Mathematics
Mathematical Research
In recent decades, the complexity of the challenges facing the industry has grown increasingly, so that the concrete application of mathematical methods has become imperative in many areas. TWT has been investing in the development of mathematics since its foundation in 1986. The Mathematical Research Group, consisting of more than 30 mathematicians and people interested in mathematics, the Mathematics Circle, takes this fact into account in several ways. Through the broadly spread expertise of the staff, we offer expertise from all major areas of mathematics, such as analysis, algebra, number theory, numerics, stochastics, data analytics, and operations research. Among other things, we use this knowledge to investigate and solve fundamental mathematical problems at the highest academic level. We then use the knowledge gained in this way in the sense of technical-scientific transfer to give our customers the necessary edge in the increasingly complex engineering, physical and technical problems that will ensure their competitiveness in the industry of the future. In order to live up to this claim, we maintain a constantly growing network of leading institutes and professors around the world, who actively advise and support us in our projects. This includes former TWT colleagues, who have returned to the university.
Basic mathematical research deals with abstract problems, which often -at first sight- have no direct application in industrial projects. The importance of basic mathematical research lies mainly in the development of new methods and the extension of existing knowledge. The full potential of these research results is often only realized later and very often in surprising and unforeseen ways. In this sense, basic mathematical research is an important contribution to future innovations.
Prime numbers and fractals
The phenomenon of prime numbers has always been a fascinating subject to study for Mathematicians all over the world. A particular aspect is their distribution within the positive real numbers. For this purpose, the function ?(x) is defined, which gives to each real number x the number ?(x) of primes p for which p ? x holds. A first estimate was shown independently in 1896 by Jacques Salomon Hadamard and Charles-Jean de La Valée Poussin: ?(x) behaves like x/ln(x) for x towards infinity. According to his own recollection, Carl Friedrich Gauss had already made the same conjecture in 1792 and 93, respectively. An improvement of this estimate succeeds by Euler's logarithmic integral Li(x)=?dt/ln(t), which is integrated from 2 to x. This conjecture was first mentioned by Carl Friedrich Gauss in a letter to Encke in 1849 and formulated by Legendre in 1798.
A sketch of the proof was described by Terrence Tao in 2007 in his lecture "Structure and randomness in the prime numbers", which could be called the music of primes: One applies the von Mangoldt function vanishing on non-prime(powers) and makes the resulting "tones" audible by applying the Fourier transform.
Ganz im Geiste dieser Skizze wird hier die Zeichnung eines Bildes der Primzahlen angestrebt, dessen Ergebnis rechts dargestellt ist. Dies geschieht, indem ein weiterer Zugang zur Approximation der Primzahl zählenden Funktion ?(x) verfolgt wird, der auf einer additiven Funktion beruht. Zusammen mit Fourier-Polygonen, die im Kontext von regularisierenden Polygontransformationen zur Anwendung kommen, können fraktale Primzahlpolygone and fraktale Primzahlkurven abgeleitet werden.
A detailed account of the connections and all precise citations can be found in:
Dimitris Vartziotis, Joachim Wipper: The fractal nature of an approximate prime counting function, https://arxiv.org/abs/1611.01949 |
Number Theory
A current project from basic research at TWT deals with the symmetry of finite sums of complex exponential functions. The exponential function is of fundamental importance in mathematics and occurs, for example, as an elementary building block in Fourier series. If one interprets the graph of a complex exponential function as a parameterized curve, the result is the unit circle. However, if one adds several different complex exponential functions, fascinating graphs arise whose symmetry depends on the generating parameters. In the current work, the symmetry groups of these graphs could be determined as a function of the generating parameters.
PAUSINGER, Florian; VARTZIOTIS, Dimitris. On the symmetry of finite sums of exponentials. Elemente der Mathematik, 2021, 76. Jg., Nr. 2, S. 62-73, Elemente der Mathematik, 2021, 76. Jg., Nr. 2, S. 62-73, |
Game Theory
Typically, game theory is applied in economics. The question of the timing and price of a product at market launch or the identification of power relationships are classic problems. In a specific assignment, for example, we have already advised our client on how to ideally allocate the award of projects to external partners, taking into account several internal stakeholders with their own priorities regarding the award. Here, we were particularly interested in the decision optimum, but also in the likely behavior patterns of the stakeholders, in order to be able to derive optimal strategies.
For some years now, game theory methods have been increasingly used in other areas, such as route planning. We are also interested in further developing game theory at the academic level and thus advancing new areas, as this published research sketch shows:
VARTZIOTIS, Dimitris; BOHNET, Doris; HIMPEL, Benjamin. Smoothing Game. arXiv preprint arXiv:2010.04956,, 2020, ( Link: https://arxiv.org/abs/2010.04956 ).
To improve meshes for finite element simulations, they need to be smoothed by moving the nodes. We want to introduce a new smoothing approach by treating each geometric element as a player in a game: a quest for the best element quality. In other words, each player has the goal of becoming as regular as possible.
