Error Estimates Research Articles

Optimal error estimates for non-conforming approximations of linear parabolic problems with minimal regularity

Optimal error estimates for non-conforming approximations of linear parabolic problems with minimal regularity

Meshfree RBF–FD methods for numerical simulation of PDE problems

The meshfree RBF–FD method is becoming an increasingly popular discretization method for challenging PDE problems due to its flexibility with respect to geometry and its ease of implementation. Using a combination of polyharmonic spline basis functions and polynomial basis functions has proven to be particularly successful with a convergence rate depending on the polynomial degree and good behavior at boundaries due to the contribution of the spline part. A challenge that has been observed is that derivative boundary conditions can give rise to large errors. A known remedy for this problem is to introduce one or more layers of ghost points, which improves the boundary approximations. In a recent paper, Tominec, Larsson, Heryudono (2021), it was shown that using oversampling is another effective way to handle this problem. Oversampling also allows for theoretical analysis of the method, which can then be viewed as a discretization of a continuous least-squares projection. By looking deeper into the error estimates we gain an understanding about how to best implement the method.

A comprehensive lung CT landmark pair dataset for evaluating deformable image registration algorithms.

Deformable image registration (DIR) is a key enabling technology in many diagnostic and therapeutic tasks, but often does not meet the required robustness and accuracy for supporting clinical tasks. This is in large part due to a lack of high-quality benchmark datasets by which new DIR algorithms can be evaluated. Our team was supported by the National Institute of Biomedical Imaging and Bioengineering to develop DIR benchmark dataset libraries for multiple anatomical sites, comprising of large numbers of highly accurate landmark pairs on matching blood vessel bifurcations. Here we introduce our lung CT DIR benchmark dataset library, which was developed to improve upon the number and distribution of landmark pairs in current public lung CT benchmark datasets. Thirty CT image pairs were acquired from several publicly available repositories as well as authors' institution with IRB approval. The data processing workflow included multiple steps: (1) The images were denoised. (2) Lungs, airways, and blood vessels were automatically segmented. (3) Bifurcations were directly detected on the skeleton of the segmented vessel tree. (4) Falsely identified bifurcations were filtered out using manually defined rules. (5) A DIR was used to project landmarks detected on the first image onto the second image of the image pair to form landmark pairs. (6) Landmark pairs were manually verified. This workflow resulted in an average of 1262 landmark pairs per image pair. Estimates of the landmark pair target registration error (TRE) using digital phantoms were 0.4mm±0.3mm. The data is published in Zenodo at https://doi.org/10.5281/zenodo.8200423. Instructions for use can be found at https://github.com/deshanyang/Lung-DIR-QA. The dataset library generated in this work is the largest of its kind to date and will provide researchers with a new and improved set of ground truth benchmarks for quantitatively validating DIR algorithms within the lung.

Incremental Fuzzy Clustering-Based Neural Networks Driven With the Aid of Dynamic Input Space Partition and Quasi-Fuzzy Local Models.

Fuzzy clustering-based neural networks (FCNNs) based on information granulation techniques have been shown to be effective Takagi-Sugeno (TS)-type fuzzy models. However, the existing FCNNs could not cope well with sequential learning tasks. In this study, we introduce incremental FCNNs (IFCNNs), which could dynamically update themselves whenever new learning data (e.g., single datum or block data) are incorporated into the dataset. Specifically, we employ dynamic (incremental) fuzzy C-means (FCMs) clustering algorithms to reveal a structure in data and divide the entire input space into several subregions. In the aforementioned partition, the dynamic FCM adaptively adjusts the position of its prototypes by using sequential data. Due to the time-sharing arrival of training data, compared with batch learning models, incremental learning methods may lose classification (prediction) accuracy. In order to tackle this challenge, we utilize quasi-fuzzy local models (QFLMs) based on modified Schmidt neural networks to replace the popular linear functions in TS-type fuzzy models to refine and enhance the ability to represent the behavior of fuzzy subspaces. Meanwhile, the recursive least square error (LSE) estimation is utilized to update the weights of QFLMs from one-by-one or block-by-block (fixed or varying block size) learning data. In addition, the L2 regularization is considered to ameliorate the deterioration of generalization abilities caused by potential overfitting when carrying out weight estimation. The proposed method leads to the construction of FCNNs in a new way, which can effectively deal with incremental data as well as deliver sound generalization capability. Extensive machine-learning datasets and a real-world application are employed to show the validity and performance of the presented methods. From the experimental results, we show that the proposal can maintain sound classification accuracy when effectively processing sequential data.

