Authors:Jon Eivind Vatne Pages: 213 - 223 Abstract: Abstract Acute triangles are defined by having all angles less than π/2, and are characterized as the triangles containing their circumcenter in the interior. For simplices of dimension n ≥ 3, acuteness is defined by demanding that all dihedral angles between (n−1)-dimensional faces are smaller than π/2. However, there are, in a practical sense, too few acute simplices in general. This is unfortunate, since the acuteness property provides good qualitative features for finite element methods. The property of acuteness is logically independent of the property of containing the circumcenter when the dimension is greater than two. In this article, we show that the latter property is also quite rare in higher dimensions. In a natural probability measure on the set of n-dimensional simplices, we show that the probability that a uniformly random n-simplex contains its circumcenter is 1/2 n . PubDate: 2017-06-01 DOI: 10.21136/am.2017.0187-16 Issue No:Vol. 62, No. 3 (2017)

Authors:Fei Xu; Hehu Xie Pages: 225 - 241 Abstract: Abstract A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as the linear solver for solving boundary value problems. The optimality of the computational work is also proved. Compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only needs the Lipschitz continuation in some sense of the nonlinear term. PubDate: 2017-06-01 DOI: 10.21136/am.2017.0344-16 Issue No:Vol. 62, No. 3 (2017)

Authors:Yuping Zeng; Feng Wang Pages: 243 - 267 Abstract: Abstract We derive a residual-based a posteriori error estimator for a discontinuous Galerkin approximation of the Steklov eigenvalue problem. Moreover, we prove the reliability and efficiency of the error estimator. Numerical results are provided to verify our theoretical findings. PubDate: 2017-06-01 DOI: 10.21136/am.2017.0115-16 Issue No:Vol. 62, No. 3 (2017)

Authors:Hassan S. Bakouch; Ahmed M. T. Abd El-Bar Pages: 269 - 296 Abstract: Abstract A new weighted version of the Gompertz distribution is introduced. It is noted that the model represents a mixture of classical Gompertz and second upper record value of Gompertz densities, and using a certain transformation it gives a new version of the two-parameter Lindley distribution. The model can be also regarded as a dual member of the log-Lindley-X family. Various properties of the model are obtained, including hazard rate function, moments, moment generating function, quantile function, skewness, kurtosis, conditional moments, mean deviations, some types of entropy, mean residual lifetime and stochastic orderings. Estimation of the model parameters is justified by the method of maximum likelihood. Two real data sets are used to assess the performance of the model among some classical and recent distributions based on some evaluation goodness-of-fit statistics. As a result, the variance-covariance matrix and the confidence interval of the parameters, and some theoretical measures have been calculated for such data for the proposed model with discussions. PubDate: 2017-06-01 DOI: 10.21136/am.2017.0277-16 Issue No:Vol. 62, No. 3 (2017)

Authors:Yoshiki Sugitani Pages: 1 - 18 Abstract: Abstract Numerical analysis of a model Stokes interface problem with the homogeneous Dirichlet boundary condition is considered. The interface condition is interpreted as an additional singular force field to the Stokes equations using the characteristic function. The finite element method is applied after introducing a regularization of the singular source term. Consequently, the error is divided into the regularization and discretization parts which are studied separately. As a result, error estimates of order h 1/2 in H 1 × L 2 norm for the velocity and pressure, and of order h in L 2 norm for the velocity are derived. Those theoretical results are also verified by numerical examples. PubDate: 2017-10-09 DOI: 10.21136/am.2017.0357-16

Authors:Peter Oswald Pages: 1 - 25 Abstract: Abstract Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete H 1 norm best approximation error estimates for H 2 functions hold for arbitrary triangulations. However, the constants in similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on an example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily low speed. The results complement analogous findings for conforming P1 finite elements. PubDate: 2017-10-05 DOI: 10.21136/am.2017.0150-17

Authors:Yaozong Tang; Xiaolin Li Pages: 1 - 16 Abstract: Abstract The meshless element-free Galerkin method is developed for numerical analysis of hyperbolic initial-boundary value problems. In this method, only scattered nodes are required in the domain. Computational formulae of the method are analyzed in detail. Error estimates and convergence are also derived theoretically and verified numerically. Numerical examples validate the performance and efficiency of the method. PubDate: 2017-10-03 DOI: 10.21136/am.2017.0061-17

