Meta-analysis and sensitivity analysis for multi-arm trials with selection bias
Hathaikan Chootrakool, Jian Qing Shi and Rongxian Yue
Stat. Medi., 30(2011), 1183-1198.
Multi-arm trials meta-analysis is a methodology used in combining evidence based on a synthesis of different types of comparisons from all possible similar studies and to draw inferences about the effectiveness of multiple compared-treatments. Studies with statistically significant results are potentially more likely to be submitted and selected than studies with non-significant results; this leads to false-positive results. In meta-analysis, combining only the identified selected studies uncritically may lead to an incorrect, usually over-optimistic conclusion. This problem is known as selection bias. In this paper, we first define a random-effect metaanalysis model for multi-arm trials by allowing for heterogeneity among studies. This general model is based on a normal approximation for empirical log-odds ratio. We then address the problem of publication bias by using a sensitivity analysis and by defining a selection model to the available data of a meta-analysis. This method allows for different amounts of selection bias and helps to investigate how sensitive the main interest parameter is when compared with the estimates of the standard model. Throughout the paper, we use binary data from Antiplatelet therapy in maintaining vascular patency of patients to illustrate the methods.
Pseuspectral method for quadrilaterals
Benyu Guo and Hongli Jia
J. Comp. Appl. Math., 236(2011), 962-979.
In this paper, we investigate the pseudospectral method on quadrilaterals. Some results on Legendre-Gauss-type interpolation are established, which play important roles in the pseudospectral method for partial differential equations defined on quadrilaterals. As examples of applications, we propose pseudospectral methods for two model problems and prove their spectral accuracy in space. Numerical results demonstrate the efficiency of the suggested algorithms. The approximation results and techniques developed in this paper are also applicable to other problems defined on quadrilaterals.
Some developments in spectral methods
Benyu Guo, Chao Zhang and Tao Sun
Studies Adv. Math., 51(2011), 561-574.
In this paper, we review some new developments in spectral methods. We first consider the generalized Jacobi spectral method. Then, we present the Jacobi quasi-orthogonal approximation and its applications. Next, we consider the generalized Laguerre spectral method. We also present the Laguerre quasi-orthogonal approximation and its applications.
Adini-Q1-P3 FEM for general elastic multi-structure problems
Ling Guo and Jianguo Huang
Numer. Meth. Part. Diff. Equa., 27(2011), 1092-1112.
An Adini-Q1-P3 finite element method is introduced to solve general elastic multi-structure problems, where displacements on bodies, longitudinal displacements on plates, longitudinal displacements and rotational angles on rods are discretized by conforming linear (bilinear or trilinear) elements, transverse displacements on plates and rods are discretized by Adini elements and Hermite elements of third order, respectively. The unique solvability and optimal error estimates in the energy norm are established for the discrete method, whose numerical performance is illustrated by some numerical examples.
RTE-based bioluminescence tomography: A theoretical study
Weimin Han, Joseph A. Eichholzb, Jianguo Huang and Jia Lu
Inv. Prob. Sci. Engi., 19(2011), 435-459.
Molecular imaging has become a most rapidly developing area in biomedical imaging. Bioluminescence tomography (BLT) is an emerging and promising molecular imaging technology. Light propagation within biological media is accurately described by the radiative transfer equation (RTE). However, due to the difficulties in theoretical investigation and numerical simulations, so far, the study of BLT problem has been largely based on a diffusion approximation of the RTE. In this paper, we provide a rigorous theoretical foundation for the study of the RTE based BLT. After a discussion of the forward problem of the RTE and its numerical approximation, we establish a comprehensive mathematical framework for the RTE based BLT problem through Tikhonov regularization. We show the solution existence, uniqueness and continuous dependence on the data for the regularized formulation. We then introduce stable numerical methods for the BLT reconstruction and show convergence of the numerical solutions. Finally, we present simulation results from a numerical example to demonstrate that reasonable numerical results can be expected from solving the RTE based BLT problem via regularization.
Local and parallel algorithms for fourth order problems discretized
by the Morley-Wang-Xu element method
Jianguo Huang and Xuehai Huang
Numer. Math., 119(2011), 667-697.
This paper systematically studies numerical solution of fourth order problems in any dimensions by use of the Morley-Wang-Xu (MWX) element discretization combined with two--grid methods. Since the coarse and fine finite element spaces are nonnested, two intergrid transfer operators are first constructed in any dimensions technically, based on which two classes of local and parallel algorithms are then devised for solving such problems. Following some ideas by Xu and Zhou, the intrinsic derivation of error analysis for nonconforming finite element methods of fourth order problems, and the error estimates for the intergrid transfer operators, we derive error estimates for the two classes of methods. Numerical results are performed to support the theory obtained and to compare the numerical performance of several local and parallel algorithms using different intergrid transfer operators.
