Error Estimates For Approximation Research Articles

A multiscale domain decomposition approach for parabolic equations using expanded mixed method

We study the multiscale mortar expanded mixed method for second order parabolic partial differential equations. This technique involves non-overlapping domain decomposition which restrict a large problem into smaller pieces. Each block in the domain decomposition is discretized by expanded mixed method. The flux continuity is ensured by constructing a finite element space on the interface between subdomains. A pressure variable is introduced on the interfaces which play the role of Dirichlet boundary condition for each subdomain internal boundary. We demonstrate the unique solvability of discrete problem. We considered both continuous time and discrete time formulation and established a priori error estimates for local subdomain approximations. We derived the optimal order convergence for the appropriate choice of mortar space. An error estimate for mortar pressure variable is also provided. The numerical experiments are performed to show the accuracy and effectiveness of method.

An error estimate for bilateral contact problem with nonmonotone friction between two electroelastic bodies

This work is concerned with a problem describing bilateral contact between two electroelastic bodies, with nonmonotone friction. A mixed formulation is derived allowing to state the weak formulation of the problem. A mortar finite-element method is employed to obtain finite element approximations of the problem. Using the Gauss elimination technique, we eliminate the electric potential and the resulting problem is a perturbed elastic problem. Furthermore, error estimates for both conformal and non-conformal approximations are obtained.

Approximate solutions to second-order parabolic equations: Evolution systems and discretization

<p style='text-indent:20px;'>We study the discretization of a linear evolution partial differential equation when its Green's function is known or well approximated. We provide error estimates both for the spatial approximation and for the time stepping approximation. We show that, in fact, an approximation of the Green function is almost as good as the Green function itself. For suitable time-dependent parabolic equations, we explain how to obtain good, explicit approximations of the Green function using the Dyson-Taylor commutator method that we developed in <i>J. Math. Phys.</i> <b>51</b> (2010), n. 10, 103502 (reference [<xref ref-type="bibr" rid="b15">15</xref>]). This approximation for short time, when combined with a bootstrap argument, gives an approximate solution on any fixed time interval within any prescribed tolerance.</p>

Optimal estimates of approximation errors for strongly positive linear operators on convex polytopes

In the present investigation, we introduce and study linear operators, which underestimate every strongly convex function. We call them, for brevity, sp-linear (approximation) operators. We will provide their sharp approximation errors. We show that the latter is bounded by the error approximation of the quadratic function. We use the centroidel Voronoi tessellations as a domain partition to construct best sp-linear operators. Finally, numerical examples are presented to illustrate the proposed method.

High order approximation on non-uniform meshes for generalized time-fractional telegraph equation

This paper presents a high order approximation scheme to solve the generalized fractional telegraph equation (GFTE) involving the generalized fractional derivative (GFD). The GFD is characterized by a scale function σ(t) and a weight function ω(t). Thus, we study the solution behavior of the GFTE for different σ(t) and ω(t). The scale function either stretches or contracts the solution while the weight function dramatically shifts the numerical solution of the GFTE. The time fractional GFTE is approximated using quadratic scheme in the temporal direction and the compact finite difference scheme in the spatial direction. To improve the numerical scheme’s accuracy, we use the non-uniform mesh. The convergence order of the whole discretized scheme is, O(τ2α−3,h4), where τ and h are the temporal and spatial step sizes respectively. The outcomes of the work are as follows: •The error estimate for approximation of the GFD on non-uniform meshes is established.•The numerical scheme’s stability and convergence are examined.•Numerical results for four examples are compared with those obtained using other method. The study shows that the developed scheme achieves higher accuracy than the scheme discussed in literature.

A Posteriori Error Estimates of Spectral Approximations for Second Order Partial Differential Equations in Spherical Geometries

A Posteriori Error Estimates of Spectral Approximations for Second Order Partial Differential Equations in Spherical Geometries

Kernel-independent adaptive construction of mathcal {H}^2-matrix approximations

A method for the kernel-independent construction of mathcal {H}^2-matrix approximations to non-local operators is proposed. Special attention is paid to the adaptive construction of nested bases. As a side result, new error estimates for adaptive cross approximation (ACA) are presented which have implications on the pivoting strategy of ACA.

A priori error estimates for finite element approximations to transient convection-diffusion-reaction equations in fluidized beds

Abstract In this work, a priori error estimates for finite element approximations to the governing equations of heat and mass transfer in fluidized beds are derived. These equations are time dependent strongly coupled system of five semilinear convection-diffusion-reaction equations. The a priori error estimates for all the five variables are obtained for the error measured in L ∞(L 2) and L 2 ( E ) ${L}^{2}\left(\mathcal{E}\right)$ , E $\mathcal{E}$ is the energy norm.