Authors:Jun Guo; Guozheng Yan; Jing Jin; Junhao Hu Pages: 1 - 26 Abstract: Abstract We consider the inverse scattering problem of determining the shape and location of a crack surrounded by a known inhomogeneous media. Both the Dirichlet boundary condition and a mixed type boundary conditions are considered. In order to avoid using the background Green function in the inversion process, a reciprocity relationship between the Green function and the solution of an auxiliary scattering problem is proved. Then we focus on extending the factorization method to our inverse shape reconstruction problems by using far field measurements at fixed wave number. We remark that this is done in a non intuitive space for the mixed type boundary condition as we indicate in the sequel. PubDate: 2017-10-01 DOI: 10.21136/am.2017.0194-16

Authors:Faranak Goodarzi; Mohammad Amini; Gholam Reza Mohtashami Borzadaran Pages: 1 - 15 Abstract: Abstract Nanda (2010) and Bhattacharjee et al. (2013) characterized a few distributions with help of the failure rate, mean residual, log-odds rate and aging intensity functions. In this paper, we generalize their results and characterize some distributions through functions used by them and Glaser’s function. Kundu and Ghosh (2016) obtained similar results using reversed hazard rate, expected inactivity time and reversed aging intensity functions. We also, via w(·)-function defined by Cacoullos and Papathanasiou (1989), characterize exponential and logistic distributions, as well as Type 3 extreme value distribution and obtain bounds for the expected values of selected functions in reliability theory. Moreover, a bound for the varentropy of random variable X is provided. PubDate: 2017-08-16 DOI: 10.21136/am.2017.0182-16

Authors:Koya Sakakibara Pages: 1 - 21 Abstract: Abstract The aim of this paper is to analyze mathematically the method of fundamental solutions applied to the biharmonic problem. The key idea is to use Almansi-type decomposition of biharmonic functions, which enables us to represent the biharmonic function in terms of two harmonic functions. Based on this decomposition, we prove that an approximate solution exists uniquely and that the approximation error decays exponentially with respect to the number of the singular points. We finally present results of numerical experiments, which verify the sharpness of our error estimate. PubDate: 2017-07-20 DOI: 10.21136/am.2017.0018-17

Authors:Masahito Ohta; Takiko Sasaki Pages: 1 - 28 Abstract: We consider a Strang-type splitting method for an abstract semilinear evolution equation $${\partial _t}u = Au + F\left( u \right).$$ Roughly speaking, the splitting method is a time-discretization approximation based on the decomposition of the operators A and F. Particularly, the Strang method is a popular splitting method and is known to be convergent at a second order rate for some particular ODEs and PDEs. Moreover, such estimates usually address the case of splitting the operator into two parts. In this paper, we consider the splitting method which is split into three parts and prove that our proposed method is convergent at a second order rate. PubDate: 2017-07-12 DOI: 10.21136/am.2017.0020-17

Authors:Yasunori Futamura; Takahiro Yano; Akira Imakura; Tetsuya Sakurai Pages: 1 - 23 Abstract: Abstract We consider solving complex symmetric linear systems with multiple right-hand sides. We assume that the coefficient matrix has indefinite real part and positive definite imaginary part. We propose a new block conjugate gradient type method based on the Schur complement of a certain 2-by-2 real block form. The algorithm of the proposed method consists of building blocks that involve only real arithmetic with real symmetric matrices of the original size. We also present the convergence property of the proposed method and an efficient algorithmic implementation. In numerical experiments, we compare our method to a complex-valued direct solver, and a preconditioned and nonpreconditioned block Krylov method that uses complex arithmetic. PubDate: 2017-07-07 DOI: 10.21136/am.2017.0023-17

Authors:Hongjia Chen; Akira Imakura; Tetsuya Sakurai Pages: 1 - 19 Abstract: Abstract One of the most efficient methods for solving the polynomial eigenvalue problem (PEP) is the Sakurai-Sugiura method with Rayleigh-Ritz projection (SS-RR), which finds the eigenvalues contained in a certain domain using the contour integral. The SS-RR method converts the original PEP to a small projected PEP using the Rayleigh-Ritz projection. However, the SS-RR method suffers from backward instability when the norms of the coefficient matrices of the projected PEP vary widely. To improve the backward stability of the SS-RR method, we combine it with a balancing technique for solving a small projected PEP. We then analyze the backward stability of the SS-RR method. Several numerical examples demonstrate that the SS-RR method with the balancing technique reduces the backward error of eigenpairs of PEP. PubDate: 2017-07-05 DOI: 10.21136/am.2017.0016-17

