Error Expansion Research Articles

Superconvergence analysis and extrapolation of a BDF2 fully discrete scheme for nonlinear reaction–diffusion equations

The main aim of this paper is to propose a 2-step backward differential formula (BDF2) fully discrete scheme with the bilinear Q11 finite element method (FEM) for the nonlinear reaction–diffusion equation. By use of the combination technique of the element’s interpolation and Ritz projection, and through the interpolation post-processing approach, the superclose and global superconvergence estimates with order O(h2+τ2) in H1-norm are deduced rigorously. Furthermore, with the help of the asymptotic error expansion of the Q11 element, a new suitable fully discrete scheme is developed, and the extrapolation result of order O(h3+τ2) in H1-norm is derived, which is one order higher than that of the above traditional superconvergence estimate with respect to h. Here h is the mesh size and τ is the time step. Finally, some numerical results are provided to verify the theoretical analysis. It seems that the extrapolation of the fully discrete finite element scheme has never been seen in the previous studies.

Robust model averaging approach by Mallows-type criterion.

Model averaging is an important tool for treating uncertainty from model selection process and fusing information from different models, and has been widely used in various fields. However, the most existing model averaging criteria are proposed based on the methods of ordinary least squares or maximum likelihood, which possess high sensitivity to outliers or violation of certain model assumption. For the mean regression, no optimal robust methods are developed. To fill this gap, in our paper, we propose an outlier-robust model averaging approach by Mallows-type criterion. The idea is that we first construct a generalized M (GM) estimator for each candidate model, and then build robust weighting schemes by the asymptotic expansion of the final prediction error based on the GM-type loss function. So, we can still achieve a trustworthy result even if the dataset is contaminated by outliers in response and/or covariates. Asymptotic properties of the proposed robust model averaging estimators are established under some regularity conditions. The consistency of our weight estimators tending to the theoretically optimal weight vectors is also derived. We prove that our model averaging estimator is robust in terms of having bounded influence function. Further, we define the empirical prediction influence function to evaluate the quantitative robustness of the model averaging estimator. A simulation study and a real data analysis are conducted to demonstrate the finite sample performance of our estimators and compare them with other commonly used model selection and averaging methods.

Selective Weighting and Prediction Error Expansion for High-Fidelity Images

Selective Weighting and Prediction Error Expansion for High-Fidelity Images

Prediction error expansion based reversible watermarking scheme in higher order pixels

Prediction error expansion based reversible watermarking scheme in higher order pixels

PIPE: Parallelized inference through ensembling of residual quantization expansions

Deep neural networks (DNNs) are ubiquitous in computer vision and natural language processing, but suffer from high inference cost. This problem can be addressed by quantization, which consists in converting floating point operations into a lower bit-width format. With the growing concerns on privacy rights, we focus our efforts on data-free methods. However, such techniques suffer from their lack of adaptability to the target devices, as a hardware typically only support specific bit widths. Thus, to adapt to a variety of devices, a quantization method shall be flexible enough to find good accuracy v.s. speed trade-offs for every bit width and target device. To achieve this, we propose PIPE, a quantization method that leverages residual error expansion, along with group sparsity and an ensemble approximation for better parallelization. PIPEis backed off by strong theoretical guarantees and achieves superior performance on every benchmarked application (from vision to NLP tasks), architecture (ConvNets, transformers) and bit-width (from int8 to ternary quantization).

Weak error expansion of a numerical scheme with rejection for singular Langevin process

We show expansion à la Talay-Tubaro of a numerical scheme with rejection for the Langevin process in the case of a singular potential. In order to achieve this, we provide estimates on the associated semi-group of the process. The class of admissible potentials includes the Lennard-Jones interaction with confinement, which is an important potential in molecular dynamics and served as the primary motivation for this study.

