IUSB Department of Mathematical Sciences faculty publications and conference presentations
Permanent link for this collectionhttps://hdl.handle.net/2022/19840
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Item Krylov SSP Integrating Factor Runge-Kutta WENO Methods(MDPI, 2021-06) Cheng, ShanqinWeighted essentially non-oscillatory (WENO) methods are especially efficient for numeri-cally solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear com-ponent and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method. Keywords: strong stability preserving; integrating factor; Runge–Kutta; weighted essentially non-oscillatory methods; Krylov subspace approximationItem Uniformly Accurate Discontinuous Galerkin Fast Sweeping Methods for Eikonal Equations(Society for Industrial and Applied Mathematics, 2011-01) Zhang, Yong-Tao; Chen, Shanqin; Li, Fengyan; Zhao, Hong-Kai, 1968-; Shu, Chi-WangIn [F. Li, C.-W. Shu, Y.-T. Zhang, H. Zhao, Journal of Computational Physics 227 (2008) 8191- 8208], we developed a fast sweeping method based on a hybrid local solver which is a combination of a discontinuous Galerkin (DG) finite element solver and a first order finite difference solver for Eikonal equations. The method has second order accuracy in the L1 norm and a very fast convergence speed, but only first order accuracy in the L∞ norm for the general cases. This is an obstacle to the design of higher order DG fast sweeping methods. In this paper, we overcome this problem by developing uniformly accurate DG fast sweeping methods for solving Eikonal equations. We design novel causality indicators which guide the information flow directions for the DG local solver. The values of these indicators are initially provided by the first order finite difference fast sweeping method, and they are updated during iterations along with the solution. We observe both a uniform second order accuracy in the L∞ norm (in smooth regions) and the fast convergence speed (linear computational complexity) in the numerical examples. Key Words: fast sweeping methods, discontinuous Galerkin methods, uniform accuracy, static Hamilton-Jacobi equations, Eikonal equationsItem Stabilization of Nonholonomic Systems Using Isospectral Flows(Society for Industrial and Applied Mathematics, 2006-07) Bloch, Anthony, 1955-; Drakunov, Sergey; Kinyon, MichaelIn this paper we derive and analyze a discontinuous stabilizing feedback for a Lie algebraic generalization of a class of kinematic nonholonomic systems introduced by Brockett. The algorithm involves discrete switching between isospectral and norm-decreasing flows. We include a rigorous analysis of the convergence. Keywords: Nonlinear control, nonholonomic systems, isospectral flows, Lie theoryItem Hyper-Special Valued Lattice-Ordered Groups(De Gruyter, 2000) Darnel, Michael R., 1952-; Martinez, JorgeA lattice-ordered group G is hyper-special valued if it lies in the largest torsion class which is contained in the class of special-valued lattice-ordered groups. This is precisely the class of lattice-ordered g oups G such that for each g A G, every l-homomorphic image K of the principal convex l-subgroup generated by g has the feature that each 0 < x A K is the supremum of pairwise disjoint special elements. It is shown in this article that if G is hyper-special valued, then for each g ∈ G, the space of values Y(g) of g is a compact scattered space. This property naturally gives meaning to the notion of an α-special value of g: this is a value which corresponds to an isolated point of the α-th remainder in the Cantor-Bendixson sequence of Y(g). It is shown that, for each ordinal α, the set of a-special values of G forms a disjoint union of chains, which is at once an order ideal and a dual order ideal of the root system of all values of G. If G is projectable, then in addition the set of special values of G is also a disjoint union of chains which is an order ideal and a dual order ideal. An archimedean lattice-ordered group G with weak order unit u > 0, given its Yosida representation, such that u≡1 is hyper-special valued if and only if (a) G is projectable, (b) the Yosida space Y is scattered, and (c) for each g ∈ G the image of the function g has finitely many ∞'s as well as finitely many accumulations of 0.Item Higher Genus Doubly Periodic Minimal Surfaces(Taylor and Francis, 2018) Connor, PeterWe construct Weierstrass data for higher genus embedded doubly periodic minimal surfaces and present numerical evidence that the associated period problem can be solved. In the orthogonal ends case, there previously was only one known surface for each genus g. We illustrate multiple new examples for each genus g ≥ 3. In the parallel ends case, the known examples limit as a foliation of parallel planes with nodes. We construct a new example for each genus g ≥ 3 that limit as g−1 singly periodic Scherk surfaces glued between two doubly periodic Scherk surfaces and also as a singly periodic surface with four vertical and 2g horizontal Scherk ends. 2000 Mathematics Subject Classification. Primary 53A10; Secondary 49Q05, 53C42. Keywords: Minimal surfaces, doubly periodicItem A Generalization of Moufang and Steiner Loops(Springer, 2001-11) Kinyon, Michael; Kunen, Kenneth; Phillips, Jon D.We study a variety of loops, RIF, which arise naturally from considering inner mapping groups, and a somewhat larger variety, ARIF. All Steiner and Moufang loops are RIF, and all flexible C-loops are ARIF. All ARIF loops are diassociative.Item Krylov Implicit Integration Factor WENO Method for SIR Model with Directed Diffusion(American Institute of Mathematical Sciences, 2019) Zhao, Ruijun, 1962-; Zhang, Yong-Tao; Chen, ShanqinSIR models with directed diffusions are important in describing the population movement. However, efficient numerical simulations of such systems of fully nonlinear