Authors:Guanyu Zhou; Takahito Kashiwabara; Issei Oikawa Pages: 1 - 27 Abstract: Abstract We consider the finite element method for the time-dependent Stokes problem with the slip boundary condition in a smooth domain. To avoid a variational crime of numerical computation, a penalty method is introduced, which also facilitates the numerical implementation. For the continuous problem, the convergence of the penalty method is investigated. Then we study the fully discretized finite element approximations for the penalty method with the P1/P1-stabilization or P1b/P1 element. For the discretization of the penalty term, we propose reduced and non-reduced integration schemes, and obtain an error estimate for velocity and pressure. The theoretical results are verified by numerical experiments. PubDate: 2017-07-05 DOI: 10.21136/am.2017.0328-16

Authors:Yusaku Yamamoto Pages: 1 - 13 Abstract: Abstract Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix A ∈ R m×m play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on Tr(A −1) and Tr(A −2) have attracted attention recently, because they can be computed in O(m) operations when A is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that can be computed solely from Tr(A −1) and Tr(A −2) and show that the so called Laguerre’s lower bound is the optimal one in terms of sharpness. Next, we study the gap between the Laguerre bound and the smallest eigenvalue. We characterize the situation in which the gap becomes largest in terms of the eigenvalue distribution of A and show that the gap becomes smallest when {Tr(A −1)}2/Tr(A −2) approaches 1 or m. These results will be useful, for example, in designing efficient shift strategies for singular value computation algorithms. PubDate: 2017-06-29 DOI: 10.21136/am.2017.0022-17

Authors:Roman Knobloch; Jaroslav Mlýnek; Radek Srb Pages: 1 - 12 Abstract: Abstract Differential evolution algorithms represent an up to date and efficient way of solving complicated optimization tasks. In this article we concentrate on the ability of the differential evolution algorithms to attain the global minimum of the cost function. We demonstrate that although often declared as a global optimizer the classic differential evolution algorithm does not in general guarantee the convergence to the global minimum. To improve this weakness we design a simple modification of the classic differential evolution algorithm. This modification limits the possible premature convergence to local minima and ensures the asymptotic global convergence. We also introduce concepts that are necessary for the subsequent proof of the asymptotic global convergence of the modified algorithm. We test the classic and modified algorithm by numerical experiments and compare the efficiency of finding the global minimum for both algorithms. The tests confirm that the modified algorithm is significantly more efficient with respect to the global convergence than the classic algorithm. PubDate: 2017-03-06 DOI: 10.21136/am.2017.0274-16

Authors:Ladislav Lukšan; Jan Vlček Pages: 1 - 14 Abstract: Abstract We propose a new Broyden method for solving systems of nonlinear equations, which uses the first derivatives, but is more efficient than the Newton method (measured by the computational time) for larger dense systems. The new method updates QR or LU decompositions of nonsymmetric approximations of the Jacobian matrix, so it requires O(n 2) arithmetic operations per iteration in contrast with the Newton method, which requires O(n 3) operations per iteration. Computational experiments confirm the high efficiency of the new method. PubDate: 2017-03-03 DOI: 10.21136/am.2017.0253-16

Authors:Yongfeng Wu; Andrew Rosalsky; Andrei Volodin Pages: 1 - 3 Abstract: Abstract The authors provide a correction to “Some mean convergence and complete convergence theorems for sequences of m-linearly negative quadrant dependent random variables”. PubDate: 2017-02-28 DOI: 10.21136/am.2017.0121-16

Authors:Iveta Hnětynková; Martin Plešinger; Jana Žáková Pages: 1 - 16 Abstract: Abstract The total least squares (TLS) and truncated TLS (T-TLS) methods are widely known linear data fitting approaches, often used also in the context of very ill-conditioned, rank-deficient, or ill-posed problems. Regularization properties of T-TLS applied to linear approximation problems Ax ≈ b were analyzed by Fierro, Golub, Hansen, and O’Leary (1997) through the so-called filter factors allowing to represent the solution in terms of a filtered pseudoinverse of A applied to b. This paper focuses on the situation when multiple observations b 1,..., b d are available, i.e., the T-TLS method is applied to the problem AX ≈ B, where B = [b 1,..., b d ] is a matrix. It is proved that the filtering representation of the T-TLS solution can be generalized to this case. The corresponding filter factors are explicitly derived. PubDate: 2017-02-28 DOI: 10.21136/am.2017.0228-16