Showing papers in "Applications of Mathematics in 2018"

The Virtual Element Method for Eigenvalue Problems with Potential Terms on Polytopic Meshes

Abstract: We extend the conforming virtual element method (VEM) to the numerical resolution of eigenvalue problems with potential terms on a polytopic mesh. An important application is that of the Schrodinger equation with a pseudopotential term. This model is a fundamental element in the numerical resolution of more complex problems from the Density Functional Theory. The VEM is based on the construction of the discrete bilinear forms of the variational formulation through certain polynomial projection operators that are directly computable from the degrees of freedom. The method shows a great flexibility with respect to the meshes and provides a correct spectral approximation with optimal convergence rates. This point is discussed from both the theoretical and the numerical viewpoint. The performance of the method is numerically investigated by solving the quantum harmonic oscillator problem with the harmonic potential and a singular eigenvalue problem with zero potential for the first eigenvalues.

Boubaker Hybrid Functions and their Application to Solve Fractional Optimal Control and Fractional Variational Problems

Abstract: A new hybrid of block-pulse functions and Boubaker polynomials is constructed to solve the inequality constrained fractional optimal control problems (FOCPs) with quadratic performance index and fractional variational problems (FVPs). First, the general formulation of the Riemann-Liouville integral operator for Boubaker hybrid function is presented for the first time. Then it is applied to reduce the problems to optimization problems, which can be solved by the existing method. In this way we find the extremum value of FOCPs without adding slack variables to inequality trajectories. Also we show that if the number of bases is increased, the used approximations in this method are convergent. The applicability and validity of the method are shown by numerical results of some examples, moreover, a comparison with the existing results shows the preference of this method.

Inverse Scattering Via Nonlinear Integral Equations Method for a Sound-Soft Crack with Phaseless Data

Abstract: We consider the inverse scattering of time-harmonic plane waves to reconstruct the shape of a sound-soft crack from a knowledge of the given incident field and the phaseless data, and we check the invariance of far field data with respect to translation of the crack. We present a numerical method that is based on a system of nonlinear and ill-posed integral equations, and our scheme is easy and simple to implement. The numerical implementation is described and numerical examples are presented to illustrate the feasibility of the proposed method.

A Zero-Inflated Geometric INAR(1) Process with Random Coefficient

Abstract: Many real-life count data are frequently characterized by overdispersion, excess zeros and autocorrelation. Zero-inflated count time series models can provide a powerful procedure to model this type of data. In this paper, we introduce a new stationary first-order integer-valued autoregressive process with random coefficient and zero-inflated geometric marginal distribution, named ZIGINARRC(1) process, which contains some sub-models as special cases. Several properties of the process are established. Estimators of the model parameters are obtained and their performance is checked by a small Monte Carlo simulation. Also, the behavior of the inflation parameter of the model is justified. We investigate an application of the process using a real count climate data set with excessive zeros for the number of tornados deaths and illustrate the best performance of the proposed process as compared with a set of competitive INAR(1) models via some goodness-of-fit statistics. Consequently, forecasting for the data is discussed with estimation of the transition probability and expected run length at state zero. Moreover, for the considered data, a test of the random coefficient for the proposed process is investigated.

Explicit Finite Element Error Estimates for Nonhomogeneous Neumann Problems

Abstract: The paper develops an explicit a priori error estimate for finite element solution to nonhomogeneous Neumann problems. For this purpose, the hypercircle over finite element spaces is constructed and the explicit upper bound of the constant in the trace theorem is given. Numerical examples are shown in the final section, which implies the proposed error estimate has the convergence rate as 0.5.

Some Stochastic Comparison Results for Series and Parallel Systems with Heterogeneous Pareto Type Components

Abstract: We focus on stochastic comparisons of lifetimes of series and parallel systems consisting of independent and heterogeneous new Pareto type components. Sufficient conditions involving majorization type partial orders are provided to obtain stochastic comparisons in terms of various magnitude and dispersive orderings which include usual stochastic order, hazard rate order, dispersive order and right spread order. The usual stochastic order of lifetimes of series systems with possibly different scale and shape parameters is studied when its matrix of parameters changes to another matrix in certain sense.

Computing Discrete Convolutions with Verified Accuracy Via Banach Algebras and the FFT

Abstract: We introduce a method to compute rigorous component-wise enclosures of discrete convolutions using the fast Fourier transform, the properties of Banach algebras, and interval arithmetic. The purpose of this new approach is to improve the implementation and the applicability of computer-assisted proofs performed in weighed l1 Banach algebras of Fourier/Chebyshev sequences, whose norms are known to be numerically unstable. We introduce some application examples, in particular a rigorous aposteriori error analysis for a steady state in the quintic Swift-Hohenberg PDE.

