Regularized Kernel-Based Reconstruction in Generalized Besov Spaces
| Autor: | Michael Griebel, Barbara Zwicknagl, Christian Rieger |
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| Rok vydání: | 2017 |
| Předmět: |
Series (mathematics)
Truncation Applied Mathematics 010102 general mathematics Mathematical analysis Stability (learning theory) 010103 numerical & computational mathematics 01 natural sciences Regularization (mathematics) Kernel principal component analysis Computational Mathematics Computational Theory and Mathematics Kernel embedding of distributions Kernel (statistics) 0101 mathematics Representation (mathematics) Analysis Mathematics |
| Zdroj: | Foundations of Computational Mathematics. 18:459-508 |
| ISSN: | 1615-3383 1615-3375 |
| DOI: | 10.1007/s10208-017-9346-z |
| Popis: | We present a theoretical framework for reproducing kernel-based reconstruction methods in certain generalized Besov spaces based on positive, essentially self-adjoint operators. An explicit representation of the reproducing kernel is given in terms of an infinite series. We provide stability estimates for the kernel, including inverse Bernstein-type estimates for kernel-based trial spaces, and we give condition estimates for the interpolation matrix. Then, a deterministic error analysis for regularized reconstruction schemes is presented by means of sampling inequalities. In particular, we provide error bounds for a regularized reconstruction scheme based on a numerically feasible approximation of the kernel. This allows us to derive explicit coupling relations between the series truncation, the regularization parameters and the data set. |
| Databáze: | OpenAIRE |
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