A reversible natural language watermarking for sensitive information protection

Existing methods have evolved from using synonym substitution to incorporating arbitrary word substitution to achieve reversible natural language watermarking. However, a notable limitation is that they are prone to overlook the sensitivity of information associated with the original words, with a tendency to prefer non-sensitive words for substitution. As a result, a potential risk of sensitive information leakage contained in the original text is posed. Furthermore, while aiming for reversibility, the overall performance of the watermarking method may be inadvertently compromised. In response to the above problems, this paper puts forward a novel reversible natural language watermarking method that combines a Keyword Substitution scheme and a Prediction Error Expansion algorithm (KSPEE) to protect sensitive information, verify content integrity, protect copyright, and so on. Specifically, KSPEE leverages a keyword extraction algorithm to identify important content containing sensitive information in the original text, thereby determining the potential positions for watermark information embedding. Subsequently, a masked language model is utilized to predict appropriate substitution words based on the surrounding semantic information of the embedding position. In addition, the prediction error expansion algorithm is employed to select appropriate words for substituting the original keywords, ensuring the successful embedding of watermark information while maintaining the recoverability of the original keywords. By identifying keywords and substituting them, a suitable method of protecting the original sensitive information is provided. Extensive experiments demonstrate that, under the promise of semantic distortion and lossless restoration of the original content, the proposed method KSPEE achieves outstanding watermarked text quality. A higher watermark embedding rate is achieved and strong security is shown by KSPEE. More importantly, KSPEE effectively prevents the leakage of sensitive information.

A locally conservative staggered least squares method on polygonal meshes

<abstract><p>In this paper, we propose a novel staggered least squares method for elliptic equations on polygonal meshes. Our new method can be flexibly applied to rough grids and allows hanging nodes, which is of particular interest in practical applications. Moreover, it offers the advantage of not having to deal with inf-sup conditions and yielding positive definite discrete problems. Optimal a priori error estimates in energy norm are derived. In addition, a superconvergent estimates in energy norm are also developed by employing variational error expansion. The main difficulty involved here is to show the $ L^2 $ norm error estimates for the potential variable, where duality argument and the superconvergent estimates are the key ingredients. The single valued flux over the outer boundary of the dual partition enables us to construct a locally conservative flux. Numerical experiments confirm the theoretical findings and the performance of the adaptive mesh refinement guided by the least squares functional estimator are also displayed.</p></abstract>

Production of nouns and adjectives of children with cochlear implants and of children with typical hearing

This analytical cross-sectional study aimed to investigate the production of nouns and adjectives in 62 children between the ages 5 and 7, with 31 children having Cochlear Implants (CIs) and 31 children having Typical Hearing (TH). The study compaired their performance in a picture naming test of nouns and adjectives. Poisson regression models were fitted to compare the responses of both groups of children, and intra-subject differences between responses to the noun and adjective naming tasks were also analyzed. The results showed that both groups of children produced the same number of non-responses of nouns and of adjectives and a higher number of correct productions of nouns than of adjectives. However, children with CIs produced more errors when naming adjectives than when naming nouns, while this difference is not observed in children with TH. The comparative analysis between both groups of children indicates that children with CIs produced a higher proportion of non-responses when naming nouns, but the same proportion as children with TH when naming adjectives. Children with CIs also produced fewer correct nouns and adjectives and more errors than children with TH. Vocabulary expansion and repair of production errors in children with CIs should be targeted by speech-language pathologists in intervention programs.

BE-BDF2 Time Integration Scheme Equipped with Richardson Extrapolation for Unsteady Compressible Flows

In this work we investigate the effectiveness of the Backward Euler-Backward Differentiation Formula (BE-BDF2) in solving unsteady compressible inviscid and viscous flows. Furthermore, to improve its accuracy and its order of convergence, we have equipped this time integration method with the Richardson Extrapolation (RE) technique. The BE-BDF2 scheme is a second-order accurate, A-stable, L-stable and self-starting scheme. It has two stages: the first one is the simple Backward Euler (BE) and the second one is a second-order Backward Differentiation Formula (BDF2) that uses an intermediate and a past solution. The RE is a very simple and powerful technique that can be used to increase the order of accuracy of any approximation process by eliminating the lowest order error term(s) from its asymptotic error expansion. The spatial approximation of the governing Navier–Stokes equations is performed with a high-order accurate discontinuous Galerkin (dG) method. The presented numerical results for canonical test cases, i.e., the isentropic convecting vortex and the unsteady vortex shedding behind a circular cylinder, aim to assess the performance of the BE-BDF2 scheme, in its standard version and equipped with RE, by comparing it with the ones obtained by using more classical methods, like the BDF2, the second-order accurate Crank–Nicolson (CN2) and the explicit third-order accurate Strong Stability Preserving Runge–Kutta scheme (SSP-RK3).

