Approximation of solutions to operator equations in spaces of smooth functions and in spaces of distributions

The Galerkin method, in particular, the Galerkin method with finite elements (called finite element method) is widely used for numerical solution of differential equations. The Galerkin method allows us to obtain approximations of weak solutions only. However, there arises in applications a rich variety of problems where approximations of smooth solutions and solutions in the sense of distributions have to be found. This article is devoted to the employment of the Petrov–Galerkin method for solving such problems. The article contains general results on the Petrov–Galerkin approximations of solutions to linear and nonlinear operator equations. The problem on construction of the subspaces, which ensure the convergence of the approximations, is investigated. We apply the general results to two‐dimensional (2D) and 3D problems of the elasticity, to a parabolic problem, and to a nonlinear problem of the plasticity. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 406–450, 2014     

Scaled discrete derivatives of singularly perturbed elliptic problems

Numerical approximations to the solution of a singularly perturbed elliptic convection–diffusion problem in two space dimensions are generated using a monotone finite difference operator on a tensor product of piecewise‐uniform Shishkin meshes. The bilinear interpolants of these numerical approximations are parameter‐uniformly convergent to the solution of the continuous problem, in the pointwise maximum norm. In this article, discrete approximations to the first derivatives of the solution are shown to be globally first‐order (up to logarithmic factors) uniformly convergent, when the errors are scaled within the analytical layers of the continuous problem. Numerical results are presented to illustrate the theoretical error bounds established in an appropriated weighted C¹–norm. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 225–252, 2015     

A unifying formulation of the discrete and continuum approximations for embedded discontinuities

A methodology for the numerical implementation of embedded discontinuities into the finite element method is developed. This is applicable for the discrete and continuum approximations of discontinuities. The variational formulation of the problem of a solid with discontinuities is established for both approximations, yielding the equations used in this methodology. Three sets of equations are obtained by applying this methodology; all are suitable to be numerically implemented. To show the application potential of this method, the numerical simulation of the formation and propagation of a discontinuity in a concrete specimen is carried out and the results are compared with those from the physical experiment, demonstrating the adequacy of the methodology and its corresponding implementations to model discontinuities. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

On scalar equilibrium problem in generalized convex spaces

In this paper we prove the existence of a solution to the scalar equilibrium problem in generalized convex space. As applications, we derive results on the best approximations and simultaneous approximations. The results of this paper generalize some known results in the literature.     

Fundamental mode exact schemes for unsteady problems

The problem of increasing the accuracy of an approximate solution is considered for boundary value problems for parabolic equations. For ordinary differential equations (ODEs), nonstandard finite difference schemes are in common use for this problem. They are based on a modification of standard discretizations of time derivatives and, in some cases, allow to obtain the exact solution of problems. For multidimensional problems, we can consider the problem of increasing the accuracy only for the most important components of the approximate solution. In the present work, new unconditionally stable schemes for parabolic problems are constructed, which are exact for the fundamental mode. Such two‐level schemes are designed via a modification of standard schemes with weights using Padé approximations. Numerical results obtained for a model problem demonstrate advantages of the proposed fundamental mode exact schemes.     

Solution of nonlinear optimal control problems in Hilbert spaces by means of linear programming techniques

Optimal control problems in Hilbert spaces are considered in a measure-theoretical framework. Instead of minimizing a functional defined on a class of admissible trajectory-control pairs, we minimize one defined on a set of measures; this set is defined by the boundary conditions and the differential equation of the problem. The new problem is an infinite-dimensionallinear programming problem; it is shown that it is possible to approximate its solution by that of a finite-dimensional linear program of sufficiently high dimensions, while this solution itself can be approximated by a trajectory-control pair. This pair may not be strictly admissible; if the dimensionality of the finite-dimensional linear program and the accuracy of the computations are high enough, the conditions of admissibility can be said to be satisfied up to any given accuracy. The value given by this pair to the functional measuring the performance criterion can be about equal to theglobal infimum associated with the classical problem, or it may be less than this number. It appears that this method may become a useful technique for the computation of optimal controls, provided the approximations involved are acceptable.     

