Authors:Antti Hannukainen; Sergey Korotov; Michal Křížek Pages: 1 - 13 Abstract: Abstract The maximum angle condition of J. L. Synge was originally introduced in interpolation theory and further used in finite element analysis and applications for triangular and later also for tetrahedral finite element meshes. In this paper we present some of its generalizations to higher-dimensional simplicial elements. In particular, we prove optimal interpolation properties of linear simplicial elements in ℝ d that degenerate in some way. PubDate: 2017-02-01 DOI: 10.21136/am.2017.0132-16 Issue No:Vol. 62, No. 1 (2017)

Authors:Béla J.~Szekeres; Ferenc Izsák Pages: 15 - 36 Abstract: Abstract Numerical solution of fractional order diffusion problems with homogeneous Dirichlet boundary conditions is investigated on a square domain. An appropriate extension is applied to have a well-posed problem on R2 and the solution on the square is regarded as a localization. For the numerical approximation a finite difference method is applied combined with the matrix transformation method. Here the discrete fractional Laplacian is approximated with a matrix power instead of computing the complicated approximations of fractional order derivatives. The spatial convergence of this method is proved and demonstrated by some numerical experiments. PubDate: 2017-02-01 DOI: 10.21136/am.2017.0385-15 Issue No:Vol. 62, No. 1 (2017)

Authors:María Isabel García-Planas Pages: 37 - 47 Abstract: Abstract In recent years there has been growing interest in the descriptive analysis of complex systems, permeating many aspects of daily life, obtaining considerable advances in the description of their structural and dynamical properties. However, much less effort has been devoted to studying the controllability of the dynamics taking place on them. Concretely, for complex systems it is of interest to study the exact controllability; this measure is defined as the minimum set of controls that are needed in order to steer the whole system toward any desired state. In this paper, we focus the study on the obtention of the set of all B making the system (A,B) exact controllable. PubDate: 2017-02-01 DOI: 10.21136/am.2017.0427-15 Issue No:Vol. 62, No. 1 (2017)

Authors:Petr Vaněk; Ivana Pultarová Pages: 49 - 73 Abstract: Abstract We extend the analysis of the recently proposed nonlinear EIS scheme applied to the partial eigenvalue problem. We address the case where the Rayleigh quotient iteration is used as the smoother on the fine-level. Unlike in our previous theoretical results, where the smoother given by the linear inverse power method is assumed, we prove nonlinear speed-up when the approximation becomes close to the exact solution. The speed-up is cubic. Unlike existent convergence estimates for the Rayleigh quotient iteration, our estimates take advantage of the powerful effect of the coarse-space. PubDate: 2017-02-01 DOI: 10.21136/am.2017.0101-16 Issue No:Vol. 62, No. 1 (2017)

Authors:Yun-Bo Yang; Qiong-Xiang Kong Pages: 75 - 100 Abstract: Abstract A new error correction method for the stationary Navier-Stokes equations based on two local Gauss integrations is presented. Applying the orthogonal projection technique, we introduce two local Gauss integrations as a stabilizing term in the error correction method, and derive a new error correction method. In both the coarse solution computation step and the error computation step, a locally stabilizing term based on two local Gauss integrations is introduced. The stability and convergence of the new error correction algorithm are established. Numerical examples are also presented to verify the theoretical analysis and demonstrate the efficiency of the proposed method. PubDate: 2017-02-01 DOI: 10.21136/am.2017.0119-16 Issue No:Vol. 62, No. 1 (2017)

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:Miloslav Vlasák Pages: 1 - 35 Abstract: Abstract The aim of this work is to give an introductory survey on time discretizations for liner parabolic problems. The theory of stability for stiff ordinary differential equations is explained on this problem and applied to Runge-Kutta and multi-step discretizations. Moreover, a natural connection between Galerkin time discretizations and Runge-Kutta methods together with order reduction phenomenon is discussed. PubDate: 2017-03-06 DOI: 10.21136/am.2017.0268-16

Authors:Jiří Hozman; Tomáš Tichý Pages: 1 - 25 Abstract: Abstract Option pricing models are an important part of financial markets worldwide. The PDE formulation of these models leads to analytical solutions only under very strong simplifications. For more general models the option price needs to be evaluated by numerical techniques. First, based on an ideal pure diffusion process for two risky asset prices with an additional path-dependent variable for continuous arithmetic average, we present a general form of PDE for pricing of Asian option contracts on two assets. Further, we focus only on one subclass—Asian options with floating strike—and introduce the concept of the dimensionality reduction with respect to the payoff leading to PDE with two spatial variables. Then the numerical option pricing scheme arising from the discontinuous Galerkin method is developed and some theoretical results are also mentioned. Finally, the aforementioned model is supplemented with numerical results on real market data. PubDate: 2017-03-04 DOI: 10.21136/am.2017.0273-16

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

Authors:Michele Colturato Pages: 623 - 650 Abstract: Abstract We consider a phase-field system of Caginalp type perturbed by the presence of an additional maximal monotone nonlinearity. Such a system arises from a recent study of a sliding mode control problem. We prove the existence of strong solutions. Moreover, under further assumptions, we show the continuous dependence on the initial data and the uniqueness of the solution. PubDate: 2016-12-01 DOI: 10.1007/s10492-016-0150-x Issue No:Vol. 61, No. 6 (2016)