Approximately well-balanced Discontinuous Galerkin methods using bases enriched with Physics-Informed Neural Networks

This work concerns the enrichment of Discontinuous Galerkin (DG) bases, so that the resulting scheme provides a much better approximation of steady solutions to hyperbolic systems of balance laws. The basis enrichment leverages a prior – an approximation of the steady solution – which we propose to compute using a Physics-Informed Neural Network (PINN). To that end, after presenting the classical DG scheme, we show how to enrich its basis with a prior. Convergence results and error estimates follow, in which we prove that the basis with prior does not change the order of convergence, and that the error constant is improved. To construct the prior, we elect to use parametric PINNs, which we introduce, as well as the algorithms to construct a prior from PINNs. We finally perform several validation experiments on four different hyperbolic balance laws to highlight the properties of the scheme. Namely, we show that the DG scheme with prior is much more accurate on steady solutions than the DG scheme without prior, while retaining the same approximation quality on unsteady solutions.

Study on applicable coverage extension of theory-based generalization errors bounds to the variants of RVFL network and ELM

This study investigates the coverage extendibility of generalized prediction error estimation for the random vector functional link (RVFL) network and generalized extreme learning machine (GELM). GELM is an ELM applying the limiting behavior of Moore–Penrose generalized inverse (M–P GI) to the output matrix of the last hidden layer of ELM. The RVFL network, the ELM, and their variants are currently employed in various areas due to their fast learning and are expected to be adopted in more applications. Additionally, obtaining prediction error bounds quickly for these networks could greatly increase their utilities. Therefore, exploring the error bounds of the variants is crucial. Recently, the upper and lower bounds of generalized prediction error and their associated tail probabilities for the RVFL network and GELM were analyzed. This study aims to extend the coverage of the existing error bounds to variants of RVFL and ELM. To this end, we investigate each solution of the objective functions for the existing RVFL and ELM variants, compare it with the solution obtained by using the limiting behavior of M–P GI, and then discuss the applicability of the probability inequalities for the prediction error to the variant. Specifically, we conceptualize a restricted solution vector and prove that the restricted solution of a variant of RVFL is identical to that of GELM, with an appropriately selected regularization parameter. The mechanism used in the proof can be applied to other variants, which have different objective functions but have similar forms of solution. Experiments are performed with the CAIDA (Center for Applied Internet Data Analysis) dataset. Numerical results verify the analysis, showing that the generalization prediction error bounds and their associated tail probabilities for ELM (RVFL) variants in applicability coverage are almost identical to those of the GELM (RVFL). In addition, performance-based network similarity is introduced and computed with the obtained results. Based on this analysis, we can conclude that the probability inequalities for prediction error derived can be directly applied to some variants of the RVFL network and ELM, such as robust regularized RVFL and ridge regression ELM. However, it cannot apply directly to the networks that use information obtained during the learning process in its learning, such as RVFL+ and its variants. The significance of this study is that it enables us to figure out the relationship between the size-restricted final weight vector of RVFL and the final weight vectors of ELM, the effect of the relationship on generalized prediction error estimation, and the performance-based similarity between networks. This study provides a theoretical basis for the generalized prediction error's upper and lower bounds and their associated tail probabilities for ELM (RVFL) variants.

