Research Interests: Mathematics and Graph
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The aim of this paper is to present partial differential equations (PDEs) on surface to the community of methods of fundamental solutions (MFS). First, we present an embedding formulation to embed surface PDEs into a domain so that MFS... more
The aim of this paper is to present partial differential equations (PDEs) on surface to the community of methods of fundamental solutions (MFS). First, we present an embedding formulation to embed surface PDEs into a domain so that MFS can be applied after the PDEs is homogenized with a particular solution. Next, we discuss how the domain-MFS method can be used to directly collocate surface PDEs. Some numerical demonstrations were included to study the effect of basis functions and source point locations. 1. Partial differential equations on surfaces. In this paper, we focus on second-order elliptic partial differential equations (PDEs) posed on some sufficiently smooth, connected, and compact surface S ⊂ R with bounded geometry. Without loss of generality, we assume dim(S) = d − 1, a.k.a., S has co-dimension 1. We denote the unit outward normal vector at x ∈ S as n = n(x) and the corresponding projection matrix to the tangent space of S at x as P(x) = [~ P1, . . . , ~ Pd](x) := Id ...
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Distributed stochastic gradient descent (SGD) algorithms are widely deployed in training large-scale deep learning models, while the communication overhead among workers becomes the new system bottleneck. Recently proposed gradient... more
Distributed stochastic gradient descent (SGD) algorithms are widely deployed in training large-scale deep learning models, while the communication overhead among workers becomes the new system bottleneck. Recently proposed gradient sparsification techniques, especially Top-$k$ sparsification with error compensation (TopK-SGD), can significantly reduce the communication traffic without an obvious impact on the model accuracy. Some theoretical studies have been carried out to analyze the convergence property of TopK-SGD. However, existing studies do not dive into the details of Top-$k$ operator in gradient sparsification and use relaxed bounds (e.g., exact bound of Random-$k$) for analysis; hence the derived results cannot well describe the real convergence performance of TopK-SGD. To this end, we first study the gradient distributions of TopK-SGD during the training process through extensive experiments. We then theoretically derive a tighter bound for the Top-$k$ operator. Finally, ...
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Partial differential equations (PDEs) on surfaces arise in a variety of application areas including biological systems, medical imaging, fluid dynamics, mathematical physics, image processing and computer graphics. In this paper, we... more
Partial differential equations (PDEs) on surfaces arise in a variety of application areas including biological systems, medical imaging, fluid dynamics, mathematical physics, image processing and computer graphics. In this paper, we propose a radial basis function (RBF) discretization of the closest point method. The corresponding localized meshless method may be used to approximate diffusion on smooth or folded surfaces. Our method has the benefit of having an a priori error bound in terms of percentage of the norm of the solution. A stable solver is used to avoid the ill-conditioning that arises when the radial basis functions (RBFs) become flat.