In their recent work, Gentili and Struppa proposed a different quaternionic analogue of the notion of holomorphic functions in the complex plane, called slice regular functions, which has led to several analogues of c...
One of the weaknesses of classical (fuzzy) rough sets is their sensitivity to noise, which is particularly undesirable for machine learning applications. One approach to solve this issue is by making use of fuzzy quan...
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Bayesian approaches are one of the primary methodologies to tackle an inverse problem in high dimensions. Such an inverse problem arises in hydrology to infer the permeability field given flow data in a porous media. ...
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computational models are increasingly used for diagnosis and treatment of cardiovascular disease. To provide a quantitative hemodynamic understanding that can be effectively used in the clinic, it is crucial to quanti...
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Deep neural networks (DNNs) are famous for their high prediction accuracy, but they are also known for their black-box nature and poor interpretability. We consider the problem of variable selection, that is, selectin...
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In this paper, we focus on the finite difference approximation of nonlinear degenerate parabolic equations, a special class of parabolic equations where the viscous term vanishes in certain regions. This vanishing giv...
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In this paper, we focus on the finite difference approximation of nonlinear degenerate parabolic equations, a special class of parabolic equations where the viscous term vanishes in certain regions. This vanishing gives rise to additional challenges in capturing sharp fronts, beyond the restrictive CFL conditions commonly encountered with explicit time discretization in parabolic equations. To resolve the sharp front, we adopt the high-order multi-resolution alternative finite difference WENO (A-WENO) methods for the spatial discretization, which is designed to effectively suppress oscillations in the presence of large gradients and achieve nonlinear stability. To alleviate the time step restriction from the nonlinear stiff diffusion terms, we employ the exponential time differencing Runge-Kutta (ETD-RK) methods, a class of efficient and accurate exponential integrators, for the time discretization. However, for highly nonlinear spatial discretizations such as high-order WENO schemes, it is a challenging problem how to efficiently form the linear stiff part in applying the exponential integrators, since direct computation of a Jacobian matrix for high-order WENO discretizations of the nonlinear diffusion terms is very complicated and expensive. Here we propose a novel and effective approach of replacing the exact Jacobian of high-order multi-resolution A-WENO scheme with that of the corresponding high-order linear scheme in the ETD-RK time marching, based on the fact that in smooth regions the nonlinear weights closely approximate the optimal linear weights, while in non-smooth regions the stiff diffusion degenerates. The algorithm is described in detail, and numerous numerical experiments are conducted to demonstrate the effectiveness of such a treatment and the good performance of our method. The stiffness of the nonlinear parabolic partial differential equations (PDEs) is resolved well, and large time-step size computations of ∆t ∼ O(∆x) are *** Codes 65
Discovering patterns of the complex high-dimensional data is a long-standing problem. Dimension Reduction (DR) and Intrinsic Dimension Estimation (IDE) are two fundamental thematic programs that facilitate geometric u...
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Three aspects of applying homotopy continuation, which is commonly used to solve parameterized systems of polynomial equations, are investigated. First, for parameterized systems which are homogeneous, we investigate ...
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The monodromy group is an invariant for parameterized systems of polynomial equations that encodes structure of the solutions over the parameter space. Since the structure of real solutions over real parameter spaces ...
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