# Section for Scientific Computing

We define scientific computing as the combined interdisciplinary activity bridging applications, matematical modeling, theory, and the associated numerical algorithms and their implementations.

In other words: Scientific Computing is the science of using computers and mathematics to solve problems from science and engineering. The challenging problems facing society and science today requires sophisticated mathematical methods and computational algorithms as well as the use of modern computer architectures. We develop both.

This section's expertise includes many of the aspects of Scientific Computing: from the modeling of physical phenomena to developing, analyzing, and implementing methodologies for the solution of real-life problems.

## Research areas

#### Computational Uncertainty Quantification

Uncertainty Quantification (UQ) is a branch of scientific computing where we characterize and study the sensitivity of a solution considering 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.

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#### Inverse Problems

Inverse problems arise in areas like tomography when reconstructing the interior structures of an object from x-ray measurements. In inverse problems it is not possible to observe a phenomenon directly, only the 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. Solutions to inverse problems are extremely sensitive to errors in the data, and there may not be a unique solution to the problem. It is therefore 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.

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#### Mathematical Modelling, Analysis, and Simulation

We study general boundary value problems controlled by boundary input, particularly 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 the behaviour with respect to non-smooth data and domains. In the MECOBS network (Modelling, Estimation and Control of Biotechnological Systems) we study various PDE models related to mathematical biology. We develop methods for topological and fast structural comparison of protein structures and investigate metric training of knowledge based potentials and its application to protein structure prediction.

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#### Optimization and Control

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 optimized while satisfying a number of design goals and/or operational requirements.  We also develop new generations of algorithms and toolboxes for applied model predictive control (MPC) in science and industry. This has applications in smart energy systems, reservoir engineering, medical systems, and process control.

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#### Computational Mathematics

Computational mathematics lies at the interface between research in mathematics and computer science, along with their areas of application. With the availability of increased computational resources, the importance of advanced computer simulations is becoming more central as a driver for new scientific discoveries, engineering analysis, and decision making. Computational mathematics enables these advanced simulations based on numerical solution of complex equations.