Convergence of an adaptive mixed finite element method
for Kirchhoff plate bending problems
Jianguo Huang, Xuehai Huang and Yifeng Xu
SIAM J. Numer. Anal., 49(2011), 574-607.
Some reliable and efficient a posteriori error estimators are produced for a mixed finite element method (the Hellan-Herrmann-Johnson (H-H-J) method) for Kirchhoff plate bending problems. Based on these results with k=0,1, where k denotes the polynomial order of the discrete moment-field space, an adaptive mixed finite element method (AMFEM) is set up and its convergence and complexity are studied thoroughly. The key points of the theoretical analysis include achieving a discrete Helmholtz decomposition and a discrete inf-sup condition, which serve as the main tool to deduce the quasi-orthogonality of the moment field and the discrete reliability of the estimator. It is shown that the AMFEM is a contraction for the sum of the moment-field error in an energy norm and the scaled error estimator between two consecutive adaptive loops. Moreover, an estimate for the AMFEM's complexity via the number of elements is developed.
Multidomain pseudospectral methods for nonlinear convection-diffusion equations
Yuanyuan Ji, Hua Wu, Heping Ma and Benyu Guo,
Appl. Math. Mech., 32(2011), 1255–1268.
Multidomain pseudospectral approximations to nonlinear convection-diffusion equations are considered. The schemes are formulated with the Legendre-Galerkin method, but the nonlinear term is collocated at the Legendre/Chebyshev-Gauss-Lobatto points inside each subinterval. Appropriate base functions are introduced so that the matrix of the system is sparse, and the method can be implemented efficiently and in parallel. The stability and the optimal rate of convergence of the methods are proved. Numerical results are given for both the single domain and the multidomain methods to make a comparison.
Asymptotic stability of neutral differential systems with many delays
Jiaoxun Kuang, Hongjiong Tian and Kaiting Shan
Appl. Math. Comp., 217 (2011), 10087-10094
We are concerned with delay-independent asymptotic stability of linear system of neutral differential equations. We first establish a sufficient and necessary condition for the system to be delay-independently asymptotically stable, and then give some equivalent stability conditions. This paper improves many recent results on the asymptotic stability in the literature. One example is given to show that the sufficient and necessary condition is easy to verify.
Optimality criteria for multiresponse linear models
based on predictive ellipsoids
Xin Liu, Rongxian Yue and Fred J. Hickernell
Statistica Sinica, 21(2011), 421-432.
This paper proposes a new class of optimum design criteria for the linear regression model with r responses based on the volume of the predictive ellipsoid. This is referred to as -optimality. The -optimality criterion is invariant with respect to different parameterizations of the model, and reduces to -optimality as proposed by Dette and O’Brien (1999) in single response situations. An equivalence theorem for -optimality is provided and used to verify -optimality of designs, and this is illustrated by several examples.
Solving rational eigenvalue problems via linearization
Yangfeng Su and Zhaojun Bai
SIAM J. Matrix Anal. Appl., 32(1) (2011), 201-216.
The rational eigenvalue problem is an emerging class of nonlinear eigenvalue problems arising from a variety of physical applications. In this paper, we propose a linearization-based method to solve the rational eigenvalue problem. The proposed method converts the rational eigenvalue problem into a well-studied linear eigenvalue problem, and meanwhile, exploits and preserves the structure and properties of the original rational eigenvalue problem. For example, the low-rank property leads to a trimmed linearization. We show that solving a class of rational eigenvalue problems is just as convenient and efficient as solving linear eigenvalue problems.
Asymptotic stability of linear neutral delay differential-algebraic equations and linear multistep methods
Hongjiong Tian, Quanhong Yu and Jiaoxun Kuang
SIAM J. Numer. Anal., 49(2011), 608-618.
This paper is concerned with delay-independent asymptotic stability of linear neutral delay differential-algebraic equations and linear multistep methods. We first give some sufficient conditions for the delay-independent asymptotic stability of these equations. Then we study and derive a sufficient and necessary condition for the delay-independent asymptotic stability of numerical solutions obtained by linear multistep methods combined with Lagrange interpolation. Finally, one numerical example is performed to confirm our theoretical result.
Continuous block implicit hybrid one-step methods forordinary and delay differential equations
Hongjiong Tian, Quanhong Yu and Cilai Jin
Appl. Numer. Math., 61 (2011), 1289–1300.