Explicit Estimation of Error Constants Appearing in Non-Conforming Linear Triangular Finite Element Method

Abstract: The non-conforming linear (P1) triangular FEM can be viewed as a kind of the discontinuous Galerkin method, and is attractive in both the theoretical and practical purposes. Since various error constants must be quantitatively evaluated for its accurate a priori and a posteriori error estimates, we derive their theoretical upper bounds and some computational results. In particular, the Babuska-Aziz maximum angle condition is required just as in the case of the conforming P1 triangle. Some applications and numerical results are also included to see the validity and effectiveness of our analysis.

A Free Boundary Problem for a Predator-Prey Model with Nonlinear Prey-Taxis

Abstract: This paper deals with a reaction-diffusion system modeling a free boundary problem of the predator-prey type with prey-taxis over a one-dimensional habitat. The free boundary represents the spreading front of the predator species. The global existence and uniqueness of classical solutions to this system are established by the contraction mapping principle. With an eye on the biological interpretations, numerical simulations are provided which give a real insight into the behavior of the free boundary and the stability of the solutions.

On Interpolation Error on Degenerating Prismatic Elements

Abstract: We propose an analogue of the maximum angle condition (commonly used in finite element analysis for triangular and tetrahedral meshes) for the case of prismatic elements. Under this condition, prisms in the meshes may degenerate in certain ways, violating the so-called inscribed ball condition presented by P.G.Ciarlet (1978), but the interpolation error remains of the order O(h) in the H1-norm for sufficiently smooth functions.

Optimization approaches to some problems of building design

Abstract: Advanced building design is a rather new interdisciplinary research branch, combining knowledge from physics, engineering, art and social science; its support from both theoretical and computational mathematics is needed. This paper shows an example of such collaboration, introducing a model problem of optimal heating in a low-energy house. Since all particular function values, needed for optimization are obtained as numerical solutions of an initial and boundary value problem for a sparse system of parabolic partial differential equations of evolution with at least two types of physically motivated nonlinearities, the usual gradient-based methods must be replaced by the downhill simplex Nelder-Mead approach or its quasi-gradient modifications. One example of the real low-energy house in Moravian Karst is demonstrated with references to other practical applications.

A comparison of deterministic and Bayesian inverse with application in micromechanics

Abstract: The paper deals with formulation and numerical solution of problems of identification of material parameters for continuum mechanics problems in domains with heterogeneous microstructure. Due to a restricted number of measurements of quantities related to physical processes, we assume additional information about the microstructure geometry provided by CT scan or similar analysis. The inverse problems use output least squares cost functionals with values obtained from averages of state problem quantities over parts of the boundary and Tikhonov regularization. To include uncertainties in observed values, Bayesian inversion is also considered in order to obtain a statistical description of unknown material parameters from sampling provided by the Metropolis-Hastings algorithm accelerated by using the stochastic Galerkin method. The connection between Bayesian inversion and Tikhonov regularization and advantages of each approach are also discussed.

Reliable Numerical Modelling of Malaria Propagation

Abstract: We investigate biological processes, particularly the propagation of malaria. Both the continuous and the numerical models on some fixed mesh should preserve the basic qualitative properties of the original phenomenon. Our main goal is to give the conditions for the discrete (numerical) models of the malaria phenomena under which they possess some given qualitative property, namely, to be between zero and one. The conditions which guarantee this requirement are related to the time-discretization step-size. We give a sufficient condition for some explicit methods. For implicit methods we prove that the above property holds unconditionally.

Stochastic Affine Evolution Equations with Multiplicative Fractional Noise

Abstract: A stochastic affine evolution equation with bilinear noise term is studied, where the driving process is a real-valued fractional Brownian motion with Hurst parameter greater than 1/2. Stochastic integration is understood in the Skorokhod sense. The existence and uniqueness of weak solution is proved and some results on the large time dynamics are obtained.

