Space-Filling Design and Kernels
Theory and Algorithms
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Beschreibung
This book provides a unified and comprehensive treatment of space-filling design and kernel methods, bringing together theoretical foundations, practical algorithms, and extensive numerical studies. It examines how finite point sets can be distributed efficiently over bounded domains and offers a systematic account of the principal criteria used to assess their quality, including packing and covering radii, quantization error, discrepancy, and model-based prediction and integration errors.
A distinctive feature of the book is its use of kernels to formulate, analyse, and relax classical space-filling criteria. It develops insightful connections between energy, potential theory, maximum mean discrepancy, reproducing kernel Hilbert spaces, and kernel-based approximation and integration. Particular attention is given to nested designs, in which points are added sequentially while preserving good space-filling properties—an important requirement in applications where observations are expensive or collected progressively.
The presentation combines rigorous mathematical results on optimal distribution of points, convergence, and approximation with implementable greedy algorithms and clear practical recommendations. Extensive numerical comparisons reveal the strengths and limitations of competing criteria and construction methods. The book also addresses the substantial geometric and computational challenges arising in high-dimensional cubes and proposes specialized approaches for high and very high dimensions.
By connecting ideas from approximation theory, potential theory, statistics, machine learning, numerical analysis, and experimental design, the book offers both a valuable reference and a practical guide. It will be useful to researchers, graduate students, and practitioners working with kernel methods, computer experiments, uncertainty quantification, numerical integration, sequential sampling, and efficient design construction.
This book provides a unified and comprehensive treatment of space-filling design and kernel methods, bringing together theoretical foundations, practical algorithms, and extensive numerical studies. It examines how finite point sets can be distributed efficiently over bounded domains and offers a systematic account of the principal criteria used to assess their quality, including packing and covering radii, quantization error, discrepancy, and model-based prediction and integration errors.
A distinctive feature of the book is its use of kernels to formulate, analyse, and relax classical space-filling criteria. It develops insightful connections between energy, potential theory, maximum mean discrepancy, reproducing kernel Hilbert spaces, and kernel-based approximation and integration. Particular attention is given to nested designs, in which points are added sequentially while preserving good space-filling properties—an important requirement in applications where observations are expensive or collected progressively.
The presentation combines rigorous mathematical results on optimal distribution of points, convergence, and approximation with implementable greedy algorithms and clear practical recommendations. Extensive numerical comparisons reveal the strengths and limitations of competing criteria and construction methods. The book also addresses the substantial geometric and computational challenges arising in high-dimensional cubes and proposes specialized approaches for high and very high dimensions.
By connecting ideas from approximation theory, potential theory, statistics, machine learning, numerical analysis, and experimental design, the book offers both a valuable reference and a practical guide. It will be useful to researchers, graduate students, and practitioners working with kernel methods, computer experiments, uncertainty quantification, numerical integration, sequential sampling, and efficient design construction.
Unifies geometric, probabilistic, and kernel-based approaches to space-filling design Develops practical algorithms for nested designs, supported by extensive numerical comparisons Provides specialized methods and practical guidance for space-filling in high dimensions
Autor*in
Toni S. Karvonen
Themen in »Space-Filling Design and Kernels«
maximum mean discrepancy Space-filling design Kernel methods Nested designs Packing radius Covering radius Quantization error Maximum mean discrepancy Energy minimization Potential theory Reproducing kernel Hilbert spaces Experimental design Gaussian processes Greedy algorithms High-dimensional design