Preface

This book contains well-written monographs within the broad spectrum of applied mathematics, with the aim of offering an interesting reading of current trends and problems in this fascinating and critically important field of mathematics to a broad category of researchers and practitioners. Recent developments in high-performance computing are radically changing the way we do numerics as applied mathematicians. Because of the impressive advances in computer technology and the introduction of fast methods that require less algorithmic cost and fewer memory resources, nowadays a rigorous numerical solution of many difficult computational science applications has become possible. In the future we will be solving much bigger problems, and even more factors will need to be considered than in the past when attempting to identify the optimal solution approach. The gap between fast and slow algorithms is rapidly growing. Methods that do more operations per grid node, cell, or element, such as higher-order and discontinuous Galerkin discretization schemes and spectral element methods, are becoming very attractive to use against more traditional techniques such as finite element discretization schemes. Structured data are already coming back, because they may achieve a better load balance than unstructured grids on computers with hundreds of thousands of processors. Novel classes of numerical methods with reduced computational complexity will need to be found to solve large-scale problems arising in an industrial setting.

The book is structured in three distinct parts, according to the aims and methodologies used by the authors in the development of their studies, ranging from optimization techniques to graph-oriented approaches and approximation theory, providing overall a good mix of both theory and practice. Chapters 1–2 present an overview of unconstrained optimization techniques, covering both line search and trust-region methods that are essential ingredients to guarantee global convergence of descent schemes. Numerical optimization is the primary tool used in Chapter 3 to analyze the shape factor of exceedance probability curves, which is a critical analysis tool to assess risks, e.g., in the study of natural disasters such as floods, hurricanes, and earthquakes. Chapters 4–5 describe graph-oriented approaches. Chapter 4 develops a graph-based model for the topological design of the wide area network using dynamic programming and dynamic programming with state-space relaxation methodologies. Chapter 5 uses graph and subgraph models to speed up the computations of scalar multiplication algorithms on elliptic curves defined over finite fields, which is one central and time-consuming operation in elliptic curve cryptography. Finally, the contributions of the last two chapters deal with some aspects of functional approximation. Chapter 6 proposes a study of different forms of bounded variation sequence spaces of invariant means with the help of ideal operators and functions such as Orlicz function and modulus function. The results show the potential of the new theoretical tools to deal with the convergence problems of sequences in sigma-bounded variation occurring in many branches of science, engineering, and applied mathematics. Chapter 7 is devoted to an overview of the mathematics of special polynomials showing how to obtain them in a simple and straightforward approach using basic linear algebra concepts. Overall, the collection of contributions demonstrates the highly interdisciplinary character of the

discipline, and emphasizes the continuing need for close cooperation between applied mathematicians, experimental physicists, engineers, and computer scientists in modern applied data science.

We express appreciation to all those who helped in the preparation of this book, and in particular to Luka Cvjetković at IntechOpen for his tireless editorship assistance.

> Bruno Carpentieri Free University of Bozen-Bolzano, Bolzano, Italy
