Ανάπτυξη και θεμελίωση νέων μεθόδων υπολογιστικών μαθηματικών στην υπολογιστική νοημοσύνη

Τwo very important scientific fields, Computational Intelligence and Computational Mathematics are useful for addressing efficiently and effectively complex real-world problems. A variety of methods and techniques have been developed based on the two aforementioned scientific fields, detaching the attention of the scientific community. It is often observed algorithms and techniques, which are adapted to a particular problem without being able to respond equally well to other, similar problems. The ’collaboration’ of the two abovementioned disciplines can provide mathematical algorithms in order to reliably deal with a plethora of problems. The algorithms and the models that are included in the field of Computational Intelligence and are based on Computational Mathematics (Numerical Solution of OrdinaryDifferential Equations) are an important part of this thesis. Computational Intelligence as a new scientific field, describes computational methods that are inspired by nature and biological systems and can combine scientific fields, such as Computational Mathematics and Computer Science, and it has several extensions and applications in almost all science. Many and varied applications affecting several areas of research and technology arise or can be addressed as Optimization problems. Several methods of Computational Intelligence refer to a special class of optimization methods that involves the study and the creation of computational methods that effectively deal with difficult problems of real-world. These methods are very important as they approach optimization problems without limitations, which are more common in most problems. Nevertheless, due to the amount of applications, and in particular the development of new technologies, new problems that require solution and are related to different kinds of optimization, such as multi-objective or constrained optimization, have been arisen. To handle these difficult problems, Computational Intelligence methods are used, and in particular techniques and algorithms belonging to the field of Machine Learning and Artificial Neural Networks. In the context of the above collaborative approach, it is necessary to study the pre-requisites of these algorithms. Thus, before applying these techniques, it is necessary to prepare and preprocess the available data set in order to create reliable models with good performance and good generalization ability. Then, these methods can be applied to tackle problems in several fields, such as engineering, physics, biology, mathematics, medicine, computer science, industry, music, economics, cryptography etc. Modern literature presents a huge number of algorithms and techniques related to the above directions. However, the majority of them are related to a specific framework that is tailored to the needs of the problem. The algorithms are usually limited to statistical models and parameters that satisfy a particular problem, thus excluding the successful application to any other problem or even to the same problem with different needs. A reliable solution to this major problem is provided by the development of algorithms with mathematical foundation and mathematical proof. An algorithm with a strong mathematical background can be used in many applications with greater success. Towards this direction is based our thesis, that is the mathematical foundation of algorithmic techniques, which will address as many problems as possible and thus, will fill the existing gaps in the literature concerning Computational Intelligence and Computational Mathematics. We believe and expect that the results of our thesis will contribute to the creation of reliable methods that may, among the others, enable scientists in others scientific fields to reliably deal with large-scale and complex problems that often occur in many scientific, industrial and financial fields with obvious benefits.