Book contents
- Frontmatter
- Dedication
- Reviews
- Contents
- Figures
- Tables
- Preface
- Acknowledgments
- 1 Introduction
- 2 Positive Definite Kernels and Functions
- 3 RBF Interpolation and Polynomial Precision
- 4 Error Estimates and Examples of Kernels
- 5 On Uncertainty Principles and Shape Parameter Selection
- 6 Rational Kernels
- 7 Partition of Unity
- 8 Efficient Reduced-Basis Approximation
- 9 Approximating via Variably Scaled Discontinuous Kernels
- 10 Application of VSDKs to Magnetic Particle Imaging
- 11 Applications to Astronomical Imaging
- 12 Kernels in Machine Learning
- 13 Support Vector Machines and other Methods
- 14 Local-to-global SVMs
- 15 Learning Data-Driven Optimized Kernels
- 16 Choosing and Learning the Scaling Function for the VSKs
- 17 Kernels for Persistent Homology
- 18 Software
- 19 Conclusions
- References
- Index
7 - Partition of Unity
Published online by Cambridge University Press: 08 August 2026
Book contents
- Frontmatter
- Dedication
- Reviews
- Contents
- Figures
- Tables
- Preface
- Acknowledgments
- 1 Introduction
- 2 Positive Definite Kernels and Functions
- 3 RBF Interpolation and Polynomial Precision
- 4 Error Estimates and Examples of Kernels
- 5 On Uncertainty Principles and Shape Parameter Selection
- 6 Rational Kernels
- 7 Partition of Unity
- 8 Efficient Reduced-Basis Approximation
- 9 Approximating via Variably Scaled Discontinuous Kernels
- 10 Application of VSDKs to Magnetic Particle Imaging
- 11 Applications to Astronomical Imaging
- 12 Kernels in Machine Learning
- 13 Support Vector Machines and other Methods
- 14 Local-to-global SVMs
- 15 Learning Data-Driven Optimized Kernels
- 16 Choosing and Learning the Scaling Function for the VSKs
- 17 Kernels for Persistent Homology
- 18 Software
- 19 Conclusions
- References
- Index
Summary
The Partition of Unity (PU) method addresses computational complexity and stability issues in global kernel approximation by decomposing the domain into overlapping subdomains. Local interpolants are computed independently on each subdomain and combined via weighted sums. Theoretical analysis shows that PU interpolants preserve local approximation orders in the global fit, with error bounds matching those of local schemes. Efficient data structures like integer-based partitioning enable fast point location and neighbor searches. We demonstrate the method's effectiveness for implicit surface reconstruction from 3D point clouds.
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- Kernel-Based MethodsMilestones and New Trends, pp. 48 - 55Publisher: Cambridge University PressPrint publication year: 2026
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