# Research on Applied Mathematics

The research in Applied Mathematics is divided into eight areas which are Mathematical modeling in continuum mechanics, Functional analysis, Geometric analysis and general relativity, Applied Fourier analysis, Spatial and spatio-temporal statistics, Machine learning, Non-linear PDEs and Numerical linear algebra and model fitting.

## Spatial and spatio-temporal statistics

Spatial statistics is dedicated to analyzing data that displays spatial dependencies, involving point pattern data or geostatistical data.

Point pattern data manifests at various spatial scales, encompassing observations such as the random locations of point-like defects in silicon crystals, the locations of tumor cells and the immune system in microscopic images of lymph nodes, crime scene locations, the locations and times of infected animals, as well as the epicenters and timing of earthquakes. Essential hypotheses involve understanding how the occurrence of points correlates with environmental variables and determining whether point patterns exhibit inhomogeneity, clustering, or regularity. For example, in the context of crime data, a pivotal question is how crime events are related to socio-economic variables or policing strategies. In the analysis of forest stand data, there is an interest in exploring the relationship between tree locations and covariates such as terrain elevation, soil quality, and understanding how tree patterns vary due to climate change.

Geostatistical data comprises geographic observations of a variable of interest, such as measurements of soil nutrients in a field where the spatial locations of soil samples are intrinsic to the data. In geostatistical data, sampling locations are typically considered fixed, and the objective is to analyze variations in the measured variable across space.

Research within the applied mathematics division primarily centers on spatial and spatio-temporal point pattern data. Key research areas involve the creation of realistic parametric statistical models tailored for spatial and spatio-temporal point patterns, investigating the computational and theoretical facets of parameter estimation, and employing inference methods for these models. Additionally, there is a notable focus on non-parametric summary statistics, offering the flexibility to explore scientific hypotheses without the need to prescribe a specific model for the data. In this context, we are also exploring the integration of machine learning techniques into spatial statistics.

## Geometric analysis and general relativity

Geometric analysis is in some sense the study of curvature. Since curvature is essentially a second derivative, the field lies at the intersection of differential geometry and the analysis of partial differential equations (PDEs). In other words, geometric analysis can be viewed as the study of geometry through the analysis of PDEs.

General relativity is arguably Einstein’s greatest contribution to science and is our current accepted model of both gravity and the very universe itself. The cornerstone of the theory an equation that relates the curvature of space-time to the physical matter content in the universe, and leads us to the conclusion that gravity is not a force in the usual sense of physics, but rather an effect due to the curvature of space and time. This means that understanding both gravity and the large scale structure of the universe comes down to understanding curvature – that is, it falls into the purview of geometric analysis.

Interestingly, it is not only the mathematics that guides an understanding of the physics but also the physics can lead us to a better understanding of geometric problems that may a priori seem totally unrelated to physics. Most of the research in this area at LTU focuses on this interplay between physics and geometry through the relationship between energy and scalar curvature. More specifically, our research focuses on geometric inequalities in the context of initial data for the Einstein equations, and on the problem of quasi-local mass in general relativity.

## Applied Fourier Analysis

In recent years, research has mainly focused on signal processing applications for various types of vibration analysis. In collaboration with the Division of Structural and Fire Engineering, as well as external stakeholders from industry and society, vibration measurements on bridges and other building structures have been analyzed. Using finite element models and various optimization methods, new damage detection methods have been developed, where so-called sparse representations have been employed to exploit the fact that the damages that occur usually begin as very localized.

Another current application is a collaboration with the steel industry and KTH for measuring the stirring of the steel during steel production. The stirring is performed by blowing various noble gases into the molten steel, but due to inevitable leakage at the nozzle, the amount of gas entering the molten steel is unknown, and the high temperatures make it extremely difficult to lower measurement instruments into the molten steel. Therefore, we have developed and tested new methods for measuring stirring through the analysis of vibration measurements on the ladles (or converter) and from the surrounding steel mill. We have evaluated different independent component analysis-methods for separating vibrations related to stirring from vibrations from the surrounding steel mill. Previous methods using so-called "vibration index" have been combined with machine learning methods based on sparse representations for this application to better identify vibrations related to the actual stirring.

Previous research has also included signal processing for signal transmission of teh types used in mobile phone networks as well as for VDSL signals in the fixed telephone network.

## Numerical Linear Algebra and Model Fitting

We develop and analyze computational algorithms to determining unknown parameters in mathematical models, ensuring that the models are well fitted to known data. We consider linear as well as nonlinear problems and use standard least squares fitting criteria as well as more robust criteria. Emphasis is on developing algorithms that leverage the structure of the problem in order to achieve efficient and accurate methods. Some examples are:

• Fitting of 3D rigid body movements from sets of 3D coordinates
• Fitting of Non Uniform Rational B-splines (Nurbs)

Both examples above are utilized in a project, together with experimental mechanics at LTU, were we developed a system for contact free shape verification used in the production industry.

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