Showing papers in "Applications of Mathematics in 2008"
TL;DR: In this article, a proper regularization is introduced which guarantees that the semi-smooth Newton method is superlinearly convergent for each regularized problem, and a shift motivated by an augmented Lagrangian framework, to the regularization term is feasible.
Abstract: Semi-smooth Newton methods are analyzed for the Signorini problem. A proper regularization is introduced which guarantees that the semi-smooth Newton method is superlinearly convergent for each regularized problem. Utilizing a shift motivated by an augmented Lagrangian framework, to the regularization term, the solution to each regularized problem is feasible. Convergence of the regularized problems is shown and a report on numerical experiments is given.
34 citations
TL;DR: In this article, the authors consider a phase-field model of grain structure evolution and show the existence of solutions in a generalized sense, and also show the uniqueness of solutions for the model in one-dimensional space.
Abstract: We consider a phase-field model of grain structure evolution, which appears in materials sciences. In this paper we study the grain boundary motion model of Kobayashi-Warren-Carter type, which contains a singular diffusivity. The main objective of this paper is to show the existence of solutions in a generalized sense. Moreover, we show the uniqueness of solutions for the model in one-dimensional space.
27 citations
TL;DR: In contrast to the theory of non-homogeneous incompressible fluids where one has only one velocity field, here we have two densities and two velocity fields assigned to each species of the fluid as mentioned in this paper.
Abstract: We consider mixtures of compressible viscous fluids consisting of two miscible species. In contrast to the theory of non-homogeneous incompressible fluids where one has only one velocity field, here we have two densities and two velocity fields assigned to each species of the fluid. We obtain global classical solutions for quasi-stationary Stokes-like system with interaction term.
24 citations
TL;DR: In this article, a one-dimensional model for phase transition phenomena in thermoviscoelastic materials is presented, where the model is coupled in such a way as to take into account the fact that the properties of viscous and elastic parts influence the phase transition phenomenon in different ways.
Abstract: This paper is devoted to the analysis of a one-dimensional model for phase transition phenomena in thermoviscoelastic materials. The corresponding parabolic-hyperbolic PDE system features a strongly nonlinear internal energy balance equation, governing the evolution of the absolute temperature ϑ, an evolution equation for the phase change parameter χ, including constraints on the phase variable, and a hyperbolic stress-strain relation for the displacement variable u. The main novelty of the model is that the equations for χ and u are coupled in such a way as to take into account the fact that the properties of the viscous and of the elastic parts influence the phase transition phenomenon in different ways. However, this brings about an elliptic degeneracy in the equation for u which needs to be carefully handled.
22 citations
TL;DR: In this paper, the large time asymptotic behavior of solutions of the doubly degenerate parabolic equation with an initial condition u(x, 0) = u0(x) was studied.
Abstract: We study the large time asymptotic behavior of solutions of the doubly degenerate parabolic equation ut = div(um−1|Du|p−2Du) − uq with an initial condition u(x, 0) = u0(x). Here the exponents m, p and q satisfy m + p ⩾ 3, p > 1 and q > m + p − 2.
22 citations
TL;DR: In this paper, a robust version of exponential and Holt-Winters smoothing is proposed, where the idea of M-estimation is used to obtain an easy-to-use recursive formula for its computation.
Abstract: To obtain a robust version of exponential and Holt-Winters smoothing the idea of M-estimation can be used. The difficulty is the formulation of an easy-to-use recursive formula for its computation. A first attempt was made by Cipra (Robust exponential smoothing, J. Forecast. 11 (1992), 57–69). The recursive formulation presented there, however, is unstable. In this paper, a new recursive computing scheme is proposed. A simulation study illustrates that the new recursions result in smaller forecast errors on average. The forecast performance is further improved upon by using auxiliary robust starting values and robust scale estimates.
12 citations
TL;DR: In this paper, the authors show stability and consistency of the linear semi-implicit complementary volume numerical scheme for solving the regularized, in the sense of Evans and Spruck, mean curvature flow equation in the level set formulation.
Abstract: We show stability and consistency of the linear semi-implicit complementary volume numerical scheme for solving the regularized, in the sense of Evans and Spruck, mean curvature flow equation in the level set formulation. The numerical method is based on the finite volume methodology using the so-called complementary volumes to a finite element triangulation. The scheme gives the solution in an efficient and unconditionally stable way.
11 citations
TL;DR: In this paper, a new class of robust regression estimators based on the two-step least weighted squares (2S-LWS) estimator was proposed, which employs data-adaptive weights determined from the empirical distribution or quantile functions of regression residuals obtained from an initial robust fit.
