Scattered Data Approximation: Conditionally positive definite functions

By Holger Wendland. 2004

997 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Numerical Methods & Scientific Computing

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Citations in the corpus

Reconciling modern machine-learning practice and the classical bias–variance trade-off (2018, background): “Applying Theorem 11.22 in [40] to f yields the result.”
Hilbert Space Embeddings and Metrics on Probability Measures (2009, background): “ve deﬁnite function. We provide a detailed explanation for Corollary 4 in Remark 5. Before stating the results, we need a famous result due to Bochner, that characterizes ψ. We quote this result from Wendland (2005, Theorem 6.6). Theorem 3 (Bochner) A continuous function ψ: Rd→ Ris positive deﬁnite ”
Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions (2016, background): “Let us mention that (cid:70) ψ is exactly the unique positive, symmetric, ﬁnite measure appearing in Bochner’s theorem (Wendland, 2004, Thm.6.6), and whose (usual) Fourier transform is ψ .”
Aerodynamic optimisation of a camber morphing aerofoil (2015, methodology): “Radial basis function methods [14] are favoured by some authors [15–18], especially for FSI simulations where they provide not only a framework to deform the aerodynamic and structural meshes, but a way to interpolate the forces and moments between them, as the two meshes will likely not be coincide”
Some covariance models based on normal scale mixtures (2011, background): “, along the way, introduces a new class of variograms, γ(h)=(khka +1)b −1, a∈(0,2],b∈(0,1]. This class generalizes the class of variograms of fractal Brownian motion and that of multiquadric kernels (Wendland (2005)). Example 9. In the context of modelling rainfall, Cox and Isham (1988) proposed in ”
Topology optimization of shell-infill structures using a distance regularized parametric level-set method (2018, positive): “…a vector of spatial shape functions and a vector of timedependent expansion coefficients φ xð Þ ¼ φ1 xð Þ φ2 xð Þ ⋯ φN xð Þ½ T ð6Þ α tð Þ ¼ α1 tð Þ α2 tð Þ ⋯ αN tð Þ½ T ð7Þ with φi(x) is Wendland’s CSRBF with C2 continuity (Wendland 2004) φi xð Þ ¼ max 0; 1−rð Þð Þ4⋅ 4r þ 1ð Þ ð8Þ r x−xik kð Þ…”
On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs (2010, background): “Further detail on RBF can be found in two excellent books, by Buhmann [26] and Wendland [27].”
Convergence Types and Rates in Generic Karhunen-Loève Expansions with Applications to Sample Path Properties (2014, background): “kernel Hilbert spaces (RKHSs), and their interaction with measures from [68]. To this end, let (T,B,ν) be a measure 5 space and k : T × T → Rbe a measurable (reproducing) kernel with RKHS H, see e.g. [7, 75, 66] for more information about these spaces. Recall that in this case the RKHS H consists of”