Asymptotic expansion of the error of the numerical method for solving wave equation with functional delay

A wave equation with functional delay is considered. The problem is discretized. Constructions of the difference method with weights with piecewise linear interpolation are given. A basic method with weights with piecewise cubic interpolation is constructed. The order of the residual is studied without interpolation of the basic method, and the expansion coefficients of the residual with respect to time-steps and space-steps are written out. It is proved that the weighted method with piecewise cubic interpolation converges with order 2 in the energy norm. An equation is written for the main term of the asymptotic expansion of the global error of the basic method. Under certain assumptions, the validity of the application of the Richardson extrapolation procedure is substantiated, and the corresponding numerical method is constructed, that has the fourth order of convergence with respect to time-steps and space-steps. The validity of Runge's formulas for practical estimation of the error is proved. The results of numerical experiments on a test example are presented.

Numerical algorithm with fifth‐order accuracy for axisymmetric Laplace equation with linear boundary value problem

AbstractIn order to obtain the numerical solutions of the axisymmetric Laplace equation with linear boundary problem in three dimensions, we have developed a quadrature method to solve the problem. Firstly, the problem can be transformed to a integral equation with weakly singular operator by using the Green's formula. Secondly, A quadrature method is constructed by combing the mid‐rectangle formula with a singular integral formula to solve the integral equation, which has the accuracy of and low computational complexity. Thirdly, the convergence of the numerical solutions is proved based on the theory of compact operators and the single parameter asymptotic expansion of errors with odd power is got. From the expansion, we construct an extrapolation algorithm (EA) to further improve the accuracy of the numerical solutions. After one extrapolation, the accuracy of the numerical solutions can reach the accuracy of . Finally, two numerical examples are presented to demonstrate the efficiency of the method.

Adaptive Reversible 3D Model Hiding Method Based on Convolutional Neural Network Prediction Error Expansion

Although reversible data hiding technology is widely used, it still faces several challenges and issues. These include ensuring the security and reliability of embedded secret data, improving the embedding capacity, and maintaining the quality of media data. Additionally, irregular data types, such as three-dimensional point clouds and triangle mesh-represented 3D models, lack an ordered structure in their representation. As a result, embedding these irregular data into digital media does not provide sufficient information for the complete recovery of the original data during extraction. To address this issue, this paper proposes a method based on convolutional neural network prediction error expansion to enhance the embedding capacity of carrier images while maintaining acceptable visual quality. The triangle mesh representation of the 3D model is regularized in a two-dimensional parameterization domain, and the regularized 3D model is reversibly embedded into the image. The process of embedding and extracting confidential information in carrier images is symmetrical, and the regularization and restoration of 3D models are also symmetrical. Experiments show that the proposed method increases the reversible embedding capacity, and the triangle mesh can be conveniently subjected to reversible hiding.

A review of different prediction methods for reversible data hiding

In recent times, prediction error expansion (PEE) based reversible data hiding (RDH) schemes have gained significant traction due to their performance in terms of embedding capacity and image quality. However, the major part of their performance is dependent on how good the prediction has been. For a good prediction, various predictors such as median edge detection (MED), rhombus mean, least square, convolution neural network based predictor (CNNP) have been introduced. In this paper, a review of the working predictors being used in PEE-RDH is presented and discussed. In addition, a new predictor using extreme gradient boosting (XGBoost) is introduced in reversible data hiding. The XGBoost predictor makes use of a machine learning algorithm, where several optimization techniques are combined to get accurate results. To evaluate the performance comprehensively, experimental results considering different test images have been used and analyzed. From the analysis, it has been found that the XGBoost provides better prediction accuracy than some of the existing predictors. However, its performance is not up to the level of some other popular predictors such as least square, CNNP.

Colour image steganography through channel transformation approach

The interest in information security strategies is expanding due the quick expansion in the use of media content and transmission over the web employments. Accordingly, there is a need of special embedding procedure for steganography. The creator presents an original reversible data hiding (RDH) algorithm for colour pictures that further develops the inserting execution by applying a channel transformation function and a versatile expectation blunder extension prediction error expansion structure. The proposed calculation will bring the first colour picture with no information misfortune from the inserted picture (which is unique in addition to target picture). The evaluation is done here at the zero for the given base place for the histogram that marginally changes the pixel esteems for inserting the information. It can insert more information when contrasted with the majority of the current data hiding calculations. A hypothetical confirmation and various examinations state that the embedding capability of the proposed model which consistently is further noteworthy in comparison with other reversible data hiding techniques. The calculation has been applied to a wide scope of various colour pictures effectively. Some exploratory outcomes are introduced to show the legitimacy of the calculation.