Kriging surrogate model for optimizing outlet temperature distribution in low-emission combustors without dilution holes

Designing advanced combustors that operate at high temperatures and produce little pollution, especially in the absence of primary and dilution holes, is a difficult task that may bring significant challenges. In this regard, this paper introduces a Kriging surrogate model approach to optimize the outlet temperature distribution of the combustor to achieve such advanced low-pollution combustors. Building upon previous research, this study explores the effects of the swirler blade installation angle on the outlet temperature distribution of the combustor without primary or dilution holes. Traditional methods, such as the control variable method using computational fluid dynamics (CFD) for numerical simulation, are limited in application due to the complex coupling of flow, heat transfer, mass transfer, and combustion processes. In contrast, surrogate models, especially the Kriging model, offer a rapid and efficient alternative to extensive CFD simulations that provide accurate predictions and error estimates for the solution of the problem. In summary, this paper details the process of generating sample points through three-dimensional numerical simulations, develops a Kriging surrogate model through Latin hypercube sampling, and optimizes the model to identify the most uniform outlet temperature distribution achievable by adjusting the installation angle of the swirl blade. The optimal design parameters, which are quickly obtained through the Kriging model, showed a significant reduction in the overall temperature distribution function and the radial temperature distribution function by 21% and 27.14%, respectively.

Mixed virtual element methods for elliptic optimal control problems with boundary observations in L2(Γ)

In this paper, we study the mixed virtual element approximation to an elliptic optimal control problem with boundary observations. The objective functional of this type of optimal control problem contains the outward normal derivatives of the state variable on the boundary, which reduces the regularity of solutions to the optimal control problems. We construct the mixed virtual element discrete scheme and derive an a priori error estimate for the optimal control problem based on the variational discretization for the control variable. Numerical experiments are carried out on different meshes to support our theoretical findings.

Semi analytical scheme for the presentation of solution to Fractional Fokker–Planck Equation

In this work, we have also proposed new semi analytical scheme combining least-squares and asymptotic homotopy perturbation methods AHPM for the solution of Fractional Fokker–Planck Equation (FFPE). The Caputo version of derivative is used. The finding of this work is fast convergence of the AHPM to the solutions of linear and nonlinear applied problems. AHPM is a modified form of the HPM and is particularly useful for solving problems with small or large parameters. This can be seen as a clear advantage of this new semi analytical method over existing methods because its implementation and the calculation involved in it are much simpler and easier. The proposed method solved the problems without using Ji-Huan He and Adomian polynomials. Numerical solutions obtained in this way show that this approach is simple and computationally very attractive. An error estimate for the solution is also provided. Approximate solutions of AHPM are tabulated and graphically displayed to show that AHPM is effective and accurate.

Unconditionally optimal error estimates of linearized Crank-Nicolson virtual element methods for quasilinear parabolic problems on general polygonal meshes

In this paper, we construct, analyze, and numerically validate a linearized Crank-Nicolson virtual element method (VEM) for solving quasilinear parabolic problems on general polygonal meshes. In particular, we consider the more general nonlinear term a(x, u), which does not require Lipschitz continuity or uniform ellipticity conditions. To ensure that the fully discrete solution remains bounded in L∞-norm, we construct two novel elliptic projections and apply a new error splitting technique. With the help of the boundedness of numerical solution and delicate analysis of the nonlinear term, we derive the optimal error estimates for any k-order VEMs without any time-step restrictions. Numerical experiments on various polygonal meshes validate the accuracy of theoretical analysis and the unconditional convergence of the proposed scheme.

