Numerical approximations with tensor-based techniques for high-dimensional problems

Resumen

The idea of following a sequence of steps to achieve a desired result is inherent in human nature: from the moment we start walking, following a cooking recipe or learning a new card game. Since ancient times, this scheme has been followed to organize laws, correct writings, and even assign diagnoses. In mathematics, this way of thinking is called an algorithm. Formally, an algorithm is a set of defined and unambiguous instructions, ordered and finite, that allows for solving a problem. From childhood, we face them when we learn to multiply or divide, and as we grow, these structures will enable us to solve different increasingly complex problems: linear systems, differential equations, optimization problems, etc. There is a multitude of algorithms that allow us to deal with this type of problem, such as iterative methods, where we find the famous Newton Method to find roots; search algorithms to locate an element with specific properties in a more extensive set; or matrix decompositions, such as the LU decomposition to solve some linear systems. However, these classical approaches have limitations when faced with large-dimensional problems, a problem known as the `curse of dimensionality'. The advancement of technology, the use of social networks and, in general, the new problems that have appeared with the development of Artificial Intelligence, have revealed the need to handle large amounts of data, which requires the design of new mechanisms that allow its manipulation. This fact has aroused interest in the scientific community in tensor structures since they allow us to work efficiently with large-dimensional problems. However, most of the classic methods are not designed to be used together with these operations, so specific tools are required to allow their treatment, which motivates work like this. This work is divided as follows: after reviewing some definitions necessary for its understanding, in Chapter 3, the theory of a new tensor decomposition for square matrices is developed. Next, Chapter 4 shows an application of said decomposition to regular graphs and small-world networks. In Chapter 5, an efficient implementation of the algorithm provided by the new matrix decomposition is proposed, and some order two PDEs are studied as an application. Finally, Chapters 6 and 7 present some brief conclusions and list some of the references consulted.