Optimization and Graph Theory
In practice, relevant key figures for a process or model often need to be improved. We use state-of-the-art algorithms from derivative-free, global, AI-assisted or even nonlinear optimization. Regardless of whether the problem is discrete or continuous, we investigate it for solvability and propose robust algorithms. An example can be found in our product Veris® (Link: https://twt-innovation.de/en/produkte) which, among other things, performs optimization in route planning. In this environment, we conduct research in the area of quantum algorithms, which we presented at the Digital Product Forum 2022 at Mercedes-Benz, among others:
With Cubic AI https://cubicai.twt-gmbh.de/home we also take the optimization of highly complex systems to a whole new level. Almost like a genie in a bottle, you can wish for the behavior of your system and the modern AI will determine the best approximation to your specification in real time, taking into account the physical limitations. Cubic AI was also presented at the Digital Product Forum 2022.
Numerical Methods
Do you work with complex physical processes? We simulate them using the latest numerical algorithms from the world of multibody simulation, fluid dynamics and others, using artificial intelligence for simulation acceleration. For one of our clients, for example, we investigated why their solver - used for a high-dimensional system of equations which originated from a brake simulation - failed to find a solution in certain scenarios. The complex model, which included discontinuous equations, was adapted using an appropriate solution strategy so that the system of equations could be solved for physically relevant parameters.
Another example of the pioneering application of numerical methods, especially in the field of finite elements, can be found in our product GETMe, the Geometric Element Transformation Method.
The GETMe - Geometric element transformation
High quality meshes play a key role in many applications based on digital modeling and simulation. The finite element method is a prime example of such an approach, and it is well known that high-quality meshes can significantly improve the computational efficiency and solution accuracy of this method. As a result, much work has been invested in methods to improve mesh quality. These range from simple geometric approaches such as Laplace smoothing, which results in high computational efficiency but potentially low mesh quality, to global optimization-based methods that result in excellent mesh quality at the cost of increased computational and implementation complexity.
The geometric element transform (GETMe) method bridges the gap between these two approaches
It is based on geometric mesh element transformations, where polygonal and polyhedral elements are iteratively transformed into their regular counterparts or into elements with a given shape. GETMe combines computational efficiency similar to Laplace smoothing with effectiveness approaching global optimization methods.
GETme as a pioneering method is now indispensable in the field of mesh smoothing and is attracting increasing attention in the relevant literature:
„A breakthrough was made when Vartziotis et al. (2008) proposed the GETMe method, which is purely a geometric process to move the nodes of a triangle so as to improve its quality. (...) GETMe is the most interesting node-smoothing scheme as it is purely geometric in nature in transforming elements into regular forms without a direct link to shape quality measure. It is robust in removing inverted and poorly shaped elements rapidly and is consistent in diverse applications to produce quality meshes." |
An open source implementation of GETMe is available on Github: https://github.com/twt-gmbh/getme
A detailed description of the mathematical theory as well as numerical studies of GETMe can be found in the book
VARTZIOTIS, Dimitris; WIPPER, Joachim. The GETMe mesh smoothing framework: A geometric way to quality finite element meshes.. CRC Press, 2018, |
and the associated publications, such as
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Data Science
The hot topic of Data Science is also represented by us. With our TWT product ZAMRIS, we offer a quality assurance framework which, in addition to an intuitive user interface and professional quality rule management, provides machine learning algorithms for quality assurance. Thereby, an intelligent time series analysis of arbitrary data sets is performed by means of Machine Learning and Pattern Recognition. ZAMRIS can be used both as a stand-alone application and integrated into existing simulation processes, thus enabling efficient and quality-assured development even with an increasing number of variants to be examined.
The use of Data Science is also relevant in our research projects. For example, in OPsTIMAL www.opstimal.de air traffic is optimized holistically from the perspective of the aviation company. Additionally, we use modern tools like image recognition with Convolutional Neural Networks, self-learning neural networks, or state estimation with Kalman filters in other projects.
The goal of the KARLI project is to develop adaptive, responsive and level-compliant interaction in the vehicle of the future. To this end, customer-relevant AI functions are being developed in KARLI that detect driver states and design interactions for different stages on the way to an automated vehicle (automation level).
These AI functions are developed in KARLI from empirical and synthetically generated data. The data will be collected and used in KARLI in such a way that the project results are scalable to Big Data from production vehicles that will be available in the future.
Breaking Boundaries
"Mathematics compares the most diverse phenomena and discovers the secret analogies that connect them." - Joseph Fourier, French mathematician and physicist |
Loosely based on this quote, we too are constantly trying to find new connections between phenomena or models that apparently have no connection. For example, would you have thought that there is a direct analogy between the transformation of triangular meshes in a CAD model and the placement of so-called quantum wells in the quantum computing environment? We are exploring these and other analogies in order to apply findings from one area to another and gain more understanding of the structure inherent in the problem. If you are interested in this topic, please feel free to read the following publication:
VARTZIOTIS, Dimitris; HIMPEL, Benjamin; PFEIL, Markus. Creation of higher-energy superposition quantum states motivated by geometric transformations. arXiv preprint arXiv:1712.07963, 2017, https://arxiv.org/abs/1712.07963. |
We propose a way to generate higher energy superposition states in a circular system of quantum wells. This is inspired by a connection to convergence results for geometric transformations of polygons with circulant Hermitian matrices.
Contact
Have we aroused your interest? For research collaborations or industrial projects in the field of mathematics, please contact us at mathematics@twt-gmbh.de.
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