A class of high order continuous block implicit hybrid one-step methods has been proposed to solve numerically initial value problems for ordinary and delay differential equations. The convergence and -stability of the continuous block implicit hybrid methods for ordinary differential equations are studied. Alternative form of continuous extension is constructed such that the block implicit hybrid one-step methods can be used to solve delay differential equations and have same convergence order as for ordinary differential equations. Some numerical experiments are conducted to illustrate the efficiency of the continuous methods.
A modified accelerated monotone iterative method for finite difference reaction-diffusion-convection equations
J. Comp. Appl. Math., 235 (2011), 3646-3660.
This paper is concerned with monotone algorithms for the finite difference solutions of a class of nonlinear reaction-diffusion-convection equations with nonlinear boundary conditions. A modified accelerated monotone iterative method is presented to solve the finite difference systems for both the time-dependent problem and its corresponding steady-state problem. This method leads to a simple and yet efficient linear iterative algorithm. It yields two sequences of iterations that converge monotonically from above and below, respectively, to a unique solution of the system. The monotone property of the iterations gives concurrently improving upper and lower bounds for the solution. It is shown that the rate of convergence for the sum of the two sequences is quadratic. Under an additional requirement, quadratic convergence is attained for one of these two sequences. In contrast with the existing accelerated monotone iterative methods, our new method avoids computing local maxima in the construction of these sequences. An application using a model problem gives numerical results that illustrate the effectiveness of the proposed method.
On Numerov’s method for a class of strongly nonlinear two-point boundary value problems
Appl. Numer. Math., 61(2011) , 38-52.
The purpose of this paper is to give a numerical treatment for a class of strongly nonlinear two-point boundary value problems. The problems are discretized by fourth-order Numerov’s method, and a linear monotone iterative algorithm is presented to compute the solutions of the resulting discrete problems. All processes avoid constructing explicitly an inverse function as is often needed in the known treatments. Consequently, the full potential of Numerov’s method for strongly nonlinear two-point boundary value problems is realized. Some applications and numerical results are given to demonstrate the high efficiency of the approach.
Error and extrapolation of a compact LOD method for parabolic differential equations
J. Comp. Appl. Math., 235 (2011) , 1367-1382.
This paper is concerned with a compact locally one-dimensional (LOD) finite difference method for solving two-dimensional nonhomogeneous parabolic differential equations. An explicit error estimate for the finite difference solution is given in the discrete infinity norm. It is shown that the method has the accuracy of the second-order in time and the fourth-order in space with respect to the discrete infinity norm. A Richardson extrapolation algorithm is developed to make the final computed solution fourth-order accurate in both time and space when the time step equals the spatial mesh size. Numerical results demonstrate the accuracy and the high efficiency of the extrapolation algorithm.
A fourth-order compact finite difference method for nonlinear higher-order multi-point boundary value problems
Yuanming Wang, Wenjia Wu and Ravi P. Agarwal
Comp. Math. Appl., 61 (2011), 3226-3245.
A fourth-order compact finite difference method is proposed for a class of nonlinear 2nth-order multi-point boundary value problems. The multi-point boundary condition under consideration includes various commonly discussed boundary conditions, such as the three- or four-point boundary condition, (n+2)-point boundary condition and 2(n-m)-point boundary condition. The existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. The convergence and the fourth-order accuracy of the method are proved. An efficient monotone iterative algorithm is developed for solving the resulting nonlinear finite difference systems. Various sufficient conditions for the construction of upper and lower solutions are obtained. Some applications and numerical results are given to demonstrate the high efficiency and advantages of this new approach.
A block monotone iterative method for numerical solutions of nonlinear elliptic boundary value problems
Yuanming Wang, Cuixia Liang and Ravi P. Agarwal
Numer. Meth. Part. Diff. Equa., 27(2011), 680-701.
The aim of this article is to develop a new block monotone iterative method for the numerical solutions of a nonlinear elliptic boundary value problem. The boundary value problem is discretized into a system of nonlinear algebraic equations, and a block monotone iterative method is established for the system using an upper solution or a lower solution as the initial iteration. The sequence of iterations can be computed in a parallel fashion and converge monotonically to a maximal solution or a minimal solution of the system. Three theoretical comparison results are given for the sequences from the proposed method and the block Jacobi monotone iterative method. The comparison results show that the sequence from the proposed method converges faster than the corresponding sequence given by the block Jacobi monotone iterative method. A simple and easily verified condition is obtained to guarantee a geometric convergence of the block monotone iterations. The numerical results demonstrate advantages of this new approach.
Global asymptotic stability of Lotka-Volterra competition reaction-diffusion systems with time delays
Math. Comp. Model., 53(2011), 337-346.