Abstract: The paper studies a new class of robust regression estimators based on the two-step least weighted squares (2S-LWS) estimator which employs data-adaptive weights determined from the empirical distribution or quantile functions of regression residuals obtained from an initial robust fit. Just like many existing two-step robust methods, the proposed 2S-LWS estimator preserves robust properties of the initial robust estimate. However, contrary to the existing methods, the first-order asymptotic behavior of 2S-LWS is fully independent of the initial estimate under mild conditions. We propose data-adaptive weighting schemes that perform well both in the cross-section and time-series data and prove the asymptotic normality and efficiency of the resulting procedure. A simulation study documents these theoretical properties in finite samples.
11 citations
TL;DR: In this paper, Kendall's tau-type rank statistics are employed for statistical inference, avoiding most of parametric assumptions to a greater extent, and applications in microarray data models are stressed.
Abstract: High-dimensional data models abound in genomics studies, where often inadequately small sample sizes create impasses for incorporation of standard statistical tools. Conventional assumptions of linearity of regression, homoscedasticity and (multi-) normality of errors may not be tenable in many such interdisciplinary setups. In this study, Kendall’s tau-type rank statistics are employed for statistical inference, avoiding most of parametric assumptions to a greater extent. The proposed procedures are compared with Kendall’s tau statistic based ones. Applications in microarray data models are stressed.
10 citations
TL;DR: In this paper, the asymptotic distribution of covariate values in the quantile regression basic solution under weak assumptions is considered, and a diagnostic procedure for assessing homogeneity of the conditional densities is also proposed.
Abstract: We consider the asymptotic distribution of covariate values in the quantile regression basic solution under weak assumptions. A diagnostic procedure for assessing homogeneity of the conditional densities is also proposed.
9 citations
Journal Article•
TL;DR: In this paper, the authors study the generation of analytic semigroups in the L-2 topology by second order elliptic operators in divergence form, that may degenerate at the boundary of the space domain.
Abstract: We study the generation of analytic semigroups in the L-2 topology by second order elliptic operators in divergence form, that may degenerate at the boundary of the space domain. Our results, that hold in two space dimensions, guarantee that the solutions of the corresponding evolution problems support integration by parts. So, this paper provides the basis for deriving Carleman type estimates for degenerate parabolic operators.
TL;DR: In this paper, a multiscale stochastic homogenization for convection-diffusion problems is studied, and it is shown that the sequence of solutions converges in the sense of G-convergence of parabolic operators to the solution u of the homogenized problem ∂u/∂t − div (B(t)∇u= f).
Abstract: Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form ∂uɛω / ∂t+1 / ɛ3C(T3(x/ɛ3)ω3) · ∇uɛω − div(α(T2(x/ɛ2)ω2, t) ∇uɛω) = f. It is shown, under certain structure assumptions on the random vector field C(ω3) and the random map α(ω1, ω2, t), that the sequence {uɛω} of solutions converges in the sense of G-convergence of parabolic operators to the solution u of the homogenized problem ∂u/∂t − div (B(t)∇u= f).
TL;DR: In this article, the authors performed a nonparametric inference on the Pareto tail-index of wet and dry durations at each of those spatial scales, based on the same data, and compared it with conclusions based on a classical Hill estimator.
Abstract: Durations of rain events and drought events over a given region provide important information about the water resources of the region. Of particular interest is the shape of upper tails of the probability distributions of such durations. Recent research suggests that the underlying probability distributions of such durations have heavy tails of hyperbolic type, across a wide range of spatial scales from 2 km to 120 km. These findings are based on radar measurements of spatially averaged rain rate (SARR) over a tropical oceanic region. The present work performs a nonparametric inference on the Pareto tail-index of wet and dry durations at each of those spatial scales, based on the same data, and compares it with conclusions based on the classical Hill estimator. The results are compared and discussed.
TL;DR: In this paper, the Stam type inequalities are obtained for the Fisher information in a multivariate observation depending on a univariate location parameter and for the variance of the Pitman estimator of the latter.
Abstract: Zamir showed in 1998 that the Stam classical inequality for the Fisher information (about a location parameter)
$$ 1/I(X + Y) \geqslant 1/I(X) + 1/I(Y) $$
for independent random variables X, Y is a simple corollary of basic properties of the Fisher information (monotonicity, additivity and a reparametrization formula). The idea of his proof works for a special case of a general (not necessarily location) parameter. Stam type inequalities are obtained for the Fisher information in a multivariate observation depending on a univariate location parameter and for the variance of the Pitman estimator of the latter.
TL;DR: In this article, the unsaturated flow of an incompressible liquid carrying a bacterial population through a porous medium contaminated with some pollutant is studied and an existence theorem for it is proved.
