion of system performance (a mathematical model) must be optimzed while satisfying a number of design goals and/or operational requirements. Advances in optimization, the availability of high-quality software tools, and the fast growth of computing power have all propelled the widespread adoption of optimization techniques in areas such as inverse problems, imaging signal processing, computational structural biology, and optimal control. In the Scientific Computing section we teach and conduct research in optimization techniques and formulations with a diverse set of applications, including:- Interior-point methods for conic optimization
- Algorithms for tomographic image reconstruction
- Model predictive control
- Enhanced oil recovery and production optimization
**Applied model predictive control (MPC) in science and industry**
We develop a new generation of algorithms and toolboxes for MPC, in collaboration with software companies and industry, and with applications in smart energy systems, reservoir engineering, medical systems, and process control. We will maintain our international recognition and visibility. The goal is to use that basis to be recognized as a world leader within computational aspects of MPC and selected applications.

*People: Martin S. Andersen, John Bagterp Jørgensen, Yiqiu Dong*

**Computational uncertainty quantification **

Uncertainty Quantification (UQ) is a branch of scientific computing where we characterize and study the sensitivity of a solution taking into account errors and inaccuracies in the data, models, algorithms, etc.

The systematic quantification of the uncertainties affecting computer models and the characterization of the uncertainty of their outcomes are critical for engineering design and analysis, where risks must be reduced as much as possible. Uncertainties stem naturally from our limitations in measurements, predictions and manufacturing.

In numerical computations, the curse of dimensionality is a problem that afflicts many problems. One approach to dealing with this is to use novel and efficient spectral (high-order) algorithms.

In the CUQI project we develop the mathematical, statistical and computational framework for applying UQ to inverse problems such as deconvolution, image deblurring, tomographic imaging, source reconstruction, and fault inspection. The goal is to create a computational platform, suited for non-experts, which can be used by many different industrial and academic end users.

*People: Allan Engsig-Karup, Per Christian Hansen*