On Generalized Conditional Cumulative Past Inaccuracy Measure

Abstract: The notion of cumulative past inaccuracy (CPI) measure has recently been proposed in the literature as a generalization of cumulative past entropy (CPE) in univariate as well as bivariate setup. In this paper, we introduce the notion of CPI of order α and study the proposed measure for conditionally specified models of two components failed at different time instants, called generalized conditional CPI (GCCPI). Several properties, including the effect of monotone transformation and bounds of GCCPI are discussed. Furthermore, we characterize some bivariate distributions under the assumption of conditional proportional reversed hazard rate model. Finally, the role of GCCPI in reliability modeling has also been investigated for a real-life problem.

Homogenization of a Linear Parabolic Problem with a Certain Type of Matching Between the Microscopic Scales

Abstract: This paper is devoted to the study of the linear parabolic problem $$\varepsilon\partial_t{u_\varepsilon}(x,t)- abla\cdot(a(x/\varepsilon, t/\varepsilon^3) abla{u_\varepsilon}(x,t))=f(x,t)$$ by means of periodic homogenization. Two interesting phenomena arise as a result of the appearance of the coefficient e in front of the time derivative. First, we have an elliptic homogenized problem although the problem studied is parabolic. Secondly, we get a parabolic local problem even though the problem has a different relation between the spatial and temporal scales than those normally giving rise to parabolic local problems. To be able to establish the homogenization result, adapting to the problem we state and prove compactness results for the evolution setting of multiscale and very weak multiscale convergence. In particular, assumptions on the sequence {ue} different from the standard setting are used, which means that these results are also of independent interest.

Model of Shell Metal Mould Heating in the Automotive Industry

Abstract: This article focuses on heat radiation intensity optimization on the surface of a shell metal mould. Such moulds are used in the automotive industry in the artificial leather production (the artificial leather is used, e.g., on car dashboards). The mould is heated by infrared heaters. After the required temperature is attained, the inner mould surface is sprinkled with special PVC powder. The powder melts and after cooling down it forms the artificial leather. A homogeneous temperature field of the mould is a necessary prerequisite for obtaining a uniform colour shade and material structure of the artificial leather. The article includes a description of a mathematical model that allows to calculate the heat radiation intensity on the outer mould surface for each fixed positioning of the infrared heaters. Next, we use this mathematical model to optimize the locations of the heaters to provide approximately the same heat radiation intensity on the whole outer mould surface during the heating process. The heat radiation intensity optimization is a complex task, because the cost function may have many local minima. Therefore, using gradient methods to solve this problem is not suitable. A differential evolution algorithm is applied during the optimization process. Asymptotic convergence of the algorithm is shown. The article contains a practical example including graphical outputs. The calculations were performed by means of Matlab code written by the authors.

Application of Calderón’s inverse problem in civil engineering

Abstract: In specific fields of research such as preservation of historical buildings, medical imaging, geophysics and others, it is of particular interest to perform only a non-intrusive boundary measurements. The idea is to obtain comprehensive information about the material properties inside the considered domain while keeping the test sample intact. This paper is focused on such problems, i.e. synthesizing a physical model of interest with a boundary inverse value technique. The forward model is represented here by time dependent heat equation with transport parameters that are subsequently identified using a modified Calderon problem which is numerically solved by a regularized Gauss-Newton method. The proposed model setup is computationally verified for various domains, loading conditions and material distributions.

A Multilevel Newton’s Method for Eigenvalue Problems

Abstract: We propose a new type of multilevel method for solving eigenvalue problems based on Newton’s method. With the proposed iteration method, solving an eigenvalue problem on the finest finite element space is replaced by solving a small scale eigenvalue problem in a coarse space and a sequence of augmented linear problems, derived by Newton step in the corresponding sequence of finite element spaces. This iteration scheme improves overall efficiency of the finite element method for solving eigenvalue problems. Finally, some numerical examples are provided to validate the efficiency of the proposed numerical scheme.

Estimation of Vibration Frequencies of Linear Elastic Membranes

Abstract: The free motion of a thin elastic linear membrane is described, in a simplyfied model, by a second order linear homogeneous hyperbolic system of partial differential equations whose spatial part is the Laplace Beltrami operator acting on a Riemannian 2- dimensional manifold with boundary. We adapt the estimates of the spectrum of the Laplacian obtained in the last years by several authors for compact closed Riemannian manifolds. To make so, we use the standard technique of the doubled manifold to transform a Riemannian manifold with nonempty boundary (M, ∂M, g) to a compact Riemannian manifold ( $$M\# M,\tilde g$$ ) without boundary. An easy numerical investigation on a concrete semi-ellipsoidic membrane with clamped boundary tests the sharpness of the method.

