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Variational Spline Theory
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Book Synopsis Variational Theory of Splines by : Anatoly Yu. Bezhaev
Download or read book Variational Theory of Splines written by Anatoly Yu. Bezhaev and published by Springer Science & Business Media. This book was released on 2013-04-18 with total page 291 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a systematic description of the variational theory of splines in Hilbert spaces. All central aspects are discussed in the general form: existence, uniqueness, characterization via reproducing mappings and kernels, convergence, error estimations, vector and tensor hybrids in splines, dimensional reducing (traces of splines onto manifolds), etc. All considerations are illustrated by practical examples. In every case the numerical algorithms for the construction of splines are demonstrated.
Book Synopsis Variational Spline Theory by : A. Yu Bežaev
Download or read book Variational Spline Theory written by A. Yu Bežaev and published by . This book was released on 1993 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Splines and Variational Methods by : P. M. Prenter
Download or read book Splines and Variational Methods written by P. M. Prenter and published by Courier Corporation. This book was released on 2013-11-26 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: One of the clearest available introductions to variational methods, this text requires only a minimal background in calculus and linear algebra. Its self-contained treatment explains the application of theoretic notions to the kinds of physical problems that engineers regularly encounter. The text’s first half concerns approximation theoretic notions, exploring the theory and computation of one- and two-dimensional polynomial and other spline functions. Later chapters examine variational methods in the solution of operator equations, focusing on boundary value problems in one and two dimensions. Additional topics include least squares and other Galerkin methods. Many helpful definitions, examples, and exercises appear throughout the book. A classic reference in spline theory, this volume will benefit experts as well as students of engineering and mathematics.
Book Synopsis On Ridge Splines, Neural Networks, and Variational Problems in Radon-domain BV Spaces by : Rahul Parhi (Ph.D.)
Download or read book On Ridge Splines, Neural Networks, and Variational Problems in Radon-domain BV Spaces written by Rahul Parhi (Ph.D.) and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Deep neural networks are not well understood mathematically and their success in many science and engineering applications is usually only backed by empirical evidence. In this dissertation, we study neural networks from first principles, beginning with the simplest architecture of shallow feedforward neural networks. We use tools from variational spline theory to mathematically understand neural networks. In particular, we view neural networks as a type of spline. We propose and study a new family of Banach spaces, which are bounded variation (BV) spaces defined via the Radon transform. These are the "native spaces" for neural networks. We show that finite-width neural networks are solutions to data-fitting variational problems over these spaces. Moreover, these variational problems can be recast as finite-dimensional neural network training problems with regularization schemes related to weight decay and path-norm regularization, giving theoretical insight into these common regularization schemes as well as providing several new, principled forms of regularization for (deep) neural networks. The Radon-domain BV spaces are also interesting from the perspective of functional analysis and statistical estimation. The best approximation and estimation error rates of these spaces are (essentially) independent of the input dimension, while the best linear approximation and estimation error rates suffer the curse of dimensionality. The Radon-domain BV spaces contain functions that are very smooth in all directions except (perhaps) a few directions. The anisotropic nature of these spaces distinguishes them from classical function spaces. This dissertation provides a first step towards a mathematical theory of neural networks through the lens of spline theory and functional analysis.
Book Synopsis Multidimensional Minimizing Splines by : R. Arcangéli
Download or read book Multidimensional Minimizing Splines written by R. Arcangéli and published by Springer. This book was released on 2013-05-08 with total page 263 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is of interest to mathematicians, geologists, engineers and, in general, researchers and post graduate students involved in spline function theory, surface fitting problems or variational methods. From reviews: The book is well organized, and the English is very good. I recommend the book to researchers in approximation theory, and to anyone interested in bivariate data fitting." (L.L. Schumaker, Mathematical Reviews, 2005).
Book Synopsis The Theory of Splines and Their Applications by : J. H. Ahlberg
Download or read book The Theory of Splines and Their Applications written by J. H. Ahlberg and published by Elsevier. This book was released on 2016-06-03 with total page 297 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Theory of Splines and Their Applications discusses spline theory, the theory of cubic splines, polynomial splines of higher degree, generalized splines, doubly cubic splines, and two-dimensional generalized splines. The book explains the equations of the spline, procedures for applications of the spline, convergence properties, equal-interval splines, and special formulas for numerical differentiation or integration. The text explores the intrinsic properties of cubic splines including the Hilbert space interpretation, transformations defined by a mesh, and some connections with space technology concerning the payload of a rocket. The book also discusses the theory of polynomial splines of odd degree which can be approached through algebraically (which depends primarily on the examination in detail of the linear system of equations defining the spline). The theory can also be approached intrinsically (which exploits the consequences of basic integral relations existing between functions and approximating spline functions). The text also considers the second integral relation, raising the order of convergence, and the limits on the order of convergence. The book will prove useful for mathematicians, physicist, engineers, or academicians in the field of technology and applied mathematics.
