Testing heteroscedasticity by wavelets in a nonparametric regression model

In the nonparametric regression models, a homoscedastic structure is usually assumed. However, the homoscedasticity cannot be guaranteed a priori. Hence, testing the heteroscedasticity is needed. In this paper we propose a consistent nonparametric test for heteroscedasticity, based on wavelets. The empirical wavelet coefficients of the conditional variance in a regression model are defined first. Then they are shown to be asymptotically normal, based on which a test statistic for the heteroscedasticity is constructed by using Fan's wavelet thresholding idea. Simulations show that our test is superior to the traditional nonparametric test.     

New applications of Picard?s successive approximations

The iterative method of successive approximations, originally introduced by Émile Picard in 1890, is a basic tool for proving the existence of solutions of initial value problems regarding ordinary first order differential equations. In the present paper, it is shown that this method can be modified to get estimates for the growth of solutions of linear differential equations of the typef(k)+Ak−1(z)f(k−1)+?+A₁(z)f^′+A₀(z)f=0 with analytic coefficients. A short comparison to the growth results in the literature, obtained by means of different methods, is also given. It turns out that many known results can be proved by applying Picard?s successive approximations in an effective way. Self-contained considerations are carried out in the complex plane and in the unit disc, and some remarks about solutions of real linear differential equations are made.     

A θ-stable approximations of abstract Cauchy problems

Summary We study the approximation of linear parabolic Cauchy problems by means of Galerkin methods in space andA -stable multistep schemes of arbitrary order in time. The error is evaluated in the norm ofL _t ² (H _x ¹ ) L _t (L _x ² ).     

Cone-constrained eigenvalue problems: theory and algorithms

Equilibria in mechanics or in transportation models are not always expressed through a system of equations, but sometimes they are characterized by means of complementarity conditions involving a convex cone. This work deals with the analysis of cone-constrained eigenvalue problems. We discuss some theoretical issues like, for instance, the estimation of the maximal number of eigenvalues in a cone-constrained problem. Special attention is paid to the Paretian case. As a short addition to the theoretical part, we introduce and study two algorithms for solving numerically such type of eigenvalue problems.     

Testing optimality for quadratic 0–1 unconstrained problems

This paper analyses a necessary and sufficient optimality condition for quadratic pseudo-Boolean unconstrained problems. It is proved that in general testing any necessary and sufficient optimality condition is a difficult task for anyNP-hard problem. An-optimality condition is derived together with an approximation scheme to test it.This work has been supported by the Progetto Finalizzato Trasporti 2, 93.01799.PF74.Professor Paolo Carraresi died unexpectedly on March 5, 1994. At the time of his death this paper had been completed. While undertaking the final revision, the other two authors were reminded just how much they were indebted to Professor Carraresi after many years of common work together.     

Time-marching algorithms for initial-boundary value problems based upon “approximate approximations”

New time marching algorithms for solving initial-boundary value problems for semi-linear parabolic and hyperbolic equations are described. With respect to the space variable the discretization is based upon a method of approximate approximation proposed by the second author. We use approximate approximations of the fourth order. In time the algorithms are finite-difference schemes of either first or second approximation order. At each time step the approximate solution is represented by an explicit analytic formula. The algorithms are stable under mild restrictions to the time step which come from the non-linear part of the equation. Some computational results and hints on crucial implementation issues are provided.Supported by the Center for Applied and Industrial Mathematics, Department of Mathematics, Linköping University, Sweden.     

Interior penalty discontinuous approximations of convection–diffusion problems with parabolic layers

Summary A nonsymmetric discontinuous Galerkin finite element method with interior penalties is considered for two–dimensional convection–diffusion problems with regular and parabolic layers. On an anisotropic Shishkin–type mesh with bilinear elements we prove error estimates (uniformly in the perturbation parameter) in an integral norm associated with this method. On different types of interelement edges we derive the values of discontinuity–penalization parameters. Numerical experiments complement the theoretical results.     

A posteriori error estimates for approximations of evolutionary convection–diffusion problems

We derive computable upper bounds for the difference between an exact solution of the evolutionary convection-diffusion problem and an approximation of this solution. The estimates are obtained by certain transformations of the integral identity that defines the generalized solution. These estimates depend on neither special properties of the exact solution nor its approximation and involve only global constants coming from embedding inequalities. The estimates are first derived for functions in the corresponding energy space, and then possible extensions to classes of piecewise continuous approximations are discussed. Bibliography: 7 titles.     

Solving chance-constrained combinatorial problems to optimality

The aim of this paper is to provide new efficient methods for solving general chance-constrained integer linear programs to optimality. Valid linear inequalities are given for these problems. They are proved to characterize properly the set of solutions. They are based on a specific scenario, whose definition impacts strongly on the quality of the linear relaxation built. A branch-and-cut algorithm is described to solve chance-constrained combinatorial problems to optimality. Numerical tests validate the theoretical analysis and illustrate the practical efficiency of the proposed approach.     

Tools of mathematical modeling of arbitrary object packing problems

The article reviews the concept of and further develops phi-functions (Φ-functions) as an efficient tool for mathematical modeling of two-dimensional geometric optimization problems, such as cutting and packing problems and covering problems. The properties of the phi-function technique and its relationship with Minkowski sums and the nofit polygon are discussed. We also describe the advantages of phi-functions over these approaches. A clear definition of the set of objects for which phi-functions may be derived is given and some exceptions are illustrated. A step by step procedure for deriving phi-functions illustrated with examples is provided including the case of continuous rotation.     