The estimation of approximation error using inverse problem and a set of numerical solutions

In this paper, we consider the inverse problem for the estimation of a point-wise approximation error occurring at the discretization of the system of partial differential equations. We analyse the set of the solutions, obtained by the numerical algorithms of the dissimilar structures on the same grid. The differences between the numerical solutions are used as the input data for the inverse problem, which is posed in the variational statement with the zero-order Tikhonov regularization. The numerical tests, performed for the two-dimensional inviscid compressible flows corresponding to Edney-I and Edney-VI shock wave interference modes, are provided. The comparison of the estimated error and the exact error, obtained by subtraction of numerical and analytic solutions, is presented.

A study of defect-based error estimates for the Krylov approximation of \u03c6-functions

Prior recent work, devoted to the study of polynomial Krylov techniques for the approximation of the action of the matrix exponential etAv, is extended to the case of associated φ-functions (which occur within the class of exponential integrators). In particular, a posteriori error bounds and estimates, based on the notion of the defect (residual) of the Krylov approximation are considered. Computable error bounds and estimates are discussed and analyzed. This includes a new error bound which favorably compares to existing error bounds in specific cases. The accuracy of various error bounds is characterized in relation to corresponding Ritz values of A. Ritz values yield properties of the spectrum of A (specific properties are known a priori, e.g., for Hermitian or skew-Hermitian matrices) in relation to the actual starting vector v and can be computed. This gives theoretical results together with criteria to quantify the achieved accuracy on the fly. For other existing error estimates, the reliability and performance are studied by similar techniques. Effects of finite precision (floating point arithmetic) are also taken into account.

A FOM/ROM Hybrid Approach for Accelerating Numerical Simulations

The basis generation in reduced order modeling usually requires multiple high-fidelity large-scale simulations that could take a huge computational cost. In order to accelerate these numerical simulations, we introduce a FOM/ROM hybrid approach in this paper. It is developed based on an a posteriori error estimation for the output approximation of the dynamical system. By controlling the estimated error, the method dynamically switches between the full-order model and the reduced-oder model generated on the fly. Therefore, it reduces the computational cost of a high-fidelity simulation while achieving a prescribed accuracy level. Numerical tests on the non-parametric and parametric PDEs illustrate the efficacy of the proposed approach.

A General Error Estimate For Parabolic Variational Inequalities

Abstract The gradient discretisation method (GDM) is a generic framework designed recently, as a discretisation in spatial space, to partial differential equations. This paper aims to use the GDM to establish a first general error estimate for numerical approximations of parabolic obstacle problems. This gives the convergence rates of several well-known conforming and non-conforming numerical methods. Numerical experiments based on the hybrid finite volume method are provided to verify the theoretical results.

Analysis of a Galerkin finite element method for the Maxwell–Schrödinger system under temporal gauge

Abstract This paper is concerned with the numerical solution of the Maxwell–Schrödinger system under the temporal gauge, which describes light–matter interactions. We first propose a semidiscrete finite element scheme for the system and establish stability estimates for the finite element solution. Due to the lack of control over its divergence we cannot get $\textbf{H}^{1}$$a \;priori$ estimates for the vector potential, making it difficult to obtain error estimates by usual techniques. We apply an exhaustion argument to overcome this difficulty and derive error estimates for the finite element approximation. An energy-conserving time-stepping scheme is proposed to solve the semidiscrete system.

Optimal stochastic Bernstein polynomials in Ditzian–Totik type modulus of smoothness

We introduce a family of symmetric stochastic Bernstein polynomials based on order statistics, and establish the order of convergence in probability in terms of the second order Ditzian–Totik type modulus of smoothness on the interval [0,1], which epitomizes an optimal pointwise error estimate for the classical Bernstein polynomial approximation. Monte Carlo simulation results (presented at the end of the article) show that this new approximation scheme is efficient and robust.

Exact Artificial Boundary Conditions for Quasi-Linear Problems in Semi-Infinite Strips

In this paper, the exact artificial boundary conditions for quasi-linear problems in semi-infinite strips are investigated. Based on the Kirchhoff transformation, the exact and approximate boundary conditions on a segment artificial boundary are derived. The error estimate for the finite element approximation with the artificial boundary condition is obtained. Some numerical examples show the efficiency of this method.

