Numerical and Symbolic Scientific Computing [electronic resource] : Progress and Prospects /
Langer, Ulrich.editor.
Paule, Peter.editor.
The book presents the state of the art, new results, and it also includes articles pointing to future developments. Most of the articles center around the theme of partial differential equations. Major aspects are fast solvers in elastoplasticity, symbolic analysis for boundary problems, symbolic treatment of operators, computer algebra, and finite element methods, a symbolic approach to finite difference schemes, cylindrical algebraic decomposition and local Fourier analysis, and white noise analysis for stochastic partial differential equations. Further numerical-symbolic topics range from applied and computational geometry to computer algebra methods used for total variation energy minimization.
Description based upon print version of record.
Numerical and Symbolic Scientific Computing: Progress and Prospects; Preface; Contents; Approximate Implicitization of Space Curves; 1 Introduction; 2 Preliminaries; 3 Approximate Implicitization; 3.1 Direct Techniques; 3.1.1 Dokken's method; 3.1.2 Dokken's weak method; 3.1.3 Algebraic curve and surface fitting; 3.2 Iterative (Evolution-Based) Techniques; 4 Approximate Implicitization of Space Curves; 4.1 Fitting Two Implicitly Defined Surfaces; 4.2 Regularization; 4.3 Putting Things Together; 4.4 Examples; 5 Conclusion; References
Sparsity Optimized High Order Finite ElementFunctions on Simplices1 Introduction; 2 Properties of Jacobi Polynomials with Weight (1-x)a; 3 Preliminary Definitions; 4 The L2 Orthogonal Basis Functions of Dubiner; 5 Sparsity Optimized H1-Conforming Basis Functions; 5.1 Sparse H1-Conforming Basis Functions on the Triangle; 5.2 Sparse H1-Conforming Basis Functions on the Tetrahedron; 6 Sparsity Optimization of H(div)-Conforming Basis Functions; 7  Sparsity Optimized H(curl)-Conforming Basis Functions; 8 Integration by Rewriting; References
Fast Solvers and A Posteriori Error Estimates in Elastoplasticity1 Introduction; 2 Modeling of Elastoplasticity; 3 The Incremental Elastoplasticity Problems and Solvers; 4 Adaptive h-FEM and A Posteriori Error Estimates for Elastoplasticity; 5 High Order FEM for Elastoplasticity: hp-FEM and BC-FEM; 6 Conclusion; References; A Symbolic-Numeric Algorithm for Genus Computation; 1 Introduction; 2 The Genus Computation Problem; 2.1 Genus of Plane Algebraic Curves; 2.2 Strategy for Solving the Problem; 3 Why Knot? Alternative Solution to the Genus Computation Problem
3.1 Computing the Singularities of the Algebraic Curve3.2 Computing the Algebraic Link of an Isolated Singularity; 3.3 Computing the Alexander Polynomial of an Algebraic Link; 3.3.1 Knot Theory and the Alexander Polynomial; 3.3.2 Alexander Polynomial and Computational Geometry; 3.4 Computing the Delta-Invariant of an Isolated Singularity; 3.5 Computing the Genus of the Algebraic Curve; 4 What Precisely Means ``Approximate Computation''?; 5 Numerical Experiments; 6 Conclusion and Future Work; References; The ``Seven Dwarfs'' of Symbolic Computation; 1 Introduction
2 Exact Linear Algebra, Integer Lattices3 Exact Polynomial and Differential Algebra, Gröbner Bases; 4 Inverse Symbolic Problems, e.g., Interpolation and Parameterization; 5 Tarski's Algebraic Theory of Real Geometry; 6 Hybrid Symbolic-Numeric Computation; 7 Computation of Closed Form Solutions; 8 Rewrite Rule Systems and Computational Group Theory; References; Computer Algebra Meets Finite Elements: An Efficient Implementation for Maxwell's Equations; 1 Introduction; 2 FEM Formulation of Maxwell's Equations; 2.1 Numerical Flux; 2.2 Numerical Examples (Spherical Vacuum Resonator)
3 Computational Aspects
Includes bibliographical references.
The book presents the state of the art, new results, and it also includes articles pointing to future developments. Most of the articles center around the theme of partial differential equations. Major aspects are fast solvers in elastoplasticity, symbolic analysis for boundary problems, symbolic treatment of operators, computer algebra, and finite element methods, a symbolic approach to finite difference schemes, cylindrical algebraic decomposition and local Fourier analysis, and white noise analysis for stochastic partial differential equations. Further numerical-symbolic topics range from applied and computational geometry to computer algebra methods used for total variation energy minimization.
Computer mathematics.
Algebra.
Applied mathematics.
Engineering mathematics.
Computational Mathematics and Numerical Analysis.
Algebra.
Mathematical and Computational Engineering.
Texts and Monographs in Symbolic Computation