A Higher-Order Defect Correction Method Over an Adaptive Bakhvalov–Shishkin Mesh for Advection–Diffusion Equations

The paper presents a defect correction method based on finite difference discretizations over a Bakhvalov–Shishkin mesh for a class of convection-diffusion equations. The technique combines a stable, low-order accurate, computationally inexpensive upwind difference scheme with a higher-order less stable modified central difference operator at several grid points. We prove that the method is second-order uniformly convergent at all mesh points. We do not use asymptotic expansions of discretization errors in our analysis. The method has no directional bias, and it can be used in an adaptive procedure to refine the mesh in the non-smooth parts. Numerical results are presented, which indicate that defect correction on layer-adapted meshes is an excellent method to achieve higher-order accuracy for singular perturbation problems.

Efficient reversible data hiding via two layers of double-peak embedding

Reversible data hiding continues to attract significant attention in recent years. In particular, an increasing number of authors focus on the higher significant bit (HSB) plane of an image which can yield more redundant space. On the other hand, the lower significant bit planes are often ignored for embedding in existing schemes due to their harm to the embedding rate. This paper proposes an efficient reversible data hiding scheme via a double-peak two-layer embedding (DTLE) strategy with prediction error expansion. The higher six-bit planes of the image are assigned as the HSB plane, and double prediction error peaks are applied in either embedding layer. This makes fuller use of the redundancy space of images compared with the one error peak strategy. Moreover, we carry out the median-edge detector pre-processing for complex images to reduce the size of the auxiliary information. A series of experimental results show that our DTLE approach makes better performance on the scenarios of high capacity embedding and achieves up to 83% higher embedding rate on real-world datasets, BOSSbase, BOWS2 and UCID, than the state-of-the-art approaches while guaranteeing better image quality.

On the rate of convergence of an exponential scheme for the non-linear stochastic Schrödinger equation with finite-dimensional state space

We address the numerical solution of the finite-dimensional non-linear stochastic Schrödinger equation, which is a locally Lipschitz stochastic differential equation modeling, for instance, quantum measurement processes. We study the rate of weak convergence of an exponential scheme that reproduces the norm of the desired solution by projecting onto the unit sphere. This justifies the use of the Talay-Tubaro extrapolation procedure in the numerical simulation of open quantum systems. In particular, we prove that an Euler-Exponential scheme converges with weak-order one, and we obtain the leading order term of its weak error expansion with respect to the step-size. Then, applying the Talay-Tubaro extrapolation procedure to the Euler-Exponential scheme under consideration we get a second-order method for computing the mean values of smooth functions of the solution of the non-linear stochastic Schrödinger equation. We also prove that the exponential scheme under study has order of strong convergence 1/2, which gives theoretical support to the use of the multilevel Monte Carlo method in simulating open quantum systems. We present a numerical experiment with a quantized electromagnetic field in interaction with a reservoir that illustrates the good performance of the weak second-order method, and the multilevel Monte Carlo method.

Visually meaningful image encryption combining with reversible data hiding in encrypted images

At present, most image encryption schemes directly change plaintext images into ciphertext images without visual significance, such ciphertext images can be detected and therefore subject to various attacks. To protect the content security and visual safety of images, a visually meaningful image encryption (VMIE) algorithm combining with Reversible Data Hiding in Encrypted Images (RDHEI) is proposed. First, the secret image is encrypted and additional information is embedded by Chen hyperchaotic system and prediction error expansion (PEE) method. Then the source image is encrypted by a Lorentz hyperchaotic system. Finally, bit-xor operation is performed on the resulting encrypted image and the result is saved in a cloud database to generate an indexing key using SHA-256,which enables the receiver to recover the secret image with the resource image and the key alone. The scheme combines with the RDHEI, improving the transmission efficiency and has no image quality problems.The experimental results and security analysis show that the image encryption scheme has sufficient key space, strong key sensitivity and wide applicability.

Numerical algorithm with high accuracy for the modified Helmholtz equation with Robin boundary value problem

In order to obtain the numerical solutions of the modified Helmholtz equation with Robin boundary conditions in two dimensions, we have developed a numerical algorithm to solve the problem, which is based on a Nyström discretization. First, the problem is transformed to a weakly-singular integral equation. Then we construct a quadrature method that achieve a convergence rate of O(h3), which has the characteristics of simple calculation and high precision. Further, the convergence of the numerical solutions is proved based on the collectively compact operators theory and the single parameter asymptotic expansion of errors with odd power O(h3) is got. From the expansion, we construct an extrapolation algorithm (EA) to further improve the accuracy of the numerical solutions. Finally, some numerical examples are presented to demonstrate the efficiency of the method.