Strongly coupled nonlinear phenomena such as those described by Earth System Models (ESM) are com... more Strongly coupled nonlinear phenomena such as those described by Earth System Models (ESM) are composed of multiple component models with independent mesh topologies and scalable numerical solvers. A common operation in ESM is to remap or interpolate results from one component's numerical mesh to another, e.g., from the atmosphere to the ocean, during the temporal integration of the coupled system. Several remapping schemes are currently in use or available for ESM. However, a unified approach to compare the properties of these different schemes has not been attempted previously. We present a rigorous methodology for the evaluation and intercomparison of remapping methods through an independently implemented suite of metrics that measure the ability of a method to adhere to constraints such as grid independence, monotonicity, global conservation, and local extrema or feature preservation. A comprehensive set of numerical evaluations can be conducted based on a progression of scal...
A new adaptive Particle-in-Cloud (AP-Cloud) method for obtaining optimal numerical solutions to t... more A new adaptive Particle-in-Cloud (AP-Cloud) method for obtaining optimal numerical solutions to the Vlasov-Poisson equation has been proposed. The traditional particle-in-cell (PIC) method, commonly used for solving this problem, is not optimal in terms of the balance of errors of the differential operator discretization and source integration; it is also inaccurate when the particle distribution is highly non-uniform. Our method replaces the Cartesian grid in the traditional PIC with adaptive computational nodes or particles, to which the charges from the physical macroparticles are assigned by a weighted least-square approximations. The partial differential equation is then discretized using a generalized finite difference (GFD) method and solved with fast linear solvers. The density of computational particles is chosen adaptively, so that the error from GFD and that from the source integration are balanced and the total error is approximately minimized. The method is independent ...
Numerical integration on spheres, including the computation of the areas of spherical triangles, ... more Numerical integration on spheres, including the computation of the areas of spherical triangles, is a core computation in geomathematics. The commonly used techniques sometimes suffer from instabilities and significant loss of accuracy. We describe a new algorithm, called ARPIST, for accurate and stable integration of functions on spherical triangles. ARPIST is based on an easy-to-implement transformation to the spherical triangle from its corresponding linear triangle via radial projection to achieve high accuracy and efficiency. More importantly, ARPIST overcomes potential instabilities in computing the Jacobian of the transformation, even for poorly shaped triangles that may occur at poles in regular longitude-latitude meshes, by avoiding potential catastrophic rounding errors. We compare our proposed technique with L'Huilier's Theorem for computing the area of spherical triangles, and also compare it with the recently developed LSQST method (J. Beckmann, H.N. Mhaskar, an...
Network structured data is ubiquitous in natural and social science applications. Graph Convoluti... more Network structured data is ubiquitous in natural and social science applications. Graph Convolutional Neural Network (GCN) has attracted significant attention recently due to its success in representing, modeling, and predicting large-scale network data. Various types of graph convolutional filters were proposed to process graph signals to boost the performance of graph-based semi-supervised learning. This paper introduces a novel spectral learning technique called EigLearn, which uses residual learning to perturb the eigenvalues of the graph filter matrix to optimize its capability. EigLearn is relatively easy to implement, yet thorough experimental studies reveal that it is more effective and efficient than prior works on the specific issue, such as LanczosNet and FisherGCN. EigLearn only perturbs a small number of eigenvalues and does not require a complete eigendecomposition. Our investigation shows that EigLearn reaches the maximal performance improvement by perturbing about 30...
Strongly coupled nonlinear phenomena such as those described by Earth System Models (ESM) are com... more Strongly coupled nonlinear phenomena such as those described by Earth System Models (ESM) are composed of multiple component models with independent mesh topologies and scalable numerical solvers. A common operation in ESM is to remap or interpolate results from one component's numerical mesh to another, e.g., from the atmosphere to the ocean, during the temporal integration of the coupled system. Several remapping schemes are currently in use or available for ESM. However, a unified approach to compare the properties of these different schemes has not been attempted previously. We present a rigorous methodology for the evaluation and intercomparison of remapping methods through an independently implemented suite of metrics that measure the ability of a method to adhere to constraints such as grid independence, monotonicity, global conservation, and local extrema or feature preservation. A comprehensive set of numerical evaluations can be conducted based on a progression of scal...
A new adaptive Particle-in-Cloud (AP-Cloud) method for obtaining optimal numerical solutions to t... more A new adaptive Particle-in-Cloud (AP-Cloud) method for obtaining optimal numerical solutions to the Vlasov-Poisson equation has been proposed. The traditional particle-in-cell (PIC) method, commonly used for solving this problem, is not optimal in terms of the balance of errors of the differential operator discretization and source integration; it is also inaccurate when the particle distribution is highly non-uniform. Our method replaces the Cartesian grid in the traditional PIC with adaptive computational nodes or particles, to which the charges from the physical macroparticles are assigned by a weighted least-square approximations. The partial differential equation is then discretized using a generalized finite difference (GFD) method and solved with fast linear solvers. The density of computational particles is chosen adaptively, so that the error from GFD and that from the source integration are balanced and the total error is approximately minimized. The method is independent ...
Numerical integration on spheres, including the computation of the areas of spherical triangles, ... more Numerical integration on spheres, including the computation of the areas of spherical triangles, is a core computation in geomathematics. The commonly used techniques sometimes suffer from instabilities and significant loss of accuracy. We describe a new algorithm, called ARPIST, for accurate and stable integration of functions on spherical triangles. ARPIST is based on an easy-to-implement transformation to the spherical triangle from its corresponding linear triangle via radial projection to achieve high accuracy and efficiency. More importantly, ARPIST overcomes potential instabilities in computing the Jacobian of the transformation, even for poorly shaped triangles that may occur at poles in regular longitude-latitude meshes, by avoiding potential catastrophic rounding errors. We compare our proposed technique with L'Huilier's Theorem for computing the area of spherical triangles, and also compare it with the recently developed LSQST method (J. Beckmann, H.N. Mhaskar, an...
Network structured data is ubiquitous in natural and social science applications. Graph Convoluti... more Network structured data is ubiquitous in natural and social science applications. Graph Convolutional Neural Network (GCN) has attracted significant attention recently due to its success in representing, modeling, and predicting large-scale network data. Various types of graph convolutional filters were proposed to process graph signals to boost the performance of graph-based semi-supervised learning. This paper introduces a novel spectral learning technique called EigLearn, which uses residual learning to perturb the eigenvalues of the graph filter matrix to optimize its capability. EigLearn is relatively easy to implement, yet thorough experimental studies reveal that it is more effective and efficient than prior works on the specific issue, such as LanczosNet and FisherGCN. EigLearn only perturbs a small number of eigenvalues and does not require a complete eigendecomposition. Our investigation shows that EigLearn reaches the maximal performance improvement by perturbing about 30...
Uploads