Error Estimates For Approximate Solutions Research Articles

Error Estimates for Approximate Solutions to Seminlinear Elliptic Optimal Control Problems with Nonlinear and Mixed Constraints

This paper gives some sufficient conditions for convergence of approximate solutions to seminlinear elliptic optimal control problems with mixed pointwise constraints. We build discrete optimal control problems by the finite element method in type of the full control discretization. We show that if the strictly second-order sufficient condition is valid, then some error estimates between approximate solutions of discrete optimal control problems and optimal solutions of the original problem are obtained.

A non-exponential extension of Sanov’s theorem via convex duality

AbstractThis work is devoted to a vast extension of Sanov’s theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of independent and identically distributed samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include new results on the rate of convergence of empirical measures in Wasserstein distance, uniform large deviation bounds, and variational problems involving optimal transport costs, as well as an application to error estimates for approximate solutions of stochastic optimization problems. The proofs build on the Dupuis–Ellis weak convergence approach to large deviations as well as the duality theory for convex risk measures.

Эффективные алгоритмы вычисления глобальной и локальной апостериорной оценки точности решений линейных некорректных задач

We consider extremal problems introduced and investigated earlier by the author for calculating global and local a posteriori error estimates of approximate solutions to ill-posed inverse problems. For linear inverse problems in Hilbert spaces, they consist in maximization of quadratic functionals with two quadratic constraints. The article shows how under certain conditions these problems can be reduced to a problem of maximization of a special (written analytically) differentiable functional with one constraint. New algorithms for calculating global and local a posteriori error estimates based on the solution of these problems are proposed. Their effectiveness is illustrated by numerical experiments on a posteriori error estimation of solutions to the model two-dimensional inverse problem of potential continuation. Experiments show that the proposed algorithms give a posteriori error estimates close to the true error values. Proposed algorithms for global a posteriori error estimation turn out to be more rapid (3 to 5 times) than the previously known algorithms.

Piecewise uniform regularization for the inverse problem of microtomography with a-posteriori error estimate

ABSTRACTAn inverse microtomography problem is under consideration in a class of functions with bounded V H variation. An algorithm for solving this problem is proposed based on Tikhonov's regularization with a special regularizer. The algorithm ensures piecewise uniform convergence of approximate solutions to exact solution of the inverse problem. In addition, the question of a-posteriori error estimate of approximate solutions obtained is considered. A new numerical algorithm for finding this estimate is proposed. Numerical experiments on solving a model inverse problem on the class of functions with bounded V H variation are presented along with the results of a-posteriori error estimate for approximate solutions obtained.

Local and parallel finite element method for solving the biharmonic eigenvalue problem of plate vibration

In this paper, we establish a new local and parallel finite element discrete scheme based on the shifted‐inverse power method for solving the biharmonic eigenvalue problem of plate vibration. We prove the local error estimation of finite element solution for the biharmonic equation/eigenvalue problem and prove the error estimation of approximate solution obtained by the local and parallel scheme. When the diameters of three grids satisfy H4 = ϑ(w2) = ϑ(h), the approximate solutions obtained by our schemes can achieve the asymptotically optimal accuracy. The numerical experiments show that the computational schemes proposed in this paper are effective to solve the biharmonic eigenvalue problem of plate vibration.

One‐level and two‐level finite element methods for the Boussinesq problem

Previous works on the convergence of numerical methods for the Boussinesq problem were conducted, while the optimal L2‐norm error estimates for the velocity and temperature are still lacked. In this paper, the backward Euler scheme is used to discrete the time terms, standard Galerkin finite element method is adopted to approximate the variables. The MINI element is used to approximate the velocity and pressure, the temperature field is simulated by the linear polynomial. Under some restriction on the time step, we firstly present the optimal L2 error estimates of approximate solutions. Secondly, two‐level method based on Stokes iteration for the Boussinesq problem is developed and the corresponding convergence results are presented. By this method, the original problem is decoupled into two small linear subproblems. Compared with the standard Galerkin method, the two‐level method not only keeps good accuracy but also saves a lot of computational cost. Finally, some numerical examples are provided to support the established theoretical analysis.

СХОДИМОСТЬ В СИЛЬНЫХ НОРМАХ ПРОЕКЦИОННО-РАЗНОСТНОГО МЕТОДА РЕШЕНИЯ ПАРАБОЛИЧЕСКОГО УРАВНЕНИЯ С ПЕРИОДИЧЕСКИМ УСЛОВИЕМ НА РЕШЕНИЕ

A smooth soluble abstract linear parabolic equation with the periodic condition on the solution is treated in a separable Hilbert space. This problem is solved approximately by a projection-difference method using the Galerkin method in space and the implicit Euler scheme in time. Effective both in time and in space strong-norm error estimates for approximate solutions, which imply convergence of approximate solutions to the exact solution and order of convergence rate depending of the smoothness of the exact solution, are obtained.

