Numerical methods for high dimensional Hamilton-Jacobi equations using radial basis functions

By Thomas C. Cecil, Jianliang Qian, Stanley J. Osher. 2004

165 citations recorded by the source corpus.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1057ea3e0ab5bb5908e9db9d0277c2d3a72943a1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Statistical and Nonlinear Physics

Citations in the corpus

Radial basis functions and level set method for structural topology optimization (2006, background): “In real-world applications, radial basis function techniques have become extremely useful, ranging from pattern reconstruction, artificial intelligence, to simply solving mathematical PDEs based on irregular data distributions [52].”
Radial basis functions-finite differences collocation and a Unified Formulation for bending, vibration and buckling analysis of laminated plates, according to Murakami's zig-zag theory (2011, methodology): “[25] and Wright and Fornberg [26], in the so-called RBF-FD technique.”
Scattered node compact finite difference-type formulas generated from radial basis functions (2006, methodology): “This new " mesh-free " method is essentially a generalization of the classical finite difference (FD) method to scattered node layouts and appears to have been investigated independently by Shu et al. [7] [8], Tolstykh et al. [9], Cecil et al. [10], and the present authors [11].”
A local radial basis functions—Finite differences technique for the analysis of composite plates (2011, methodology): “As a result of the ill-conditioning, some authors proposed a radial basis function method with a local approach [24–28].”
Hermite WENO schemes for Hamilton-Jacobi equations (2005, unknown): “More recently, ENO schemes based on radial basis functions were constructed in [6].”
Weighted radial basis collocation method for boundary value problems (2007, methodology): “[2] gave a numerical scheme for Hamilton-Jacobi equations.”
A level set method for shape and topology optimization of large‐displacement compliant mechanisms (2008, background): “The continuity of the interpolant can be assured via the continuity of both the basis function and its partial derivatives [59].”
A level set‐based parameterization method for structural shape and topology optimization (2008, methodology): “The particular attractiveness of CS-RBFs is that the interplant can naturally inherit the continuity of the RBFs [36], which would enable us to select appropriate shape functions with favorable continuity [43, 49].”
Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks (2021, methodology): “Among meshfree methods, since construction of radial basis functions is dimension-independent, RBF-based meshfree methods have had success in solving high-dimensional problems [104].”
Meshfree Methods: Progress Made after 20 Years (2017, unknown): “A radial basis collocation method has been applied to singularity problems (Hu et al. 2005), Hamilton-Jacobi equations (Cecil et al. 2004), and fourth-order elliptic and parabolic problems (Li 2005).”
An 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions (2018, methodology): “Furthermore, the method can be used for optimizations of irregular design domains of unstructured meshes without difficulty (Cecil et al. 2004).”
Nonlocal Operators with Applications to Image Processing (2008, methodology): “Radial basis functions were used in [11] to obtain schemes in up to four dimensions.”
Structural Shape and Topology Optimization Using an Implicit Free Boundary Parametrization Method (2006, methodology): “In real-world applications, radial basis functions have become extremely useful, ranging from pattern reconstruction, artificial intelligence [Buhmann (2004)], to simply solving mathematical PDEs [Cheng, Golberg, Kansa, and Zammito (2003); Cecil, Qian, and Osher (2004)].”