Exact penalties for variational inequalities with applications to nonlinear complementarity problems

In this paper, we present a new reformulation of the KKT system associated to a variational inequality as a semismooth equation. The reformulation is derived from the concept of differentiable exact penalties for nonlinear programming. The best theoretical results are presented for nonlinear complementarity problems, where simple, verifiable, conditions ensure that the penalty is exact. We close the paper with some preliminary computational tests on the use of a semismooth Newton method to solve the equation derived from the new reformulation. We also compare its performance with the Newton method applied to classical reformulations based on the Fischer-Burmeister function and on the minimum. The new reformulation combines the best features of the classical ones, being as easy to solve as the reformulation that uses the Fischer-Burmeister function while requiring as few Newton steps as the one that is based on the minimum.     

Inverse problems and solution methods for a class of nonlinear complementarity problems

In this paper, motivated by the KKT optimality conditions for a sort of quadratic programs, we first introduce a class of nonlinear complementarity problems (NCPs). Then we present and discuss a kind of inverse problems of the NCPs, i.e., for a given feasible decision [`(x)]\bar{x} , we aim to characterize the set of parameter values for which there exists a point [`(y)]\bar{y} such that ([`(x)],[`(y)])(\bar{x},\bar{y}) forms a solution of the NCP and require the parameter values to be adjusted as little as possible. This leads to an inverse optimization problem. In particular, under ℓ _∞, ℓ ₁ and Frobenius norms as well as affine maps, this paper presents three simple and efficient solution methods for the inverse NCPs. Finally, some preliminary numerical results show that the proposed methods are very promising.     

Comment on: a two-stage fourth-order “almost” P-stable method for oscillatory problems

In Chawla and Al-Zanaidi (J. Comput. Appl. Math. 89 (1997) 115–118) a fourth-order “almost” P-stable method for y″=f(x,y) is proposed. We claim that it is possible to retrieve this combination of multistep methods by means of the theory of parameterized Runge-Kutta-Nyström (RKN) methods and moreover to generalize the method discussed by Chawla and Al-Zanaidi (J. Comput. Appl. Math. 89 (1997) 115–118).     

Nonsingular subsampling for regression S estimators with categorical predictors

Simple random subsampling is an integral part of S estimation algorithms for linear regression. Subsamples are required to be nonsingular. Usually, discarding a singular subsample and drawing a new one leads to a sufficient number of nonsingular subsamples with a reasonable computational effort. However, this procedure can require so many subsamples that it becomes infeasible, especially if levels of categorical variables have low frequency. A subsampling algorithm called nonsingular subsampling is presented, which generates only nonsingular subsamples. When no singular subsamples occur, nonsingular subsampling is as fast as the simple algorithm, and if singular subsamples do occur, it maintains the same computational order. The algorithm works consistently, unless the full design matrix is singular. The method is based on a modified LU decomposition algorithm that combines sample generation with solving the least squares problem. The algorithm may also be useful for ordinary bootstrapping. Since the method allows for S estimation in designs with factors and interactions between factors and continuous regressors, we study properties of the resulting estimators, both in the sense of their dependence on the randomness of the sampling and of their statistical performance.     

Maximum principle in linear finite element approximations of anisotropic diffusion–convection–reaction problems

A mesh condition is developed for linear finite element approximations of anisotropic diffusion–convection–reaction problems to satisfy a discrete maximum principle. Loosely speaking, the condition requires that the mesh be simplicial and \(\mathcal {O}(\Vert \varvec{b}\Vert _\infty h + \Vert c\Vert _\infty h^2)\) -nonobtuse when the dihedral angles are measured in the metric specified by the inverse of the diffusion matrix, where \(h\) denotes the mesh size and \(\varvec{b}\) and \(c\) are the coefficients of the convection and reaction terms. In two dimensions, the condition can be replaced by a weaker mesh condition (an \(\mathcal {O}(\Vert \varvec{b}\Vert _\infty h + \Vert c\Vert _\infty h^2)\) perturbation of a generalized Delaunay condition). These results include many existing mesh conditions as special cases. Numerical results are presented to verify the theoretical findings.     

Convergence analysis of finite element approximations of the Joule heating problem in three spatial dimensions

In this paper we present a finite element discretization of the Joule-heating problem. We prove existence of solution to the discrete formulation and strong convergence of the finite element solution to the weak solution, up to a sub-sequence. We also present numerical examples in three spatial dimensions. The first example demonstrates the convergence of the method in the second example we consider an engineering application.     

Generalized viscosity approximation methods in multiobjective optimization problems

In this paper, we consider an extend-valued nonsmooth multiobjective optimization problem of finding weak Pareto optimal solutions. We propose a class of vector-valued generalized viscosity approximation method for solving the problem. Under some conditions, we prove that any sequence generated by this method converges to a weak Pareto optimal solution of the multiobjective optimization problem.     

Parametrized variational inequality approaches to generalized Nash equilibrium problems with shared constraints

We consider the generalized Nash equilibrium problem (GNEP), in which each player’s strategy set may depend on the rivals’ strategies through shared constraints. A practical approach to solving this problem that has received increasing attention lately entails solving a related variational inequality (VI). From the viewpoint of game theory, it is important to find as many GNEs as possible, if not all of them. We propose two types of parametrized VIs related to the GNEP, one price-directed and the other resource-directed. We show that these parametrized VIs inherit the monotonicity properties of the original VI and, under mild constraint qualifications, their solutions yield all GNEs. We propose strategies to sample in the parameter spaces and show, through numerical experiments on benchmark examples, that the GNEs found by the parametrized VI approaches are widely distributed over the GNE set.