Authors:Owe Axelsson Pages: 537 - 559 Abstract: Abstract The paper focuses on a low-rank tensor structured representation of Slatertype and Hydrogen-like orbital basis functions that can be used in electronic structure calculations. Standard packages use the Gaussian-type basis functions which allow us to analytically evaluate the necessary integrals. Slater-type and Hydrogen-like orbital functions are physically more appropriate, but they are not analytically integrable. A numerical integration is too expensive when using the standard discretization techniques due the dimensionality of the problem. However, it can be effectively performed using the tensor representation of basis functions. Furthermore, this approach can take advantage of parallel computing. PubDate: 2017-12-05 DOI: 10.21136/am.2017.0222-17 Issue No:Vol. 62, No. 6 (2017)

Authors:Radim Blaheta; Tomáš Luber Pages: 561 - 577 Abstract: Abstract This paper deals with a nonlinear beam model which was published by D.Y.Gao in 1996. It is considered either pure bending or a unilateral contact with elastic foundation, where the normal compliance condition is employed. Under additional assumptions on data, higher regularity of solution is proved. It enables us to transform the problem into a control variational problem. For basic types of boundary conditions, suitable transformations of the problem are derived. The control variational problem contains a simple linear state problem and it is solved by the conditioned gradient method. Illustrative numerical examples are introduced in order to compare the Gao beam with the classical Euler-Bernoulli beam. PubDate: 2017-11-30 DOI: 10.21136/am.2017.0179-17 Issue No:Vol. 62, No. 6 (2017)

Authors:Vít Dolejší; Filip Roskovec Pages: 579 - 605 Abstract: Abstract Two-by-two block matrices of special form with square matrix blocks arise in important applications, such as in optimal control of partial differential equations and in high order time integration methods. Two solution methods involving very efficient preconditioned matrices, one based on a Schur complement reduction of the given system and one based on a transformation matrix with a perturbation of one of the given matrix blocks are presented. The first method involves an additional inner solution with the pivot matrix block but gives a very tight condition number bound when applied for a time integration method. The second method does not involve this matrix block but only inner solutions with a linear combination of the pivot block and the off-diagonal matrix blocks. Both the methods give small condition number bounds that hold uniformly in all parameters involved in the problem, i.e. are fully robust. The paper presents shorter proofs, extended and new results compared to earlier publications. PubDate: 2017-12-04 DOI: 10.21136/am.2017.0173-17 Issue No:Vol. 62, No. 6 (2017)

Authors:Jiří Hozman; Tomáš Tichý Pages: 607 - 632 Abstract: Abstract Poroelastic systems describe fluid flow through porous medium coupled with deformation of the porous matrix. In this paper, the deformation is described by linear elasticity, the fluid flow is modelled as Darcy flow. The main focus is on the Biot-Barenblatt model with double porosity/double permeability flow, which distinguishes flow in two regions considered as continua. The main goal is in proposing block diagonal preconditionings to systems arising from the discretization of the Biot-Barenblatt model by a mixed finite element method in space and implicit Euler method in time and estimating the condition number for such preconditioning. The investigation of preconditioning includes its dependence on material coefficients and parameters of discretization. PubDate: 2017-12-07 DOI: 10.21136/am.2017.0176-17 Issue No:Vol. 62, No. 6 (2017)

Authors:Petr Kurfürst; Jiří Krtička Pages: 633 - 659 Abstract: Abstract Bounds on the spectrum of the Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients in the analysis of many domain decomposition methods. Here we are interested in the analysis of floating clusters, i.e. subdomains without prescribed Dirichlet conditions that are decomposed into still smaller subdomains glued on primal level in some nodes and/or by some averages. We give the estimates of the regular condition number of the Schur complements of the clusters arising in the discretization of problems governed by 2D Laplacian. The estimates depend on the decomposition and discretization parameters and gluing conditions. We also show how to plug the results into the analysis of H-TFETI methods and compare the estimates with numerical experiments. The results are useful for the analysis and implementation of powerful massively parallel scalable algorithms for the solution of variational inequalities. PubDate: 2017-12-13 DOI: 10.21136/am.2017.0135-17 Issue No:Vol. 62, No. 6 (2017)

Authors:Jitka Machalová; Horymír Netuka Pages: 661 - 677 Abstract: Abstract We calculate self-consistent time-dependent models of astrophysical processes. We have developed two types of our own (magneto) hydrodynamic codes, either the operator-split, finite volume Eulerian code on a staggered grid for smooth hydrodynamic flows, or the finite volume unsplit code based on the Roe’s method for explosive events with extremely large discontinuities and highly supersonic outbursts. Both the types of the codes use the second order Navier-Stokes viscosity to realistically model the viscous and dissipative effects. They are transformed to all basic orthogonal curvilinear coordinate systems as well as to a special non-orthogonal geometric system that fits to modeling of astrophysical disks. We describe mathematical background of our codes and their implementation for astrophysical simulations, including choice of initial and boundary conditions. We demonstrate some calculated models and compare the practical usage of numerically different types of codes. PubDate: 2017-11-23 DOI: 10.21136/am.2017.0168-17 Issue No:Vol. 62, No. 6 (2017)

Authors:Martin Mrovec Pages: 679 - 698 Abstract: Abstract We deal with a posteriori error control of discontinuous Galerkin approximations for linear boundary value problems. The computational error is estimated in the framework of the Dual Weighted Residual method (DWR) for goal-oriented error estimation which requires to solve an additional (adjoint) problem. We focus on the control of the algebraic errors arising from iterative solutions of algebraic systems corresponding to both the primal and adjoint problems. Moreover, we present two different reconstruction techniques allowing an efficient evaluation of the error estimators. Finally, we propose a complex algorithm which controls discretization and algebraic errors and drives the adaptation of the mesh in the close to optimal manner with respect to the given quantity of interest. PubDate: 2017-12-12 DOI: 10.21136/am.2017.0177-17 Issue No:Vol. 62, No. 6 (2017)

Authors:Petr Vodstrčil; Jiří Bouchala; Marta Jarošová; Zdeněk Dostál Pages: 699 - 718 Abstract: Abstract The evaluation of option premium is a very delicate issue arising from the assumptions made under a financial market model, and pricing of a wide range of options is generally feasible only when numerical methods are involved. This paper is based on our recent research on numerical pricing of path-dependent multi-asset options and extends these results also to the case of Asian options with fixed strike. First, we recall the three-dimensional backward parabolic PDE describing the evolution of European-style Asian option contracts on two assets, whose payoff depends on the difference of the strike price and the average value of the basket of two underlying assets during the life of the option. Further, a suitable transformation of variables respecting this complex form of a payoff function reduces the problem to a two-dimensional equation belonging to the class of convection-diffusion problems and the discontinuous Galerkin (DG) method is applied to it in order to utilize its solving potentials. The whole procedure is accompanied with theoretical results and differences to the floating strike case are discussed. Finally, reference numerical experiments on real market data illustrate comprehensive empirical findings on Asian options. PubDate: 2017-12-11 DOI: 10.21136/am.2017.0193-17 Issue No:Vol. 62, No. 6 (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: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