On the nearest parametric approximation of a fuzzy number

Many nearest parametric approximation methods of fuzzy sets are proposed in the literature. It is clear that the specific approximations may lead to the loss of information about fuzziness. To overcome this problem, most of these methods rely on the minimization of the distance between the original fuzzy set and its approximation. But these approximations mostly are not flexible to the decision maker's choice. Hence, in this paper, we offer a parametric fuzzy approximation method based on the decision maker's strategy as an extension of trapezoidal approximation of a fuzzy number. This method comprises the selection of the form of the parametric membership function and its evaluation.     

Convergence of a monotonisation procedure for a non-monotone quasi-static model in poroplasticity

Existence theory to quasi-static initial-boundary value problem of poroplasticity is studied. The classical quasi-static Biot model for soil consolidation coupled with a nonlinear system of ordinary differential equations is considered. This article presents a convergence result for the coercive and monotone approximations to solution of the original non-coercive and non-monotone problem of poroplasticity such that the inelastic constitutive equation is satisfied in the sense of Young measures.     

Adaptive Wavelet Methods for Elliptic Operator Equations with Nonlinear Terms

We present an adaptive wavelet method for the numerical solution of elliptic operator equations with nonlinear terms. This method is developed based on tree approximations for the solution of the equations and adaptive fast reconstruction of nonlinear functionals of wavelet expansions. We introduce a constructive greedy scheme for the construction of such tree approximations. Adaptive strategies of both continuous and discrete versions are proposed. We prove that these adaptive methods generate approximate solutions with optimal order in both of convergence and computational complexity when the solutions have certain degree of Besov regularity.     

The behavior of a finite difference approximation to a singular initial boundary value problem

We develop difference approximations to a singular parabolic initial-boundary value problem and its corresponding steady-state problem. A critical value for the existence of nonnegative solutions to the discrete steady state system is established. Convergence of the computed critical values is obtained. The long time behavior for the approximated solution of the parabolic problem is investigated. It is shown that the behavior of the discrete system is consistent with that of the continuous one     

Second-order scenario approximation and refinement in optimization under uncertainty

When solving scenario-based stochastic programming problems, it is imperative that the employed solution methodology be based on some form of problem decomposition: mathematical, stochastic, or scenario decomposition. In particular, the scenario decomposition resulting from scenario approximations has perhaps the least tendency to be computationally tedious due to increases in the number of scenarios. Scenario approximations discussed in this paper utilize the second-moment information of the given scenarios to iteratively construct a (relatively) small number of representative scenarios that are used to derive bounding approximations on the stochastic program. While the sizes of these approximations grow only linearly in the number of random parameters, their refinement is performed by exploiting the behavior of the value function in the most effective manner. The implementation SMART discussed here demonstrates the aptness of the scheme for solving two-stage stochastic programs described with a large number of scenarios.This paper was presented at the IFIP Workshop onStochastic Programming: Algorithms and Models, Lillehammer, Norway, January 1994.     

Some observations on local least squares

In previous work we introduced a construction to produce biorthogonal multiresolutions from given subdivisions. The approach involved estimating the solution to a least squares problem by means of a number of smaller least squares approximations on local portions of the data. In this work we use a result by Dahlquist, et al. on the method of averages to make observational comparisons between this local least squares estimation and full least squares approximation. We have explored examples in two problem domains: data reduction and data approximation. We observe that, particularly for design matrices with a repetitive pattern of column entries, the least squares solution is often well estimated by local least squares, that the estimation rapidly improves with the size of the local least squares problems, and that the quality of the estimate is largely independent of the size of the full problem. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 93E24     

代数数的有理逼近及其在求解丢番图方程中的应用

在本文中,我们利用Ｔｈｕｅ－Ｓｉｅｐｅｌ方法研究一类代数数的有理逼近,证明了对此代数数的有效有一逼近,最后我们利用此结果研究了ｄｉｏｐｈａｎｔｉｎｅ方程。ａｘ＾２－ｂｙ＾４＝－１得出关于此方程完整的结论。     

