# Multiscale Approximation and Reproducing Kernel Hilbert Space Methods

@article{Griebel2015MultiscaleAA, title={Multiscale Approximation and Reproducing Kernel Hilbert Space Methods}, author={Michael Griebel and Christian Rieger and Barbara Zwicknagl}, journal={SIAM J. Numer. Anal.}, year={2015}, volume={53}, pages={852-873} }

We consider reproducing kernels $K:\Omega\times \Omega \to \mathbb{R}$ in multiscale series expansion form, i.e., kernels of the form $K\left(\boldsymbol{x},\boldsymbol{y}\right)=\sum_{\ell\in\mathbb{N}}\lambda_\ell\sum_{j\in I_\ell}\phi_{\ell,j}\left(\boldsymbol{x}\right)\phi_{\ell,j}\left(\boldsymbol{y}\right)$ with weights $\lambda_\ell$ and structurally simple basis functions $\left\{\phi_{\ell,i}\right\}$. Here, we deal with basis functions such as polynomials or frame systems, where, for…

## 16 Citations

### An introduction to the Hilbert-Schmidt SVD using iterated Brownian bridge kernels

- Computer ScienceNumerical Algorithms
- 2014

A class of so-called iterated Brownian bridge kernels which allow the discussion to keep the discussion as simple and accessible as possible are introduced.

### An inverse theorem for compact Lipschitz regions in ℝd using localized kernel bases

- MathematicsMath. Comput.
- 2018

Inverse estimates for finite dimensional spaces arising in radial basis function approximation and meshless methods are developed that consider control Sobolev norms of linear combinations of a localized basis by the $L_p$ norm over a bounded domain.

### Direct and Inverse Results on Bounded Domains for Meshless Methods via Localized Bases on Manifolds

- Mathematics
- 2018

This article develops direct and inverse estimates for certain finite dimensional spaces arising in kernel approximation. Both the direct and inverse estimates are based on approximation spaces…

### Regularized Kernel-Based Reconstruction in Generalized Besov Spaces

- MathematicsFound. Comput. Math.
- 2018

A theoretical framework for reproducing kernel-based reconstruction methods in certain generalized Besov spaces based on positive, essentially self-adjoint operators is presented and explicit coupling relations between the series truncation, the regularization parameters and the data set are derived.

### A stable method for the evaluation of Gaussian radial basis function solutions of interpolation and collocation problems

- Computer Science, MathematicsComput. Math. Appl.
- 2016

### Reproducing Kernel Hilbert Spaces for Parametric Partial Differential Equations

- Mathematics, Computer ScienceSIAM/ASA J. Uncertain. Quantification
- 2017

This paper presents kernel methods for the approximation of quantities of interest which are derived from solutions of parametric partial differential equations and suggests a regularized reconstruction technique from machine learning in order to approximate the quantity of interest from a finite number of point values.

### An inverse theorem on bounded domains for meshless methods using localized bases

- Mathematics
- 2014

This article develops inverse estimates for finite dimensional spaces arising in kernel approximation and meshless methods. These control Sobolev norms of linear combinations of a localized basis by…

### Existence of Unique Solutions to the Telegraph Equation in Binary Reproducing Kernel Hilbert Spaces

- MathematicsDifferential Equations and Dynamical Systems
- 2019

We demonstrate the existence of a unique solution to a nonhomogeneous telegraph initial/boundary value problem on the unit square in an appropriate binary reproducing kernel Hilbert space which…

### On the Convergence Rate of Sparse Grid Least Squares Regression

- Computer Science
- 2018

This paper presents a framework which will allow for a thorough theoretical analysis of stability properties, error decay behavior and appropriate couplings between the dataset size and the grid size and rigorously derive upper bounds on the expected error for sparse grid least squares regression.

### Error Estimates for Multivariate Regression on Discretized Function Spaces

- Computer Science, MathematicsSIAM J. Numer. Anal.
- 2017

The discretization error for the regression setting is discussed and error bounds are derived relying on the approximation properties of the discretized space and two examples based on tensor product spaces are presented which provide a suitable approach in the case of large sample sets in moderate dimensions.

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