Galerkin Method for Semilinear Parabolic Equation with Varying Time Direction

We establish the unique solvability of the first boundary value problem for a semilinear second order parabolic equation with varying time direction by using a modified Galerkin method and the regularization method. We obtain a priori error estimates for approximate solutions.

Numerical Modeling of Optical Fibers Using the Finite Element Method and an Exact Non-reflecting Boundary Condition

Abstract The original problem for eigenwaves of weakly guiding optical fibers formulated on the plane is reduced to a convenient for numerical solution linear parametric eigenvalue problem posed in a disk. The study of the solvability of this problem is based on the spectral theory of compact self-adjoint operators. Properties of dispersion curves are investigated for the new formulation of the problem. An efficient numerical method based on FEM approximations is developed. Error estimates for approximate solutions are derived. The rate of convergence for the presented algorithm is investigated numerically.

The Crank–Nicolson/Adams–Bashforth scheme for the Burgers equation with H2 and H1 initial data

In this paper, we consider the stability and convergence results of the Crank–Nicolson/Adams–Bashforth scheme for the Burgers equation with smooth and nonsmooth initial data. The spatial approximation is based on the standard conforming finite element space. The temporal treatment of the spatial discrete Burgers equation is based on the implicit Crank–Nicolson scheme for the linear term and the explicit Adams–Bashforth scheme for the nonlinear term. Firstly, we prove that the Crank–Nicolson/Adams–Bashforth scheme is almost unconditionally stable with initial data u0∈Hα (α=1,2). Secondly, the optimal error estimates of the numerical solution in L2-norm are derived with initial data u0∈H2, and the error estimates of approximate solution in L2 norm obtained with initial data u0∈H1 is reduced by 12. Finally, some numerical examples are provided to verify the established stability theory and convergence results with H2 and H1 initial data.

Vragov’s boundary value problem for an implicit equation of mixed type

We study a Vragov boundary value problem for a third-order implicit equation of mixed type with an arbitrary manifold of type switch. These Sobolev-type equations arise in many important applied problems. Given certain constraints on the coefficients and the right-hand side of the equation, we demonstrate, using nonstationary Galerkin method and regularization method, the unique regular solvability of the boundary value problem. We also obtain an error estimate for approximate solutions of the boundary value problem in terms of the regularization parameter and the eigenvalues of the Dirichlet spectral problem for the Laplace operator.

Solving a variational parabolic equation with the periodic condition by a projection-difference method with the Crank–Nicolson scheme in time

A solution to a smoothly solvable linear variational parabolic equation with the periodic condition is sought in a separable Hilbert space by an approximate projection-difference method using an arbitrary finite-dimensional subspace in space variables and the Crank–Nicolson scheme in time. Solvability, uniqueness, and effective error estimates for approximate solutions are proven. We establish the convergence of approximate solutions to a solution as well as the convergence rate sharp in space variables and time.

A functional-type a posteriori error estimate of approximate solutions for Reissner-Mindlin plates and its implementation

Paper is devoted to a numerical justification of the recent a posteriori error estimate for Reissner-Mindlin plates. This majorant provides a reliable control of accuracy of any conforming approximate solution of the problem including solutions obtained with commercial software for mechanical engineering. The estimate is developed on the basis of the functional approach and is applicable to several types of boundary conditions. To verify the approach, numerical examples with mesh refinements are provided.

Editorial preface for the special issue “Advances in security, privacy and trust technologies”