Download or read book The Trefoil written by Troy Arthur Clark and published by . This book was released on 2020 with total page 115 pages. Available in PDF, EPUB and Kindle. Book excerpt: We will consider a variational problem arising out of the localized induction equation. We are motivated by the idea of finding "fair" splines, by considering an energy functional involving the derivative of the curvature. Among the solutions to the Euler-Lagrange equations are two elastic curves and the Kiepert Trefoil. We will introduce features and properties of the trefoil. One of the features of the trefoil is that it is an algebraic curve with a simple parametrization to handle. In addition to this, we will show that the trefoil is a model for a two-parameter spline and provide examples of how pieces of the trefoil can be cut, transformed and fitted so that the resulting curve is aesthetically "fair".
Book Synopsis Spline Functions: Basic Theory by : Larry Schumaker
Download or read book Spline Functions: Basic Theory written by Larry Schumaker and published by Cambridge University Press. This book was released on 2007-08-16 with total page 524 pages. Available in PDF, EPUB and Kindle. Book excerpt: This classic work continues to offer a comprehensive treatment of the theory of univariate and tensor-product splines. It will be of interest to researchers and students working in applied analysis, numerical analysis, computer science, and engineering. The material covered provides the reader with the necessary tools for understanding the many applications of splines in such diverse areas as approximation theory, computer-aided geometric design, curve and surface design and fitting, image processing, numerical solution of differential equations, and increasingly in business and the biosciences. This new edition includes a supplement outlining some of the major advances in the theory since 1981, and some 250 new references. It can be used as the main or supplementary text for courses in splines, approximation theory or numerical analysis.
Book Synopsis Variational Regularization of 3D Data by : Hebert Montegranario
Download or read book Variational Regularization of 3D Data written by Hebert Montegranario and published by Springer Science & Business Media. This book was released on 2014-03-14 with total page 87 pages. Available in PDF, EPUB and Kindle. Book excerpt: Variational Regularization of 3D Data provides an introduction to variational methods for data modelling and its application in computer vision. In this book, the authors identify interpolation as an inverse problem that can be solved by Tikhonov regularization. The proposed solutions are generalizations of one-dimensional splines, applicable to n-dimensional data and the central idea is that these splines can be obtained by regularization theory using a trade-off between the fidelity of the data and smoothness properties. As a foundation, the authors present a comprehensive guide to the necessary fundamentals of functional analysis and variational calculus, as well as splines. The implementation and numerical experiments are illustrated using MATLAB®. The book also includes the necessary theoretical background for approximation methods and some details of the computer implementation of the algorithms. A working knowledge of multivariable calculus and basic vector and matrix methods should serve as an adequate prerequisite.
Book Synopsis Spline Variational Theory for Composite Bolted Joints by :
Download or read book Spline Variational Theory for Composite Bolted Joints written by and published by . This book was released on 1997 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The hybrid numerical approximation model for predicting stresses and strains in open-hole composites was refined. The model was also expanded to include singular full-field stress analysis of multi-interface composites under mechanical, thermal and bearing loads. The rigid fastener hole problem was also examined. This hybrid model, using spline variational theory and Reissner's variational principle, provides accurate solutions in the vicinity of the singularity. Convergence with coarse subdivisions was observed if the multiple singular terms of neighboring interfaces are included.
Book Synopsis Multivariate Polysplines by : Ognyan Kounchev
Download or read book Multivariate Polysplines written by Ognyan Kounchev and published by Academic Press. This book was released on 2001-06-11 with total page 513 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multivariate polysplines are a new mathematical technique that has arisen from a synthesis of approximation theory and the theory of partial differential equations. It is an invaluable means to interpolate practical data with smooth functions. Multivariate polysplines have applications in the design of surfaces and "smoothing" that are essential in computer aided geometric design (CAGD and CAD/CAM systems), geophysics, magnetism, geodesy, geography, wavelet analysis and signal and image processing. In many cases involving practical data in these areas, polysplines are proving more effective than well-established methods, such as kKriging, radial basis functions, thin plate splines and minimum curvature. - Part 1 assumes no special knowledge of partial differential equations and is intended as a graduate level introduction to the topic - Part 2 develops the theory of cardinal Polysplines, which is a natural generalization of Schoenberg's beautiful one-dimensional theory of cardinal splines - Part 3 constructs a wavelet analysis using cardinal Polysplines. The results parallel those found by Chui for the one-dimensional case - Part 4 considers the ultimate generalization of Polysplines - on manifolds, for a wide class of higher-order elliptic operators and satisfying a Holladay variational property
Book Synopsis Multidimensional Minimizing Splines by : R. Arcangéli
Download or read book Multidimensional Minimizing Splines written by R. Arcangéli and published by Springer Science & Business Media. This book was released on 2004-06-24 with total page 267 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is of interest to mathematicians, geologists, engineers and, in general, researchers and post graduate students involved in spline function theory, surface fitting problems or variational methods. From reviews: The book is well organized, and the English is very good. I recommend the book to researchers in approximation theory, and to anyone interested in bivariate data fitting." (L.L. Schumaker, Mathematical Reviews, 2005).