Entry-Exit Decisions with Implementation Delay Under Uncertainty

Abstract: We employ a natural method from the perspective of the optimal stopping theory to analyze entry-exit decisions with implementation delay of a project, and provide closed expressions for optimal entry decision times, optimal exit decision times, and the maximal expected present value of the project. The results in conventional research were obtained under the restriction that the sum of the entry cost and exit cost is nonnegative. In practice, we may meet cases when this sum is negative, so it is necessary to remove the restriction. If the sum is negative, there may exist two trigger prices of entry decision, which does not happen when the sum is nonnegative, and it is not optimal to enter and then immediately exit the project even though it is an arbitrage opportunity.

Reconstruction of Map Projection, its Inverse and Re-Projection

Abstract: This paper focuses on the automatic recognition of map projection, its inverse and re-projection. Our analysis leads to the unconstrained optimization solved by the hybrid BFGS nonlinear least squares technique. The objective function is represented by the squared sum of the residuals. For the map re-projection the partial differential equations of the inverse transformation are derived. They can be applied to any map projection. Illustrative examples of the stereographic and globular Nicolosi projections frequently used in early maps are involved and their inverse formulas are presented.

Existence of Solutions for Evolution Equations in Hilbert Spaces with Anti-Periodic Boundary Conditions and its Applications

Abstract: We establish the existence of solutions for evolution equations in Hilbert spaces with anti-periodic boundary conditions. The energies associated to these evolution equations are quadratic forms. Our approach is based on application of the Schaefer fixed-point theorem combined with the continuity method.

Polynomial chaos in evaluating failure probability: A comparative study

Abstract: Recent developments in the field of stochastic mechanics and particularly regarding the stochastic finite element method allow to model uncertain behaviours for more complex engineering structures. In reliability analysis, polynomial chaos expansion is a useful tool because it helps to avoid thousands of time-consuming finite element model simulations for structures with uncertain parameters. The aim of this paper is to review and compare available techniques for both the construction of polynomial chaos and its use in computing failure probability. In particular, we compare results for the stochastic Galerkin method, stochastic collocation, and the regression method based on Latin hypercube sampling with predictions obtained by crude Monte Carlo sampling. As an illustrative engineering example, we consider a simple frame structure with uncertain parameters in loading and geometry with prescribed distributions defined by realistic histograms.

A Penalty Approach for a Box Constrained Variational Inequality Problem

Abstract: We propose a penalty approach for a box constrained variational inequality problem (BVIP). This problem is replaced by a sequence of nonlinear equations containing a penalty term. We show that if the penalty parameter tends to infinity, the solution of this sequence converges to that of BVIP when the function F involved is continuous and strongly monotone and the box C contains the origin. We develop the algorithmic aspect with theoretical arguments properly established. The numerical results tested on some examples are satisfactory and confirm the theoretical approach.

Remarks on Balanced Norm Error Estimates for Systems of Reaction-Diffusion Equations

Abstract: Error estimates of finite element methods for reaction-diffusion problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the H1 seminorm leads to a balanced norm which reflects the layer behavior correctly. We discuss the difficulties which arise for systems of reaction-diffusion problems.

Lotka-Volterra Type Predator-Prey Models: Comparison of Hidden and Explicit Resources with a Transmissible Disease in the Predator Species

Abstract: The paper deals with two mathematical models of predator-prey type where a transmissible disease spreads among the predator species only. The proposed models are analyzed and compared in order to assess the influence of hidden and explicit alternative resource for predator. The analysis shows boundedness as well as local stability and transcritical bifurcations for equilibria of systems. Numerical simulations support our theoretical analysis.

Existence of Solutions for Nonlinear Nonmonotone Evolution Equations in Banach Spaces with Anti-Periodic Boundary Conditions

Abstract: The paper is devoted to the study of the existence of solutions for nonlinear nonmonotone evolution equations in Banach spaces involving anti-periodic boundary conditions. Our approach in this study relies on the theory of monotone and maximal monotone operators combined with the Schaefer fixed-point theorem and the monotonicity method. We apply our abstract results in order to solve a diffusion equation of Kirchhoff type involving the Dirichlet p-Laplace operator.

Improved convergence estimate for a multiply polynomially smoothed two-level method with an aggressive coarsening

Abstract: A variational two-level method in the class of methods with an aggressive coarsening and a massive polynomial smoothing is proposed. The method is a modification of the method of Section 5 of Tezaur, Vaněk (2018). Compared to that method, a significantly sharper estimate is proved while requiring only slightly more computational work.