An Equilibrated Error Estimator for the 2-D/1-D MSFEM T-Formulation of the Eddy Current Problem

An Equilibrated Error Estimator for the 2-D/1-D MSFEM T-Formulation of the Eddy Current Problem

Exploring the moderating effect of regulatory quality on the relationship between financial development and economic growth/economic volatility for developed and developing countries

This study examines the moderating effect of regulatory quality on the relationship between financial development and economic growth/economic volatility. We employ the Panel Corrected Standard Error estimation method, supplemented by a robustness check using the Generalized Method of Moments. Our findings offer interesting insights into the relationship between financial development and outcomes for the economy. In developed countries, regulatory quality positively moderates the relationship between banking-based financial development and economic growth. Hence, policymakers should prioritize the maintenance of a high-quality institutional and regulatory environment conducive to both banking-based and market-based growth. Continual efforts to enhance regulatory quality and adopt robust approaches are imperative. Conversely, this study does not detect a significant moderating effect of regulatory quality on the relationship between financial development and economic growth in developing countries. Policymakers in these contexts must accelerate efforts to bolster institutional and regulatory frameworks, emphasizing the enhancement of regulatory quality. Care should be taken to avoid overly stringent regulations, particularly at lower levels of financialization, to prevent potential counter-productivity. Instead, a gradual approach to improving regulatory quality is advisable. Furthermore, policymakers should recognize that regulatory quality mitigates the contribution of banking-based financial development to economic volatility, a phenomenon observed in both developed and developing countries.

An investigation of self-interstitial diffusion in α-zirconium by an on-the-fly machine learning force field

The on-the-fly machine learning force field approach, based on the Gaussian approximation potential and Bayesian error estimation, was used to study the diffusion of self-interstitial atoms in α-zirconium. Ab initio molecular dynamics simulations of lattice vibration and interstitial diffusion at different temperatures were employed to develop the force field. The radial and angular descriptors of the potential were further optimized to achieve better agreement with first-principles results. Subsequent long-term diffusion simulations were performed to assess the diffusion behavior based on the obtained force field. Tracer diffusion coefficients and diffusion anisotropy were studied at temperatures of 600–1200 K, and the Bayesian errors were estimated throughout the diffusion simulations. The mean and maximum estimated Bayesian errors of atomic force were approximately twice as large as those observed during the learning period. The basal diffusion was greatly favored compared to the interstitial diffusion along the c-axis, consistent with previous simulations based on first-principles results and classical potentials. The accuracy and applicability of the current on-the-fly machine learning approach were critically evaluated.

Numerical analysis of a nonsmooth quasilinear elliptic control problem: I. Explicit second-order optimality conditions

In this paper, we derive explicit second-order necessary and sufficient optimality conditions of a local minimizer to an optimal control problem for a quasilinear second-order partial differential equation with a piecewise smooth but not differentiable nonlinearity in the leading term. The key argument rests on the analysis of level sets of the state. Specifically, we show that if a function vanishes on the boundary and its the gradient is different from zero on a level set, then this set decomposes into finitely many closed simple curves. Moreover, the level sets depend continuously on the functions defining these sets. We also prove the continuity of the integrals on the level sets. In particular, Green’s first identity is shown to be applicable on an open set determined by two functions with nonvanishing gradients. In the second part to this paper, the explicit sufficient second-order conditions will be used to derive error estimates for a finite-element discretization of the control problem.

Performance Degradation Estimation Mechanisms for Networked Control Systems Under DoS Attacks and its Application to Autonomous Ground Vehicle.

This article examines the mechanisms by which aperiodic denial-of-service (DoS) attacks can exploit vulnerabilities in the TCP/IP transport protocol and its three-way handshake during communication data transmission to hack and cause data loss in networked control systems (NCSs). Such data loss caused by DoS attacks can eventually lead to system performance degradation and impose network resource constraints on the system. Therefore, estimating system performance degradation is of practical importance. By formulating the problem as an ellipsoid-constrained performance error estimation (PEE) problem, we can estimate the system performance degradation caused by DoS attacks. We propose a new Lyapunov-Krasovskii function (LKF) using the fractional weight segmentation method (FWSM) to examine the sampling interval and introduce a relaxed, positive definite constraint to optimize the control algorithm. We also propose a relaxed, positive definite constraint that reduces the initial constraints to optimize the control algorithm. Next, we introduce an alternate direction algorithm (ADA) to solve the optimal trigger threshold and design an integral-based event-triggered controller (IETC) to estimate the error performance of NCSs with limited network resources. Finally, we verify the effectiveness and feasibility of the proposed method using the Simulink joint platform autonomous ground vehicle (AGV) model.