Book Synopsis Variational Based Analysis and Modelling Using B-splines by :
Download or read book Variational Based Analysis and Modelling Using B-splines written by and published by . This book was released on 2005 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The use of energy methods and variational principles is widespread in many fields of engineering of which structural mechanics and curve and surface design are two prominent examples. In principle many different types of function can be used as possible trial solutions to a given variational problem but where piecewise polynomial behaviour and user controlled cross segment continuity is either required or desirable, B-splines serve as a natural choice. Although there are many examples of the use of B-splines in such situations there is no common thread running through existing formulations that generalises from the one dimensional case through to two and three dimensions. We develop a unified approach to the representation of the minimisation equations for B-spline based functionals in tensor product form and apply these results to solving specific problems in geometric smoothing and finite element analysis using the Rayleigh-Ritz method. We focus on the development of algorithms for the exact computation of the minimisation matrices generated by finding stationary values of functionals involving integrals of squares and products of derivatives, and then use these to seek new variational based solutions to problems in the above fields. By using tensor notation we are able to generalise the methods and the algorithms from curves through to surfaces and volumes. The algorithms developed can be applied to other fields where a variational form of the problem exists and where such tensor product B-spline functions can be specified as potential solutions.
Book Synopsis Variational Views in Mechanics by : Paolo Maria Mariano
Download or read book Variational Views in Mechanics written by Paolo Maria Mariano and published by Springer Nature. This book was released on 2022-02-08 with total page 315 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume provides a timely survey of interactions between the calculus of variations and theoretical and applied mechanics. Chapters have been significantly expanded since preliminary versions appeared in a special issue of the Journal of Optimization Theory and Applications (184(1), 2020) on “Calculus of Variations in Mechanics and Related Fields”. The variety of topics covered offers researchers an overview of problems in mechanics that can be analyzed with variational techniques, making this a valuable reference for researchers in the field. It also presents ideas for possible future areas of research, showing how the mastery of these foundational mathematical techniques can be used for many exciting applications. Specific topics covered include: Topology optimization Identification of material properties Optimal control Plastic flows Gradient polyconvexity Obstacle problems Quasi-monotonicity Variational Views in Mechanics will appeal to researchers in mathematics, solid-states physics, and mechanical, civil, and materials engineering.
Book Synopsis Finite Element Methods with B-splines by : Klaus Hollig
Download or read book Finite Element Methods with B-splines written by Klaus Hollig and published by SIAM. This book was released on 2003-01-01 with total page 155 pages. Available in PDF, EPUB and Kindle. Book excerpt: Finite Element Methods with B-Splines describes new weighted approximation techniques, combining the computational advantages of B-splines and standard finite elements. In particular, no grid generation is necessary, which eliminates a difficult and often time-consuming preprocessing step. The meshless methods are very efficient and yield highly accurate solutions with relatively few parameters. This is illustrated for typical boundary value problems in fluid flow, heat conduction, and elasticity. Topics discussed by the author include basic finite element theory, algorithms for B-splines, weighted bases, stability and error estimates, multigrid techniques, applications, and numerical examples.
Book Synopsis Handbook of Splines by : Gheorghe Micula
Download or read book Handbook of Splines written by Gheorghe Micula and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 622 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this book is to give a comprehensive introduction to the theory of spline functions, together with some applications to various fields, emphasizing the significance of the relationship between the general theory and its applications. At the same time, the goal of the book is also to provide new ma terial on spline function theory, as well as a fresh look at old results, being written for people interested in research, as well as for those who are interested in applications. The theory of spline functions and their applications is a relatively recent field of applied mathematics. In the last 50 years, spline function theory has undergone a won derful development with many new directions appearing during this time. This book has its origins in the wish to adequately describe this development from the notion of 'spline' introduced by 1. J. Schoenberg (1901-1990) in 1946, to the newest recent theories of 'spline wavelets' or 'spline fractals'. Isolated facts about the functions now called 'splines' can be found in the papers of L. Euler, A. Lebesgue, G. Birkhoff, J.
Book Synopsis Variational study of nonlinear spline curves by : E. H. Lee
Download or read book Variational study of nonlinear spline curves written by E. H. Lee and published by . This book was released on 1971 with total page 29 pages. Available in PDF, EPUB and Kindle. Book excerpt: The paper is an exposition of the variational and differential properties of nonlinear spline curves, based on the Euler-Bernoulli theory for the bending of thin beams or elastica. For both open and closed splines through prescribed nodal points in the Euclidean plane, various types of nodal constraints are considered, and the corresponding algebraic and differential equations relating curvature, angle, arc length, and tangential force are derived in a simple manner. The results for closed splines are apparently new, and they cannot be derived by the consideration of a constrained conservative system. There is a survey of the scanty recent literature. (Author).
