Sparse Grids and Applications - Miami 2016

Name: Sparse Grids and Applications - Miami 2016
Brand: Springer
Price: 149.79 EUR
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Jochen Garcke Dirk Pflüger Clayton G. Webster Guannan Zhang(Editor)

Springer (Publisher)

Published on 20. June 2018

X, 262 pages

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978-3-319-75426-0 (ISBN)

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Content

Intro
Preface
Contents
Contributors
Comparing Nested Sequences of Leja and PseudoGauss Points to Interpolate in 1D and Solve the Schroedinger Equation in 9D
1 Introduction
2 Interpolation
3 The Importance of Nesting
3.1 PseudoGauss Nested Points
3.2 Leja Nested Points
4 Lebesgue Constants
5 Comparison Between Leja Points and PseudoGauss Points in Collocation Calculations
6 Conclusion
References
On the Convergence Rate of Sparse Grid Least Squares Regression
1 Introduction
2 Least-Squares Regression
3 Full Grids and Sparse Grids
4 Error Analysis
4.1 Well-Posedness and Error Decay
4.2 Application to Sparse Grids
5 Numerical Experiments
5.1 Error Decay
5.2 Balancing the Error
6 Conclusion
References
Multilevel Adaptive Stochastic Collocation with Dimensionality Reduction
1 Introduction
2 Adaptivity with Sparse Grids
2.1 Interpolation on Spatially-Adaptive Sparse Grids
2.2 Interpolation with Dimension-Adaptive Sparse Grids
3 Multilevel Stochastic Collocation with Dimensionality Reduction
3.1 Generalized Polynomial Chaos
3.2 Multilevel Approaches for Generalized Polynomial Chaos
3.3 Stochastic Dimensionality Reduction
4 Numerical Results
4.1 Second-Order Linear Oscillator with External Forcing
4.2 A simple Fluid-Structure Interaction Example
5 Conclusions and Outlook
References
Limiting Ranges of Function Values of Sparse Grid Surrogates
1 Introduction
2 Sparse Grids
2.1 Hierarchical Ancestors and the Fundamental Property
2.2 Interpolation on Sparse Grids
3 Limiting Ranges of Sparse Grid Function Values
3.1 Limitation from Above and Below
3.2 Minimal Extension Set
3.3 Computing Coefficients of the Extension Set
3.4 Intersection Search
4 Approximation of Gaussians with Extended Sparse Grids
4.1 Intersection Search and Candidate Sets for Regular Sparse Grids
4.2 Extension Sets and Convergence for Regular Grids
4.3 Extension Sets for Adaptively Refined Grids
5 Conclusions
References
Scalable Algorithmic Detection of Silent Data Corruption for High-Dimensional PDEs
1 Introduction
1.1 High-Dimensional PDEs in High-Performance Computing
2 Theory of the Classical Combination Technique
3 The Combination Technique in Parallel
4 Dealing with System Faults
5 Detecting and Recovering from SDC
5.1 Method 1: Comparing Combination Solutions Pairwise via a Maximum Norm
5.2 Method 2: Comparing Combination Solutions via their Function Values Directly
5.3 Cost and Parallelization
5.4 Detection Rates
6 Numerical Tests
6.1 Experimental Setup
6.2 SDC Injection
6.3 Results: Detection Rates and Errors
6.4 Results: Scaling
6.5 Dealing with False Positives
7 Extensions to Quantities of Interest
8 Conclusion
References
Sparse Grid Quadrature Rules Based on Conformal Mappings
1 Introduction and Background
2 Transformed Quadrature Rules
2.1 Standard One-Dimensional Quadrature Rules
2.2 Sparse Quadrature for High Dimensional Integrals
3 Comparison of the Transformed Sparse Grid Quadrature Method
4 Numerical Tests of the Sparse Grid Transformed Quadrature Rules
4.1 Comparison of Maps
4.2 Effect of Dimension
5 Conclusions
References
Solving Dynamic Portfolio Choice Models in Discrete Time Using Spatially Adaptive Sparse Grids
1 Introduction
2 Dynamic Portfolio Choice Models
2.1 Approximation
2.2 Optimization
2.3 Integration
2.4 Dynamic Programming
2.5 Optimal Choices
3 Sparse Grid Approximation
3.1 Hierarchical Bases and Sparse Subspace Selection
3.2 Spatially Adaptive Refinement
4 Spatially Adaptive Sparse Grid Dynamic Programming Scheme
4.1 Spatially Adaptive Sparse Grid Dynamic Programming
4.2 Optimal Choices
4.3 Choice of Basis Functions, Extrapolation, and Refinement
5 Numerical Example
5.1 Transaction Costs Model
5.2 Error Measurement
5.3 Results
6 Conclusion
Appendix: Euler Equation Errors
References
Adaptive Sparse Grid Construction in a Context of Local Anisotropy and Multiple Hierarchical Parents
1 Introduction
2 Multidimensional Hierarchical Interpolation Strategy
2.1 Multidimensional Hierarchy of Nodes and Functions
2.2 Multidimensional Interpolation
2.3 Piece-Wise Constant Hierarchy
3 Adaptive Interpolation
4 Examples
4.1 Influence of the Type of Hierarchy
4.2 Influence of the Refinement Method
4.3 Application to Kermack-McKendrick SIR Model
5 Conclusions
References
Smolyak's Algorithm: A Powerful Black Box for the Acceleration of Scientific Computations
1 Introduction
2 Decomposition
3 Truncation
3.1 Knapsack Problem
3.2 Combination Rule
4 Convergence Analysis
4.1 Finite-Dimensional Case
4.2 Infinite-Dimensional Case
5 Applications
5.1 High-Dimensional Interpolation and Integration
5.2 Monte Carlo Path Simulation
5.3 Multilevel Quadrature
5.4 Partial Differential Equations
5.5 Uncertainty Quantification
6 Conclusion
Appendix
References
Fundamental Splines on Sparse Grids and Their Application to Gradient-Based Optimization
1 Introduction
2 B-Splines on Sparse Grids
2.1 Definition and Properties
2.2 Hierarchization with B-Splines
3 Fundamental Property
4 Sparse Grid Basis Transformations
4.1 Hierarchical Fundamental Transformation
4.2 Nodal Fundamental Transformation
4.3 Translation-Invariant Fundamental Transformation
5 Hierarchical Fundamental Splines
5.1 Definition and Properties
5.2 Modified Fundamental Splines
6 Application to Gradient-Based Optimization
6.1 Generation of the Spatially Adaptive Grid
6.2 Optimization Procedure
6.3 Test Functions and Results
6.4 Comparison of Runtime and Memory Consumption
7 Conclusion
References
Editorial Policy
Lecture Notesin Computational Scienceand Engineering
Monographs in Computational Scienceand Engineering
Texts in Computational Scienceand Engineering

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Sparse Grids and Applications - Miami 2016

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