Impact of the use of active methodology on the performance of undergraduate radiology students

Although PBL is widely used in several countries, especially in medicine courses, its application in teaching other higher education courses, which involve fundamentals applied to radiology, is still little explored. Therefore, we aim to evaluate the implementation of Problem-Based Learning (PBL) in a higher education institution's radiology and biomedicine technologist course, focusing on specific radiology-related disciplines. An interventional study was developed with 78 students. An active methodology model was created and implemented for one of the groups of participants. At the beginning of each semester, students of both groups were evaluated with pre-tests. At the end of the semesters, the students performed a post-test and a validated evaluation of the discipline methodology. Repeated measures generalized linear regressive models with robust error estimators were used to evaluate test outcomes. A significant interaction among the methodologies was found (p=0.020), with better results from students exposed to the active methodology (initial and final grades were 7.18 and 7.57 in the active methodology, respectively, and 7.45 and 6.89 in the traditional methodology, respectively). In addition, students' evaluation regarding the quality of the methodology was favorable to the active methodology with statistical significance (p<0.05) in 16 of the 22 items evaluated. The students' positive response and performance were attributed to the interaction and innovation of the methodology compared to conventional methods, highlighting the effectiveness of PBL in higher education in radiology and its potential for more participatory and contextualized learning.

Weak Galerkin finite element methods for optimal control problems governed by second order elliptic equations

This paper is concerned with the development of weak Galerkin (WG) finite element methods for optimal control problems governed by second order elliptic partial differential equations. WG methods are advantageous over other existing methods in that the control variable along the domain boundary is a natural variable by design in the numerical scheme. A convergence theory was established for the numerical solutions in various Sobolev norms. Numerical results are presented to demonstrate the efficiency and accuracy of the new scheme.

Effective numerical simulation of time fractional KdV equation with weakly singular solutions

A second-order time-stepping method for numerically solving the linearized time fractional KdV equation (TFKDVE) whose solution has initial singularity is investigated. Alikhanov’s scheme is used to discretize the Caputo fractional derivative on temporal graded mesh, and the space is approximated by a Petrov–Galerkin spectral method. Detailed stability and error estimate of the full-discrete scheme are given, and the final error bound is [Formula: see text]-robust. Numerical examples are given to illustrate the sharpness of the error analysis.

Diagnostic accuracy of estimated glomerular filtration rate in pediatric oncology patients.

Objective: To compare eGFR using different equations with 24-hour creatinine clearance Methodology: We collected data of standardized 24-hour creatinine clearance of pediatric oncology patients for a period of one year and compared the efficiency of various equations for estimation of creatinine clearance with it. For error estimation we used mean relative prediction error (MRPE) and root mean square error (RMSE) and for accuracy estimation we used Lin’s correlation. Results: Out of 100 study participants, 57 were males and 43 were females. Mean age of the population studied was 11.8 while mean BMI was 13.95. Most common tumor reported was osteosarcoma. Out of 18 equations evaluated only 5 showed an accuracy of 0.80 or more, while SEM of none of the equation is near to the desired value (zero). Minimum SEM was found to be for Schwartz (2012) equation. Conclusions: According to our study none of the equations in current clinical practices showed a good correlation with creatinine clearance adjusted according to body surface area. Keywords: eGFR, creatinine clarence, renal function, pediatric oncology.

A second-order three-point difference scheme on a posteriori mesh for a singularly perturbed convection–diffusion problem

In this paper a three-point difference scheme based on a posteriori mesh is used to solve a second-order singularly perturbed convection–diffusion problem. A hybrid difference method is constructed on an arbitrary mesh. A posteriori error analysis is developed for the three-point difference scheme on an arbitrary mesh. A solution-adaptive algorithm based on a posteriori error estimation is devised by equidistributing a monitor function. Numerical experiments illustrate that the scheme is second-order uniformly convergent.