Authors:Lam Quoc Anh; Dinh Vinh Hien Pages: 651 - 668 Abstract: Abstract In this paper we consider weak and strong quasiequilibrium problems with moving cones in Hausdorff topological vector spaces. Sufficient conditions for well-posedness of these problems are established under relaxed continuity assumptions. All kinds of wellposedness are studied: (generalized) Hadamard well-posedness, (unique) well-posedness under perturbations. Many examples are provided to illustrate the essentialness of the imposed assumptions. As applications of the main results, sufficient conditions for lower and upper bounded equilibrium problems and elastic traffic network problems to be well-posed are derived. PubDate: 2016-12-01 DOI: 10.1007/s10492-016-0151-9 Issue No:Vol. 61, No. 6 (2016)

Authors:Lam Quoc Anh; Dinh Vinh Hien Pages: 651 - 668 Abstract: Abstract In this paper we consider weak and strong quasiequilibrium problems with moving cones in Hausdorff topological vector spaces. Sufficient conditions for well-posedness of these problems are established under relaxed continuity assumptions. All kinds of wellposedness are studied: (generalized) Hadamard well-posedness, (unique) well-posedness under perturbations. Many examples are provided to illustrate the essentialness of the imposed assumptions. As applications of the main results, sufficient conditions for lower and upper bounded equilibrium problems and elastic traffic network problems to be well-posed are derived. PubDate: 2016-12-01 DOI: 10.1007/s10492-016-0151-9 Issue No:Vol. 61, No. 6 (2016)

Authors:Nguyen Vam Quang; Pham Tri Nguyen Pages: 669 - 684 Abstract: Abstract The aim of the paper is to establish strong laws of large numbers for sequences of blockwise and pairwise m-dependent random variables in a convex combination space with or without compactly uniformly integrable condition. Some of our results are even new in the case of real random variables. PubDate: 2016-12-01 DOI: 10.1007/s10492-016-0152-8 Issue No:Vol. 61, No. 6 (2016)

Authors:Ahmad Makki Pages: 685 - 725 Abstract: Abstract We consider the viscous Allen-Cahn and Cahn-Hilliard models with an additional term called the nonlinear Willmore regularization. First, we are interested in the well-posedness of these two models. Furthermore, we prove that both models possess a global attractor. In addition, as far as the viscous Allen-Cahn equation is concerned, we construct a robust family of exponential attractors, i.e. attractors which are continuous with respect to the perturbation parameter. Finally, we give some numerical simulations which show the effects of the viscosity term on the anisotropic and isotropic Cahn-Hilliard equation. PubDate: 2016-12-01 DOI: 10.1007/s10492-016-0153-7 Issue No:Vol. 61, No. 6 (2016)

Authors:Ahmad Makki Pages: 685 - 725 Abstract: Abstract We consider the viscous Allen-Cahn and Cahn-Hilliard models with an additional term called the nonlinear Willmore regularization. First, we are interested in the well-posedness of these two models. Furthermore, we prove that both models possess a global attractor. In addition, as far as the viscous Allen-Cahn equation is concerned, we construct a robust family of exponential attractors, i.e. attractors which are continuous with respect to the perturbation parameter. Finally, we give some numerical simulations which show the effects of the viscosity term on the anisotropic and isotropic Cahn-Hilliard equation. PubDate: 2016-12-01 DOI: 10.1007/s10492-016-0153-7 Issue No:Vol. 61, No. 6 (2016)

Authors:Tien Dung Nguyen Pages: 515 - 526 Abstract: Abstract Stochastic Riccati equation is a backward stochastic differential equation with singular generator which arises naturally in the study of stochastic linear-quadratic optimal control problems. In this paper, we obtain Gaussian density estimates for the solutions to this equation. PubDate: 2016-10-01 DOI: 10.1007/s10492-016-0145-7 Issue No:Vol. 61, No. 5 (2016)

Authors:Ebrahim Salehi Pages: 565 - 584 Abstract: Abstract In this paper, we consider the linear and circular consecutive k-out-of-n systems consisting of arbitrarily dependent components. Under the condition that at least n−r+1 components (r ≤ n) of the system are working at time t, we study the reliability properties of the residual lifetime of such systems. Also, we present some stochastic ordering properties of residual lifetime of consecutive k-out-of-n systems. In the following, we investigate the inactivity time of the component with lifetime T r:n at the system level for the consecutive k-out-of-n systems under the condition that the system is not working at time t > 0, and obtain some stochastic properties of this conditional random variable. PubDate: 2016-10-01 DOI: 10.1007/s10492-016-0147-5 Issue No:Vol. 61, No. 5 (2016)

Authors:Suchandan Kayal; Rajesh Moharana Pages: 585 - 605 Abstract: Abstract Doubly truncated data appear in some applications with survival and astrological data. Analogous to the doubly truncated discrimination measure defined by Misagh and Yari (2012), a generalized discrimination measure between two doubly truncated non-negative random variables is proposed. Several bounds are obtained. It is remarked that the proposed measure can never be equal to a nonzero constant which is independent of the left and right truncated points. The effect of monotone transformations on the proposed measure is discussed. Finally, a simulation study is added to provide the estimates of the proposed discrimination measure. PubDate: 2016-10-01 DOI: 10.1007/s10492-016-0148-4 Issue No:Vol. 61, No. 5 (2016)

Abstract: Abstract In recent years there has been growing interest in the descriptive analysis of complex systems, permeating many aspects of daily life, obtaining considerable advances in the description of their structural and dynamical properties. However, much less effort has been devoted to studying the controllability of the dynamics taking place on them. Concretely, for complex systems it is of interest to study the exact controllability; this measure is defined as the minimum set of controls that are needed in order to steer the whole system toward any desired state. In this paper, we focus the study on the obtention of the set of all B making the system (A, B) exact controllable. PubDate: 2016-11-01 DOI: 10.21136/AM.2016.0427-15