With the fast development of Internet technologies, the information and communication systems are continuously facing new security, trust and privacy challenges. Indeed, the potentially invasive nature of the information has implications on privacy of the users and their online activity such as transactions. Likewise, the implicit reliance on technology to make decisions on one’s behalf makes essential to study new mechanisms for expressing and reasoning about trust. This special issue brings together research articles covering diverse aspects of security, privacy and trust in computer and networked-based information systems, including privacy concerns, tools to help secure information, methods for development and assessment of trustworthy systems, analysis of vulnerabilities and attacks, trends and new developments, secure operating systems and applications, security issues in wireless networks and pervasive systems, design and test strategies for secure and survivable systems, and cryptology, among others. The papers of the special issue are selected based on strict review results, where papers went through three rounds of review and revision. As a result, nine papers were selected and are arranged in the special issue as follows. Xiong et al. in the first paper ‘‘The finite volume element method for a parameter identification problem’’ use finite volume element method to solve a parameter identification problem of parabolic equations with over-specified data. The authors provide the numerical scheme of the unknown function and control parameters and obtain the error estimates of approximate solution. The proposed method is experimentally evaluated and compared with the exact solution to confirm the good accuracy of the presented scheme. The second paper by Chen et al. ‘‘An Opinion Mining Framework For Cantonese Reviews’’, analyze opinion mining for Cantonese language. The authors motivate the need to construct a particular lexical database for Cantonese and explore some Cantonese special written-tradition rules and incorporate them into the feature-based opinion summarization system framework. The experimental results show that the proposed framework significantly outperforms the traditional Mandarin sentiment analysis method using ICTCLAS. In the third paper ‘‘A Secure Remote Data Integrity Checking Cloud Storage System from Threshold Encryption’’, Yao et al. study the confidentiality and integrity issues in cloud storage system. The authors address the privacy issue of decentralized cloud storage system using threshold cryptography. The threshold encryption scheme is therefore integrated with a secure decentralized erasure code to form a secure cloud storage system. Zhang et al. in the fourth paper ‘‘How to build a faster private information retrieval protocol’’, propose a secure multi-bit homomorphic encryption scheme based on Learning With Errors over Rings (RLWE) assumption. The authors use canonical embedding to transform ring & Fatos Xhafa fatos@cs.upc.edu

A Unified Reproducing Kernel Method and Error Estimation for Solving Linear Differential Equation with Functional Constraints

A unified reproducing kernel method for solving linear differential equations with functional constraint is provided. We use a specified inner product to obtain a class of piecewise polynomial reproducing kernels which have a simple unified description. Arbitrary order linear differential operator is proved to be bounded about the special inner product. Based on space decomposition, we present the expressions of exact solution and approximate solution of linear differential equation by the polynomial reproducing kernel. Error estimation of approximate solution is investigated. Since the approximate solution can be described by polynomials, it is very suitable for numerical calculation.

Refined error estimates for the Riccati equation with applications to the angular Teukolsky equation

We derive refined rigorous error estimates for approximate solutions of Sturm-Liouville and Riccati equations with real or complex potentials. The approximate solutions include WKB approximations, Airy and parabolic cylinder functions, and certain Bessel functions. Our estimates are applied to solutions of the angular Teukolsky equation with a complex aspherical parameter in a rotating black hole Kerr geometry.

Model reduction and dynamic iteration for coupled nonlinear systems

AbstractThe dynamical behavior of coupled systems is determined by different interconnected subsystems that are usually governed by entirely different physical laws and often act in different time and space scales. We discuss the simulation of coupled nonlinear systems using dynamic iteration combined with model order reduction. We also study the convergence of this approach and derive error estimates for approximate solutions. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The finite volume element method for a parameter identification problem

In this paper we use finite volume element method to solve a parameter identification problem of parabolic equations with overspecified-data. We provide the numerical scheme of the unknown function and control parameters and obtain the error estimates of approximate solution. The results of the numerical experiment are presented and are compared with the exact solution to confirm the good accuracy of the presented scheme.

A kinetic approach to error estimate for nonautonomous anisotropic degenerate parabolic-hyperbolic equations

This paper is devoted to the error estimate of approximate solutions for non-autonomous degenerate parabolic-hyperbolic equations, where the nonlinear convection flux, the diffusion matrix and source term depend on time $t$ explicitly. Our method is based on kinetic formulation and kinetic entropy formula. By developing the kinetic techniques, we obtain an error estimate of order $O(\sqrt{\mu})$, where $\mu$ is the artificial viscosity.

Approximate Solutions to the Discontinuous Riemann-Hilbert Problem of Elliptic Systems of First Order Complex Equations

Several approximate methods have been used to find approximate solutions of elliptic systems of first order equations. One common method is the Newton imbedding approach, i.e. the parameter extension method. In this article, we discuss approximate solutions to discontinuous Riemann-Hilbert boundary value problems, which have various applications in mechanics and physics. We first formulate the discontinuous Riemann-Hilbert problem for elliptic systems of first order complex equations in multiply connected domains and its modified well-posedness, then use the parameter extensional method to find approximate solutions to the modified boundary value problem for elliptic complex systems of first order equations, and then provide the error estimate of approximate solutions for the discontinuous boundary value problem.