second order partial differential equations (PDEs) are challenging. They are often mixed type PDEs with ill-posed or degenerate components. The solutions may develop singularities along with time evolution. Stiffness due to nonlinear diffusions in the system gives strict constraints in time step sizes for numerical methods. In this paper, we design efficient Krylov implicit integration factor (IIF) Weighted Essentially Non-Oscillatory (WENO) method to solve SIR models with directed diffusions. Numerical experiments are performed to show the good accuracy and stability of the method. Singularities in the solutions are resolved stably and sharply by the WENO approximations in the scheme. Unlike a usual implicit method for solving stiff nonlinear PDEs, the Krylov IIF WENO method avoids solving large coupled nonlinear algebraic systems at every time step. Large time step size computations are achieved for solving the fully nonlinear second-order PDEs, namely, the time step size is proportional to the spatial grid size as that for solving a pure hyperbolic PDE. Two biologically interesting cases are simulated by the developed scheme to study the finite-time blow-up time and location or discontinuity locations in the solution of the SIR model.Item Krylov Implicit Integration Factor Discontinuous Galerkin Methods on Sparse Grids for High Dimensional Reaction-diffusion Equations(Elsevier, 2019) Liu, Yuan; Cheng, Yingda; Chen, Shanqin; Zhang, Yong-TaoComputational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving effcient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained. Key words: Sparse grid; Discontinuous Galerkin methods; Implicit integration factor methods; Krylov subspace approximation; Reaction-diffusion equationsItem Bernstein Polynomial Model for Nonparametric Multivariate Density(Taylor and Francis, 2019) Wang, Tao; Guan, ZhongIn this paper, we study the Bernstein polynomial model for estimating the multivariate distribution functions and densities with bounded support. As a mixture model of multivariate beta distributions, the maximum (approximate) likelihood estimate can be obtained using EM algorithm. A change-point method of choosing optimal degrees of the proposed Bernstein polynomial model is presented. Under some conditions, the optimal rate of convergence in the mean -divergence of new density estimator is shown to be nearly parametric. The method is illustrated by an application to a real data set. Finite sample performance of the proposed method is also investigated by simulation study and is shown to be much better than the kernel density estimate but close to the parametric ones. Keywords: Approximate Bernstein polynomial model; beta mixture; maximum likelihood; multivariate density estimation; nonparametric modelItem Nonparametric Estimator of False Discovery Rate Based on Bernšteǐn Polynomials(2008) Guan, Zhong; Wu, Baolin; Zhao, HongyuUnder a local dependence assumption about the p-values, an estimator of the proportion π0 of true null hypotheses, having a closed-form expression, is derived based on Bernšteǐn polynomial density estimation. A nonparametric estimator of false discovery rate (FDR) is then obtained. These estimators are proved to be consistent, asymptotically unbiased, and normal. Confidence intervals for π0 and the FDR are also given. The usefulness of the proposed method is demonstrated through simulations and its application to a microarray dataset. Keywords: Bernsteın polynomials, bioinformatics, density estimation, false discovery rate, local dependence, microarray, mixture model, multiple comparisonItem Bayesian Adaptive Designs for Clinical Trials(BMC Medicine, 2018) Cheng, Yi; Shen, YuA Bayesian adaptive design is proposed for a comparative two-armed clinical trial using decision theoretical approaches. A loss function is specified to consider the costfor each patient, and the costs of making incorrect decisions at the end of a trial. At each interim analysis, the decision to terminate or to continue the trial is based on the expected loss function while concurrently incorporating efficacy, futility and cost. The maximum number of interim analyses is not pre-fixed but decided adaptively by the observed data. We derive explicit connections between the loss function and the frequentist error rates, so that the desired frequentist properties can be maintained for regulatory settings. The operating characteristics of the design are able to be evaluated on frequentist grounds. Extensive simulations are carried out to compare the proposed design with existing ones. The design is general enough to accommodate both continuous and discrete types of data. We illustrate the methods with an animal study evaluating a medical treatment for cardiac arrest. Keywords: adaptive designs; Decision theory; Group sequential clinical trials; Loss function; Martingale convergence theorem.Item Archimedean Closed Lattice-Ordered Groups(Rocky Mountain Mathematics Consortium, 2004-06) Chen, Yuanqian; Conrad, Paul; Darnel, Michael R., 1952-We show that, if an abelian lattice-ordered group is archimedean closed, then each principal l-ideal is also archimedean closed. This has given a positive answer to the question raised in 1965 and hence proved that the class of abelian archimedean closed lattice-ordered groups is a radical class. We also provide some conditions for lattice-ordered group F(Δ,R) to be the unique archimedean closure of Σ (Δ,R).Item Loops and Semidirect Products(Communications in Algebra 28 (2000), no. 9, 4137–4164., 1999-07) Kinyon, Michael; Jones, Oliver"In this paper we study semidirect products of loops with groups. This is a generalization of the familiar semidirect product of groups." Discusses extensive information concerning semidirect products and transversal decompositions of various kinds of loops and left loops.