Abstract: We study the unsaturated flow of an incompressible liquid carrying a bacterial population through a porous medium contaminated with some pollutant. The biomass grows feeding on the pollutant and affecting at the same time all the physics of the flow. We formulate a mathematical model in a one-dimensional setting and we prove an existence theorem for it. The so-called fluid media scaling approach, often used in the literature, is discussed and its limitations are pointed out on the basis of a specific example.
TL;DR: In this paper, the existence of positive solutions, dead core solutions and pseudodead core solutions of the singular Dirichlet problem is discussed and the existence and existence of pseudodeads are discussed.
Abstract: The paper discusses the existence of positive solutions, dead core solutions and pseudodead core solutions of the singular Dirichlet problem (ϕ(u′))′ = λf(t, u, u′), u(0) = u(T) = A. Here λ is the positive parameter, A > 0, f is singular at the value 0 of its first phase variable and may be singular at the value A of its first and at the value 0 of its second phase variable.
TL;DR: In this article, a beam on a unilateral elastic subsoil of Winkler's type and with free ends is considered and the existence, the uniqueness of the solution and the continuous dependence on the data are proved.
Abstract: The mathematical model of a beam on a unilateral elastic subsoil of Winkler’s type and with free ends is considered. Such a problem is non-linear and semi-coercive. The additional assumptions on the beam load ensuring the problem solvability are formulated and the existence, the uniqueness of the solution and the continuous dependence on the data are proved. The cases for which the solutions need not be stable with respect to the small changes of the load are described. The problem is approximated by the finite element method and the relation between the original problem and the family of approximated problems is analyzed. The error estimates are derived in dependence on the smoothness of the solution, the load and the discretization parameter of the partition.
TL;DR: A construction of the optimum kernel and optimum boundary kernel by means of the Gegenbauer and Legendre polynomials is presented.
Abstract: Kernel smoothers belong to the most popular nonparametric functional estimates used for describing data structure. They can be applied to the fix design regression model as well as to the random design regression model. The main idea of this paper is to present a construction of the optimum kernel and optimum boundary kernel by means of the Gegenbauer and Legendre polynomials.
TL;DR: In this article, local linear fitting is used for estimating unsmooth probability densities and conditional variance functions, where the unknown curve possibly has some irregularities, such as jumps or peaks at unknown locations.
Abstract: For nonparametric estimation of a smooth regression function, local linear fitting is a widely-used method. The goal of this paper is to briefly review how to use this method when the unknown curve possibly has some irregularities, such as jumps or peaks, at unknown locations. It is then explained how the same basic method can be used when estimating unsmooth probability densities and conditional variance functions.
TL;DR: In this paper, a theoretical framework for solving a class of worst scenario problems is proposed and the existence of the worst scenario is proved through the convergence of a sequence of approximate worst scenarios.
Abstract: We propose a theoretical framework for solving a class of worst scenario problems. The existence of the worst scenario is proved through the convergence of a sequence of approximate worst scenarios. The main convergence theorem modifies and corrects the relevant results already published in literature. The theoretical framework is applied to a particular problem with an uncertain boundary value problem for a nonlinear ordinary differential equation with an uncertain coefficient.
TL;DR: In this paper, the scalar Oseen equation is solved in a Sobolev space with isotropic and anisotropic weights, and existence results and regularities in L p theory are established.
Abstract: This paper solves the scalar Oseen equation, a linearized form of the Navier-Stokes equation. Because the fundamental solution has anisotropic properties, the problem is set in a Sobolev space with isotropic and anisotropic weights. We establish some existence results and regularities in L p theory.
TL;DR: In this article, Dassios and Kariotou extended a strictly analytic model and computed the magnetic field B due to a dipole in an ellipsoid, using numerical procedures for evaluating the roots of Lame polynomials of degree 5 or higher.
Abstract: The bioelectric current dipole model is important both theoretically and computationally in the study of electrical activity in the brain and stomach due to the resemblance of the shape of these two organs to an ellipsoid. To calculate the magnetic field B due to a dipole in an ellipsoid, one must evaluate truncated series expansions involving ellipsoidal harmonics \( \mathbb{E}_n^m \), which are products of Lame functions. In this article, we extend a strictly analytic model (G. Dassios and F. Kariotou, J. Math. Phys. 44 (2003), 220–241), where B was computed from an ellipsoidal harmonic expansion of order 2. The present derivations show how the field can be evaluated to arbitrary order using numerical procedures for evaluating the roots of Lame polynomials of degree 5 or higher. This can be accomplished using an optimization technique for solving nonlinear systems of equations, which allows one to acquire an understanding of the truncation error associated with the harmonic series expansion used for the calculation.