Relaxation-time limit and initial layer in the isentropic hydrodynamic model for semiconductors

This work is concerned with the relaxation-time limit in the multidimensional isentropic hydrodynamic model for semiconductors in the critical Besov space. Firstly, we construct formal approximations of the initial layer solution to the nonlinear problem by the matched expansion method. Then, assuming some regularity of the solution to the reduced problem, and proves the existence of classical solutions in the uniform time interval where the reduced problem has a smooth solution and justify the validity of the formal approximations in any fixed compact subset of the uniform time interval.     

Valuation of cash flows under random rates of interest: A linear algebraic approach

This paper reformulates the classical problem of cash flow valuation under stochastic discount factors into a system of linear equations with random perturbations. Using convergence results, a sequence of uniform approximations is developed. The new formulation leads to a general framework for deriving approximate statistics of cash flows for a broad class of models of stochastic interest rate process. We show applications of the proposed method by pricing default-free and defaultable cash flows. The methodology developed in this paper is applicable to a variety of uncertain cash flow analysis problems.     

The limits of Riemann solutions to the simplified pressureless Euler system with flux approximation

The formation of vacuum state and delta shock wave in the solutions to the Riemann problem for the simplified pressureless Euler system is considered under the linear approximations of flux functions. The method is to perturb the non‐strictly hyperbolic system into a nearby strictly hyperbolic system by introducing appropriately the linear approximations of flux functions. The solutions to the Riemann problem for the approximated system can be constructed explicitly and then the formation of vacuum state and delta shock wave can be observed by taking the perturbation parameter tend to zero in the solutions.     

Finite difference solution of Euler equations arising in variational image segmentation

This paper deals with finite-difference approximations of Euler equations arising in the variational formulation of image segmentation problems. We illustrate how they can be defined by the following steps: (a) definition of the minimization problem for the Mumford–Shah functional (MSf), (b) definition of a sequence of functionals Γ-convergent to the MSf, and (c) definition and numerical solution of the Euler equations associated to the kth functional of the sequence. We define finite difference approximations of the Euler equations, the related solution algorithms, and we present applications to segmentation problems by using synthetic images. We discuss application results, and we mainly analyze computed discontinuity contours and convergence histories of method executions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Gaussian Variational Approximation With a Factor Covariance Structure

Variational approximations have the potential to scale Bayesian computations to large datasets and highly parameterized models. Gaussian approximations are popular, but can be computationally burdensome when an unrestricted covariance matrix is employed and the dimension of the model parameter is high. To circumvent this problem, we consider a factor covariance structure as a parsimonious representation. General stochastic gradient ascent methods are described for efficient implementation, with gradient estimates obtained using the so-called “reparameterization trick.” The end result is a flexible and efficient approach to high-dimensional Gaussian variational approximation. We illustrate using robust P-spline regression and logistic regression models. For the latter, we consider eight real datasets, including datasets with many more covariates than observations, and another with mixed effects. In all cases, our variational method provides fast and accurate estimates. Supplementary material for this article is available online.     

The Global Solutions of the Scalar Nonconvex Conservation Law with Boundary Condition

Using the polygonal approximations method, we construct the global approximate solution of the initial boundary value problem (1.1)-(1.3) for the scalar nonconvex conservation law, and prove its convergence. The crux of this work is to clarify the behavior of the approximations on the boundary x = 0.     

A Stability Theorem for Constrained Optimal Control Problems

This paper presents the stability of difference approximations of an optimal control problem for a quasilinear parabolic equation with controls in the coefficients, boundary conditions and additional restrictions. The optimal control problem has been convered to one of the optimization problem using a penalty function technique. The difference approximations problem for the considered problem is obtained. The estimations of stability of the solution of difference approximations problem are proved. The stability estimation of the solution of difference approximations problem by the controls is obtained.