Mathematical and Computational Methods for Modelling, Approximation and Simulation (SEMA SIMAI Springer Series, 29)

✍ Scribed by Domingo Barrera (editor), Sara Remogna (editor), Driss Sbibih (editor)

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 261
• Category: Library

✦ Synopsis

This book contains plenary lectures given at the International Conference on Mathematical and Computational Modeling, Approximation and Simulation, dealing with three very different problems: reduction of Runge and Gibbs phenomena, difficulties arising when studying models that depend on the highly nonlinear behaviour of a system of PDEs, and data fitting with truncated hierarchical B-splines for the adaptive reconstruction of industrial models. The book includes nine contributions, mostly related to quasi-interpolation. This is a topic that continues to register a high level of interest, both for those working in the field of approximation theory and for those interested in its use in a practical context. Two chapters address the construction of quasi-interpolants, and three others focus on the use of quasi-interpolation in solving integral equations. The remaining four concern a problem related to the heat diffusion equation, new results on the notion of convexity in probabilistic metric spaces (which are applied to the study of the existence and uniqueness of the solution of a Volterra equation), the use of smoothing splines to address an economic problem and, finally, the analysis of poverty measures, which is a topic of increased interest to society. The book is addressed to researchers interested in Applied Mathematics, with particular reference to the aforementioned topics.