TL;DR: In this article, the problem of Lagrange interpolation of functions of two variables by quadratic polynomials under the condition that nodes of interpolation are vertices of a triangulation is studied.
Abstract: We study the problem of Lagrange interpolation of functions of two variables by quadratic polynomials under the condition that nodes of interpolation are vertices of a triangulation For an extensive class of triangulations we prove that every inner vertex belongs to a local six-tuple of vertices which, used as nodes of interpolation, have the following property: For every smooth function there exists a unique quadratic Lagrange interpolation polynomial and the related local interpolation error is of optimal order The existence of such six-tuples of vertices is a precondition for a successful application of certain post-processing procedures to the finite-element approximations of the solutions of differential problems
TL;DR: In this article, the Navier-Stokes equation with the initial data is considered, where u and v are two weak solutions with the same initial value, and u satisfies the usual energy inequality.
Abstract: Consider the Navier-Stokes equation with the initial data a ∈ L
2
(ℝ
d
). Let u and v be two weak solutions with the same initial value a. If u satisfies the usual energy inequality and if ∇v ∈ L
2((0, T); $$
\dot X_1
$$
(ℝ
d
)
d
) where $$
\dot X_1
$$
(ℝ
d
) is the multiplier space, then we have u = v.
TL;DR: In this article, the authors considered the problem of testing a sub-hypothesis in homoscedastic linear regression models when the covariate and error processes form independent long memory moving averages.
Abstract: This paper considers the problem of testing a sub-hypothesis in homoscedastic linear regression models when the covariate and error processes form independent long memory moving averages. The asymptotic null distribution of the likelihood ratio type test based on Whittle quadratic forms is shown to be a chi-square distribution. Additionally, the estimators of the slope parameters obtained by minimizing the Whittle dispersion is seen to be n1/2-consistent for all values of the long memory parameters of the design and error processes.
TL;DR: Three models with random coefficients and Bayesian approach are used for modelling the year average temperatures measured in Prague Klementinum, using the combination of the Metropolis-Hastings algorithm and the Gibbs sampler.
Abstract: A nonstandard approach to change point estimation is presented in this paper. Three models with random coefficients and Bayesian approach are used for modelling the year average temperatures measured in Prague Klementinum. The posterior distribution of the change point and other parameters are estimated from the random samples generated by the combination of the Metropolis-Hastings algorithm and the Gibbs sampler.
TL;DR: In this paper, a two-sample Kolmogorov-smirnov type test for a nuisance linear regression is presented, which is based on regression rank scores and provides a natural extension of the classical Kolmogi-Smirnov test.
Abstract: We derive the two-sample Kolmogorov-Smirnov type test when a nuisance linear regression is present. The test is based on regression rank scores and provides a natural extension of the classical Kolmogorov-Smirnov test. Its asymptotic distributions under the hypothesis and the local alternatives coincide with those of the classical test.
TL;DR: In this article, the Prandtl system for a nonstationary layer arising in an axially symmetric incopressible flow past a solid body is analyzed, and some drawbacks in the proof of the well-known Boundary Layer Theory in Oleinik's book are corrected.
Abstract: The purpose of this paper is to correct some drawbacks in the proof of the well-known Boundary Layer Theory in Oleinik’s book. The Prandtl system for a nonstationary layer arising in an axially symmetric incopressible flow past a solid body is analyzed.
TL;DR: Extended thermodynamics as discussed by the authors is based on a set of equations of balance which are supplemented by local and instantaneous constitutive equations so that the field equations are quasi-linear differential equations of first order.
Abstract: Extended thermodynamics is based on a set of equations of balance which are supplemented by local and instantaneous constitutive equations so that the field equations are quasi-linear differential equations of first order. If the constitutive functions are subject to the requirements of the entropy principle, one may write them in symmetric hyperbolic form by a suitable choice of fields. The kinetic theory of gases, or the moment theories based on the Boltzmann equation, provide an explicit example for extended thermodynamics. The theory proves its usefulness and practicality in the successful treatment of light scattering in rarefied gases. It would seem that extended thermodynamics is worthy of the attention of mathematicians. It may offer them a non-trivial field of study concerning hyperbolic equations, if ever they get tired of the Burgers equation.
TL;DR: In this article, the authors considered periodic solutions of first-order nonlinear functional differential equations with deviating arguments and provided sufficient conditions for the existence of the periodic solutions, extending and improving some well-known results.
Abstract: This paper is concerned with periodic solutions of first-order nonlinear functional differential equations with deviating arguments. Some new sufficient conditions for the existence of periodic solutions are obtained. The paper extends and improves some well-known results.