A priori error estimates for finite element approximations of regularized level set flows in higher norms

This paper proves error estimates for H2 conforming finite elements for elliptic equations which model the flow of surfaces by different powers of the mean curvature (this includes mean curvature flow). The scheme is based on a known regularization procedure and produces different kinds of errors, a regularization error, a finite element discretization error for the regularized problems, and a full error. While in the literature and own previous work different aspects of the aforementioned error types are treated, here, we solely and for the first time focus on the finite element discretization error in higher norms for the regularized equation and additionally analyze also the dependencies from the regularization parameter.

Uniform error estimates for artificial neural network approximations for heat equations

Abstract Recently, artificial neural networks (ANNs) in conjunction with stochastic gradient descent optimization methods have been employed to approximately compute solutions of possibly rather high-dimensional partial differential equations (PDEs). Very recently, there have also been a number of rigorous mathematical results in the scientific literature, which examine the approximation capabilities of such deep learning-based approximation algorithms for PDEs. These mathematical results from the scientific literature prove in part that algorithms based on ANNs are capable of overcoming the curse of dimensionality in the numerical approximation of high-dimensional PDEs. In these mathematical results from the scientific literature, usually the error between the solution of the PDE and the approximating ANN is measured in the $L^p$-sense, with respect to some $p \in [1,\infty )$ and some probability measure. In many applications it is, however, also important to control the error in a uniform $L^\infty $-sense. The key contribution of the main result of this article is to develop the techniques to obtain error estimates between solutions of PDEs and approximating ANNs in the uniform $L^\infty $-sense. In particular, we prove that the number of parameters of an ANN to uniformly approximate the classical solution of the heat equation in a region $ [a,b]^d $ for a fixed time point $ T \in (0,\infty ) $ grows at most polynomially in the dimension $ d \in {\mathbb {N}} $ and the reciprocal of the approximation precision $ \varepsilon> 0 $. This verifies that ANNs can overcome the curse of dimensionality in the numerical approximation of the heat equation when the error is measured in the uniform $L^\infty $-norm.

Deep-Waveform: A Learned OFDM Receiver Based on Deep Complex-Valued Convolutional Networks

The (inverse) discrete Fourier transform (DFT/ IDFT) is often perceived as essential to orthogonal frequency-division multiplexing (OFDM) systems. In this paper, a deep complex-valued convolutional network (DCCN) is developed to recover bits from time-domain OFDM signals without relying on any explicit DFT/IDFT. The DCCN can exploit the cyclic prefix (CP) of OFDM waveform for increased SNR by replacing DFT with a learned linear transform, and has the advantage of combining CP-exploitation, channel estimation, and intersymbol interference (ISI) mitigation, with a complexity of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). Numerical tests show that the DCCN receiver can outperform the legacy channel estimators based on ideal and approximate linear minimum mean square error (LMMSE) estimation and a conventional CP-enhanced technique in Rayleigh fading channels with various delay spreads and mobility. The proposed approach benefits from the expressive nature of complex-valued neural networks, which, however, currently lack support from popular deep learning platforms. In response, guidelines of exact and approximate implementations of a complex-valued convolutional layer are provided for the design and analysis of convolutional networks for wireless PHY. Furthermore, a suite of novel training techniques are developed to improve the convergence and generalizability of the trained model in fading channels. This work demonstrates the capability of deep neural networks in processing OFDM waveforms and the results suggest that the FFT processor in OFDM receivers can be replaced by a hardware AI accelerator.

Error estimate for indirect spectral approximation of optimal control problem governed by fractional diffusion equation with variable diffusivity coefficient

In this paper an indirect spectral method of an optimal control problem governed by a space-fractional diffusion equation with variable diffusivity coefficient is studied. First-order optimality conditions of the proposed model are derived and the regularity of the solutions is analyzed. Indirect spectral methods via weighted Jacobi polynomials are built up based on the “first optimize, then discretize” strategy, and a priori error estimates of the discrete optimal control problem in weighted norms are derived. As proposed indirect spectral discrete schemes are designed to accommodate the impact of the variable coefficients, which are complicated and not formulated under the variational framework, conventional error estimate techniques of the discrete space-fractional optimal control problems do not apply. Novel treatments of the discrete variational inequality are developed to resolve the aforementioned issues and to support the error estimates. Numerical examples are presented to verify the theoretical findings.

State Error Estimates for the Numerical Approximation of Sparse Distributed Control Problems in the Absence of Tikhonov Regularization

In this paper, we analyze optimal control problems of semilinear elliptic equations, where the controls are distributed. Box constraints for the controls are imposed and the cost functional does not involve the control itself, except possibly for a non-differentiable sparsity-promoting term. Under appropriate second order sufficient optimality conditions, first we estimate the difference between the discrete and continuous optimal states. Next, under an additional assumption on the optimal adjoint state, we prove error estimates for the controls and improve the estimates for the states.