Radial Basis Functions - Theory and Implementations

By M. Buhmann. Cambridge monographs on applied and computational mathematics, 2009

628 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/3a70e48cefdaf4035ad2009af02de871f0b339b1

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Citations in the corpus

• A level set method for shape and topology optimization of large‐displacement compliant mechanisms (2008, methodology): “For more details about the C6 shape function and its partial derivatives (Figure 2), refer to the related references [56, 58].”
• Review on computational methods for Lyapunov functions (2015, methodology): “Buhmann 2003 [47] and Wendland 2005 [225].”
• A level set‐based parameterization method for structural shape and topology optimization (2008, methodology): “The interplant of the level set function is uniquely determined in terms of the given interpolating data of the level set function located at the knots, owing to the property of strict positive definiteness of the CS-RBFs i (x) [43].”
• Numerical solution of fractional telegraph equation by using radial basis functions (2014, unknown): “R, according to the process of interpolation using radial basis functions [3], the interpolant of u is constructed in the following form ðSuÞðxÞ ¼ ∑ N i ¼ 1 λ i ϕðJx À x i J ÞþpðxÞ; x A R d ð2Þ where J Á J is the Euclidean norm and ϕðJ Á J Þ is a radial function.”
• Approximation of eigenfunctions in kernel-based spaces (2014, unknown): “Details can be retrieved from the monographs [2, 7, 27] and the surveys [1, 25].”
• Performance Analysis of Various Activation Functions in Generalized MLP Architectures of Neural Networks (2011, unknown): “The sigmoid functions are the most widely used functions [10-11].”
• Neural Networks and Deep Learning (2018, unknown): “An overview of radial basis functions is provided in [57].”
• Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks (2021, methodology): “First, we present a few connections of prior work on finite elements and partition-of-unity meshfree methods to better understand and place the PINN approximation and its use to solve PDEs.”