✦ Table of Contents

Preface
Contents
Editors and Contributors
About the Editors
Contributors
Part I Plenary Lectures
1 Mapped Polynomials and Discontinuous Kernels for Runge and Gibbs Phenomena
1.1 Introduction
1.2 Mitigating the Runge and Gibbs Phenomena
1.2.1 The Mapped Bases Approach
1.2.2 The Fake'' Nodes Approach 1.3 Choosing the Map S: Two Algorithms 1.3.1 Numerical Tests 1.3.1.1 Application to Runge Phenomenon 1.3.1.2 Applications to Gibbs Phenomenon 1.4 Application to Rational Approximation 1.4.1 Barycentric Polynomial Interpolation 1.4.2 Floater-Hormann Rational Interpolation 1.4.3 The AAA Algorithm 1.4.4 Mapped Bases in Barycentric Rational Interpolation 1.4.5 A Numerical Example 1.4.5.1 The FH Interpolants 1.4.5.2 The AAA Algorithm 1.5 Application to Quadrature 1.5.1 Quadrature viaFake'' Nodes
1.5.1.1 Examples
1.6 Discontinuous Kernels
1.6.1 A Brief Introduction to RBF Approximation
1.6.2 From RBF to VSK Interpolation
1.6.3 Variably Scaled Discontinuous Kernels (VSDK)
1.6.4 VSDKs: Multidimensional Case
1.7 Application to MPI
1.7.1 An Example
1.8 Conclusion and Further Works
References
2 Steady Systems of PDEs. Two Examples from Applications
2.1 Introduction: Systems of PDEs: Why Are They So Difficult?
2.2 Inverse Problems in Conductivity in the Plane
2.2.1 A Situation in Which Existence Can be Achieved
2.2.2 The Multi-Measurement Case
2.2.3 Some Numerical Tests
2.3 Some Optimal Control Problems for Soft Robots
2.3.1 The Control Problem
2.3.2 The Fiber Tension Case
2.3.3 Some Simulations
2.4 Conclusions
References
3 THB-Spline Approximations for Turbine Blade Design with Local B-Spline Approximations
3.1 Introduction
3.2 The Problem
3.3 First Stage: Local B-Spline Approximations
3.4 Second Stage: THB-Spline Approximation
3.5 Examples
Appendix
References
Part II Contributed Papers
4 A Progressive Construction of Univariate Spline Quasi-Interpolants on Uniform Partitions
4.1 Introduction
4.2 Notations and Preliminaries
4.3 A Minimization Problem
4.4 Solving the Minimization Problem
4.4.1 Case r=0
4.4.2 Case r> 0
4.5 Some Examples of Differential Quasi-Interpolants
4.5.1 Quadratic Differential Quasi-Interpolants
4.5.2 Cubic Differential QIs
4.6 Conclusion
References
5 Richardson Extrapolation of Nyström Method Associated with a Sextic Spline Quasi-Interpolant
5.1 Introduction
5.2 Sextic Spline Quasi-Interpolant
5.2.1 B-splines
5.2.2 Construction of the Discrete Spline Quasi-Interpolant
5.3 Quadrature Formula Associated with Qn
5.4 The Nyström Method
5.5 Numerical Results
5.6 Conclusion
References
6 Superconvergent Methods Based on Cubic Splines for Solving Linear Integral Equations
6.1 Introduction
6.2 PGS Cubic Spline Approximation
6.2.1 Analysis of Convergence
6.3 Superconvergent Cubic Spline Quasi-Interpolant Method
6.3.1 Quasi-Interpolation Method 1
6.3.2 Quasi-Interpolation Method 2
6.4 Numerical Examples
6.5 Conclusion
References
7 The Completely Discretized Problem of the Dual Mixed Formulation for the Heat Diffusion Equation in a Polygonal Domain by the Crank-Nicolson Scheme in Time
7.1 Introduction
7.2 The Model Problem
7.3 The Completely Discretized Problem
7.4 The Crank-Nicolson Scheme
7.4.1 Stability
7.4.2 Error Estimates on the Temperature and on the Heat Flux Density Vector
7.5 Conclusion
References
8 Economic Statistical Splicing Data Using Smoothing Quadratic Splines
8.1 Introduction
8.2 Methodology of Statistical Splicing Data
8.2.1 Splicing by Variation
8.2.2 Splicing by Linear Interpolation
8.3 Splicing by Smoothing Quadratic Splines
8.4 Computation
8.5 Case Studies
8.5.1 Splicing Data of Economics Activities for Venezuela Between 1950 and 2005
8.5.2 Approximating Some Data of Economics Activities of Morocco Between 1971 and 2015
8.5.2.1 Data of Gross Domestic Product for Morocco Between 1971 and 2015
8.5.2.2 Data of Agriculture for Morocco Between 1971 and 2015
8.5.2.3 Data of Electricity for Morocco Between 1971 and 2015
8.5.2.4 Data of Trade for Morocco Between 1971 and 2015
8.6 Error Estimate
8.7 Conclusion
References
9 Some Properties of Convexity Structure and Applications in b-Menger Spaces
9.1 Introduction and Preliminaries
9.2 Probabilistic Takahashi Convex Structure
9.3 Probabilistic Strong Convex Structure
9.4 Application to An Integral Equation
References
10 A Super-Superconvergent Cubic Spline Quasi-Interpolant
10.1 Introduction
10.2 Normalized Basis
10.2.1 Finite Element of Class C2 and Degree 3
10.2.2 Hermite B-Splines
10.2.3 Marsden's Identity in Polar Form
10.3 Spline Quasi-Interpolant in P23(I,τ1)
10.3.1 Construction of a Superconvergent Discrete Quasi-Interpolant
10.3.2 Super-Superconvergence Phenomenon
10.4 Numerical Examples
10.4.1 Superconvergence
10.4.1.1 Approximating Function Values
10.4.1.2 Approximating First Derivative Values
10.4.1.3 Approximating Second Derivative Values
10.4.2 Super-Superconvergence
10.4.2.1 Approximating Function Values
10.4.2.2 Approximating Second Derivative Values
10.5 Conclusion
References
11 Calibration Adjustment for Dealing with Nonresponse in the Estimation of Poverty Measures
11.1 Introduction
11.2 Calibrating the Distribution Function for Treating the Non-response
11.3 Poverty Measures Estimation with Missing Values
11.4 Variance Estimation for Percentile Ratio Estimators with Resampling Method
11.5 Simulation Study
11.6 Conclusion
References
12 Numerical Methods Based on Spline Quasi-Interpolating Operators for Hammerstein Integral Equations
12.1 Introduction
12.2 A Family of Discrete Spline Quasi-Interpolants
12.3 Quadrature Rules Based on Qd Defined on a Uniform Partition
12.4 Methods Based on Qd
12.4.1 Collocation Type Method and Its Iterated Version
12.4.2 Nyström Method
12.5 Error Analysis
12.5.1 Collocation and Nyström Solutions
12.5.2 Iterated Collocation Solution
12.6 Numerical Results
12.7 Conclusions
References