We propose a simple embedding method for computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on implicit surfaces.The approach follows an embedding approach for solving the surface eikonal eq...We propose a simple embedding method for computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on implicit surfaces.The approach follows an embedding approach for solving the surface eikonal equation.We replace the differential operator on the interface with a typical Cartesian differential operator in the surface neighborhood.Our proposed algorithm is easy to implement and efficient.We will give some two-and three-dimensional numerical examples to demonstrate the effectiveness of our proposed approach.展开更多
Radial Basis Function methods for scattered data interpolation and for the numerical solution of PDEs were originally implemented in a global manner. Subsequently, it was realized that the methods could be implemented...Radial Basis Function methods for scattered data interpolation and for the numerical solution of PDEs were originally implemented in a global manner. Subsequently, it was realized that the methods could be implemented more efficiently in a local manner and that the local approaches could match or even surpass the accuracy of the global implementations. In this work, three localization approaches are compared: a local RBF method, a partition of unity method, and a recently introduced modified partition of unity method. A simple shape parameter selection method is introduced and the application of artificial viscosity to stabilize each of the local methods when approximating time-dependent PDEs is reviewed. Additionally, a new type of quasi-random center is introduced which may be better choices than other quasi-random points that are commonly used with RBF methods. All the results within the manuscript are reproducible as they are included as examples in the freely available Python Radial Basis Function Toolbox.展开更多
Aims and Scope The Journal of Computational Mathematics is published bi-monthly.It is an international journal covering all branches of modern computational mathematics such as numerical linear algebra,numerical optim...Aims and Scope The Journal of Computational Mathematics is published bi-monthly.It is an international journal covering all branches of modern computational mathematics such as numerical linear algebra,numerical optimization,computational geometry,numerical PDEs and inverse problems.Papers containing new ideas,creative approaches and/or innovative applications as well as invited reviews are expected to appear regularly in the journal.展开更多
We propose a new semi-implicit level set approach to a class of curvature dependent flows.The method generalizes a recent algorithm proposed for the motion by mean curvature where the interface is updated by solving t...We propose a new semi-implicit level set approach to a class of curvature dependent flows.The method generalizes a recent algorithm proposed for the motion by mean curvature where the interface is updated by solving the Rudin-Osher-Fatemi(ROF)model for image regularization.Our proposal is general enough so that one can easily extend and apply the method to other curvature dependent motions.Since the derivation is based on a semi-implicit time discretization,this suggests that the numerical scheme is stable even using a time-step significantly larger than that of the corresponding explicit method.As an interesting application of the numerical approach,we propose a new variational approach for extracting limit cycles in dynamical systems.The resulting algorithm can automatically detect multiple limit cycles staying inside the initial guess with no condition imposed on the number nor the location of the limit cycles.Further,we also propose in this work an Eulerian approach based on the level set method to test if the limit cycles are stable or unstable.展开更多
基金supported in part by the Hong Kong RGC 16302223.
文摘We propose a simple embedding method for computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on implicit surfaces.The approach follows an embedding approach for solving the surface eikonal equation.We replace the differential operator on the interface with a typical Cartesian differential operator in the surface neighborhood.Our proposed algorithm is easy to implement and efficient.We will give some two-and three-dimensional numerical examples to demonstrate the effectiveness of our proposed approach.
文摘Radial Basis Function methods for scattered data interpolation and for the numerical solution of PDEs were originally implemented in a global manner. Subsequently, it was realized that the methods could be implemented more efficiently in a local manner and that the local approaches could match or even surpass the accuracy of the global implementations. In this work, three localization approaches are compared: a local RBF method, a partition of unity method, and a recently introduced modified partition of unity method. A simple shape parameter selection method is introduced and the application of artificial viscosity to stabilize each of the local methods when approximating time-dependent PDEs is reviewed. Additionally, a new type of quasi-random center is introduced which may be better choices than other quasi-random points that are commonly used with RBF methods. All the results within the manuscript are reproducible as they are included as examples in the freely available Python Radial Basis Function Toolbox.
文摘Aims and Scope The Journal of Computational Mathematics is published bi-monthly.It is an international journal covering all branches of modern computational mathematics such as numerical linear algebra,numerical optimization,computational geometry,numerical PDEs and inverse problems.Papers containing new ideas,creative approaches and/or innovative applications as well as invited reviews are expected to appear regularly in the journal.
基金The work of Leung was supported in part by the RGC under Grant 605612。
文摘We propose a new semi-implicit level set approach to a class of curvature dependent flows.The method generalizes a recent algorithm proposed for the motion by mean curvature where the interface is updated by solving the Rudin-Osher-Fatemi(ROF)model for image regularization.Our proposal is general enough so that one can easily extend and apply the method to other curvature dependent motions.Since the derivation is based on a semi-implicit time discretization,this suggests that the numerical scheme is stable even using a time-step significantly larger than that of the corresponding explicit method.As an interesting application of the numerical approach,we propose a new variational approach for extracting limit cycles in dynamical systems.The resulting algorithm can automatically detect multiple limit cycles staying inside the initial guess with no condition imposed on the number nor the location of the limit cycles.Further,we also propose in this work an Eulerian approach based on the level set method to test if the limit cycles are stable or unstable.