This paper is concerned with a time-delayed Lotka-Volterra competition reaction-diffusion system with homogeneous Neumann boundary conditions. Some explicit and easily verifiable conditions are obtained for the global asymptotic stability of all forms of nonnegative semitrivial constant steady-state solutions. These conditions involve only the competing rate constants and are independent of the diffusion-convection and time delays. The result of global asymptotic stability implies the nonexistence of positive steady-state solutions, and gives some extinction results of the competing species in the ecological sense. The instability of the trivial steady-state solution is also shown.
A higher-order compact ADI method with monotone iterative procedure for systems of reaction-diffusion equations
Yuanming Wang and Jie Wang
Comp. Math. Appl., 62 (2011), 2434-2451.
This paper is concerned with an existing compact finite difference ADI method, published in the paper by Liao et al. (2002) , for solving systems of two-dimensional reaction-diffusion equations with nonlinear reaction terms. This method has an accuracy of fourth-order in space and second-order in time. The existence and uniqueness of its solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear reaction terms. The convergence of the finite difference solution to the continuous solution is proved. An efficient monotone iterative algorithm is presented for solving the resulting discrete system, and some techniques for the construction of upper and lower solutions are discussed. An application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method.
Mixed spectral and pseudospectral methods for a nonlinear strongly damped wave equation in an exterior domain
Zhongqing Wang and Rong Zhang
Numer. Math. Theor. Meth. Appl., 4(2011), 255-282.
The aim of this paper is to develop the mixed spectral and pseudospectral methods for nonlinear problems outside a disc, using Fourier and generalized Laguerre functions. As an example, we consider a nonlinear strongly damped wave equation. The mixed spectral and pseudospectral schemes are proposed. The convergence is proved. Numerical results demonstrate the efficiency of this approach.
The Laguerre spectral method for solving Neumann
boundary value problems
J. Comp. Appl. Math., 235(2011), 3229-3237.
In this paper, we propose a Laguerre spectral method for solving Neumann boundary value problems. This approach differs from the classical spectral method in that the homogeneous boundary condition is satisfied exactly. Moreover, a tridiagonal matrix is employed, instead of the full stiffness matrix encountered in the classical variational formulation of such problems. For analyzing the numerical errors, some basic results on Laguerre approximations are established. The convergence is proved. The numerical results demonstrate the efficiency of this approach.
Carleman estimate for a fractional diffusion equation with half order and application
Xiang Xu, Jin Cheng and Masahiro Yamamoto
Appl. Anal., 90(2011), 1355–1371.
We consider a fractional diffusion equation in where the derivative in time t is of half order in the sense of Caputo and we establish a Carleman estimate. Since the derivatives of non-natural number orders do not satisfy the integration by parts, which is essential for establishing a Carleman estimate, we twice apply the Caputo derivative to convert the original fractional diffusion equation to a system with a usual partial differential operator: . Next we apply the Carleman estimate to prove the conditional stability in a Cauchy problem with data , .
A collocation method for initial value problems of second-order ODEs by using Laguerre functions
Jianping Yan and Benyu Guo
Numer. Math. Theo. Meth. Appl., 4(2011), 282-294.
We propose a collocation method for solving initial value problems of secondorder ODEs by using modified Laguerre functions. This new process provides global numerical solutions. Numerical results demonstrate the efficiency of the proposed algorithm.
Laguerre-Gauss collocation method for initial value problems of second-order ODEs
Jianping Yan and Benyu Guo
Appl. Math. Mech., 32(2011), 1541-1564.
This paper proposes a new collocation method for initial value problems of second order ODEs based on the Laguerre-Gauss interpolation. It provides the global numerical solutions and possesses the spectral accuracy. Numerical results demonstrate its high efficiency.
Optimal U-type design for Bayesian nonparametric multiresponse prediction
Rongxian Yue, Hong Qin and Kashinath Chatterjee
J. Stat. Plan. Infer., 141(2011), 2472-2479.
This paper presents an extension of the work of Yue and Chatterjee (2010) about U-type designs for Bayesian nonparametric response prediction. We consider nonparametric Bayesian regression model with p responses. We use U-type designs with n runs, m factors and q levels for the nonparametric multiresponse prediction based on the asymptotic Bayesian criterion. A lower bound for the proposed criterion is established, and some optimal and nearly optimal designs for the illustrative models are given.
A Markov risk model with two classes of insurance business
Fei Zhao, Rongxian Yue and Hanxin Wang
Stoch. Anal. Appl., 29(2011), 1102-1110.
A Markov risk model with two classes of insurance business is studied. In this model, the two classes of insurance business are independent. Each of the two independent claim number processes is the number of jumps of a Markov jump process from time 0 to t, whichever has not independent increments in general. An integral equation satisfied by the ruin probability is obtained and the bounds for the convergence rate of the ruin probability are given by using a generalized renewal technique.
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