Slide 27
Slide 27 text
Introduction Univariate, Low Accuracy Multivariate, Reproducing Kernel Hilbert Space Summary References
Approximation via Reproducing Kernel Hilbert Spaces (RKHSs)
X
F is a Hilbert space with reproducing kernel
K : X × X → R
K(X, X) positive definite ∀X
K(·, x) ∈ F, f(x) = K(·, x), f
F
∀x ∈ X,
e.g., K(t, x) = (1 + t − x 2
) exp − t − x 2 Matérn
Optimal (minimum norm) interpolant is
APP(X, y) = K(·, X) K(X, X) −1
y, y = f(X)
f − APP(X, y) 2
∞
K(·, ·) − K(·, X) K(X, X) −1
K(X, ·)
∞
f − APP(X, y) 2
F known
K(·, ·) − K(·, X) K(X, X) −1
K(X, ·)
∞
C2(X)
1 − C2(X) APP(X, y) 2
F
=: ERR2(X, y)
candidate set C = f ∈ F : f − APP(X, y)
F
C(X) f
F
Fasshauer, G. E. Meshfree Approximation Methods with M . (World Scientific Publishing Co., Singapore, 2007),
Fasshauer, G. E. & McCourt, M. Kernel-based Approximation Methods using MATLAB. (World Scientific Publishing Co.,
Singapore, 2015). 8/14