8 research outputs found
Modeling and Simulation in Engineering
The Special Issue Modeling and Simulation in Engineering, belonging to the section Engineering Mathematics of the Journal Mathematics, publishes original research papers dealing with advanced simulation and modeling techniques. The present book, “Modeling and Simulation in Engineering I, 2022”, contains 14 papers accepted after peer review by recognized specialists in the field. The papers address different topics occurring in engineering, such as ferrofluid transport in magnetic fields, non-fractal signal analysis, fractional derivatives, applications of swarm algorithms and evolutionary algorithms (genetic algorithms), inverse methods for inverse problems, numerical analysis of heat and mass transfer, numerical solutions for fractional differential equations, Kriging modelling, theory of the modelling methodology, and artificial neural networks for fault diagnosis in electric circuits. It is hoped that the papers selected for this issue will attract a significant audience in the scientific community and will further stimulate research involving modelling and simulation in mathematical physics and in engineering
Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory
This book aims to provide an introduction to the topic of deep learning
algorithms. We review essential components of deep learning algorithms in full
mathematical detail including different artificial neural network (ANN)
architectures (such as fully-connected feedforward ANNs, convolutional ANNs,
recurrent ANNs, residual ANNs, and ANNs with batch normalization) and different
optimization algorithms (such as the basic stochastic gradient descent (SGD)
method, accelerated methods, and adaptive methods). We also cover several
theoretical aspects of deep learning algorithms such as approximation
capacities of ANNs (including a calculus for ANNs), optimization theory
(including Kurdyka-{\L}ojasiewicz inequalities), and generalization errors. In
the last part of the book some deep learning approximation methods for PDEs are
reviewed including physics-informed neural networks (PINNs) and deep Galerkin
methods. We hope that this book will be useful for students and scientists who
do not yet have any background in deep learning at all and would like to gain a
solid foundation as well as for practitioners who would like to obtain a firmer
mathematical understanding of the objects and methods considered in deep
learning.Comment: 601 pages, 36 figures, 45 source code
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described