Element Error Research Articles

Finite Element Approximations for PDEs with Irregular Dirichlet Boundary Data on Boundary Concentrated Meshes

Abstract This paper is concerned with finite element error estimates for second order elliptic PDEs with inhomogeneous Dirichlet boundary data in convex polygonal domains. The Dirichlet boundary data are assumed to be irregular such that the solution of the PDE does not belong to H 2 ⁢ ( Ω ) {H^{2}(\Omega)} but only to H r ⁢ ( Ω ) {H^{r}(\Omega)} for some r ∈ ( 1 , 2 ) {r\in(1,2)} . As a consequence, a discretization of the PDE with linear finite elements exhibits a reduced convergence rate in L 2 ⁢ ( Ω ) {L^{2}(\Omega)} and H 1 ⁢ ( Ω ) {H^{1}(\Omega)} . In order to restore the best possible convergence rate we propose and analyze in detail the usage of boundary concentrated meshes. These meshes are gradually refined towards the whole boundary. The corresponding grading parameter does not only depend on the regularity of the Dirichlet boundary data and their discrete implementation but also on the norm, which is used to measure the error. In numerical experiments we confirm our theoretical results.

A stochastic gradient algorithm with momentum terms for optimal control problems governed by a convection–diffusion equation with random diffusivity

In this paper, we focus on a numerical investigation of a strongly convex and smooth optimization problem subject to a convection–diffusion equation with uncertain terms. Our approach is based on stochastic approximation where true gradient is replaced by a stochastic ones with suitable momentum term to minimize the objective functional containing random terms. A full error analysis including Monte Carlo, finite element, and stochastic momentum gradient iteration errors is done. Numerical examples are presented to illustrate the performance of the proposed stochastic approximations in the PDE-constrained optimization setting.

Simulation Calculation of Element Number Density in the Earth’s Atmosphere Based on X-ray Occultation Sounding

The number density profiles of elements N and O in the altitude range of 120–250 km are retrieved by simulation based on X-ray occultation. Based on the parameters of the NICER telescope, the energy spectrum forward model of the Crab Nebula in the energy range of 0.25–8 keV during the occultation is constructed, and the energy spectrum simulation data are obtained by adding noise to the energy spectrum forward model at different tangent point altitudes. The NICER energy band includes the K-shell absorption edges (0.4 keV, 0.53 keV for N, O), and there are significant differences in X-ray cross sections at the K-shell absorption edges, which provides an opportunity to retrieve the atmospheric density of each element. The MCMC algorithm is used to fit the energy spectrum forward model and simulation data, and the density profiles of elements N and O are retrieved. It is found that the retrieved error of O element in the altitude range of 120–140 km is large, which may be related to the low proportion of O in the line of sight and the low signal-to-noise ratio of simulation data. In the altitude range of 140–200 km, the retrieved error of elements N and O is small, but in the altitude range of 200–250 km, the retrieved error of elements N and O becomes larger, and the inconsistency between the retrieved results and NRLMSISE-00 model values increases. This is because the number of absorbed photons is reduced due to the thin atmospheric density at higher altitude, which introduces great uncertainty into the retrieved results. This study lays a foundation for element density retrieval based on X-ray occultation measured data in the future.

Beam physics design of a 30-MW beam transport to the target for an accelerator-driven subcritical system

The Japan Atomic Energy Agency's accelerator-driven subcritical system (JAEA-ADS) proposes the reduction of nuclear waste through the transmutation of minor actinides. The JAEA-ADS drives a 30-MW proton beam to a spallation target to produce neutrons for a subcritical 800-MWth reactor. The beam must be transported from the end of the linear accelerator (linac) to the target located inside the reactor core with high beam power stability and low peak density to ensure beam window integrity, which is a primary concern for the ADS project. Additionally, the design of the beam transport to the target (BTT) must be compatible with the established reactor design, and the elements that comprise the BTT must facilitate the maintenance and replacement of the fuel and the beam window. To this end, a robust-compact BTT design was developed through massive multiparticle simulations. First, the beam optics was optimized to guarantee beam window feasibility requirements by providing a low peak density of less than 0.3 μA/mm2. Then, beam stability was evaluated and improved by a simultaneous application of input beam and element errors. The input beam errors were based on the beam degradation obtained by implementing fast fault compensation in the linac, which is a key strategy to attain high-reliability operation. The results show that the BTT design fulfills reactor and beam window requirements for JAEA-ADS.

A priori and a posteriori error estimates for the Deep Ritz method applied to the Laplace and Stokes problem

We analyze neural network solutions to partial differential equations obtained with Physics Informed Neural Networks. In particular, we apply tools of classical finite element error analysis to obtain conclusions about the error of the Deep Ritz method applied to the Laplace and the Stokes equations. Further, we develop an a posteriori error estimator for neural network approximations of partial differential equations. The proposed approach is based on the dual weighted residual estimator. It is destined to serve as a stopping criterion that guarantees the accuracy of the solution independently of the design of the neural network training. The result is equipped with computational examples for Laplace and Stokes problems.

Geometric Quality Improvement Method of Optical Remote Sensing Satellite Images Based on Rational Function Model

The high-precision geometric positioning of optical remote sensing satellites is the prerequisite to determine the application capability of satellite image products. Its positioning accuracy is related to the observation accuracy of each link in the imaging process, including satellite attitude, orbit measurement accuracy, time synchronization accuracy, camera measurement accuracy, and so on. Untimely and inaccurate on-orbit calibration will lead to great geometric positioning errors. To optimize the positioning accuracy of satellite images with the rational function model (RFM) under low positioning accuracy, our paper proposes an improved geometric quality model based on the reorientation of internal and external orientation elements in the RFM model of remote sensing images. By establishing the rational function positioning model, the external orientation model, and the internal orientation model, the original image can be reorientated. Then, we use the improved model to generate uniformly distributed virtual ground control points. By analyzing and verifying the relationship between each rational polynomial coefficient (RPC) and its influence on geometric positioning accuracy, we propose an RPC coefficients optimization method based on image offset correction and positioning dominant coefficients. Finally, we use the small satellite “MN200Sar-1” with low geometric accuracy for experimental verification. The results show that the model can effectively eliminate the errors of internal and external elements in the on-orbit calibration, and the positioning accuracy is improved from one hundred pixels to one pixel. At the same time, the rational polynomial dominant coefficient optimization method can improve geometric positioning accuracy without introducing additional compensation parameters.

A probabilistic data assimilation framework to reconstruct finite element error fields from sparse error estimates: Application to sub‐modeling

AbstractThe present work proposes a computational approach that recovers full finite element error fields from a small number of estimates of errors in scalar quantities of interest. The approach is weakly intrusive and is motivated by large scale industrial applications wherein modifying the finite element models is undesirable and multiple regions of interest may exist in a single model. Error estimates are developed using a Zhu‐Zienkiewicz estimator coupled with the adjoint methodology to deliver goal‐oriented results. A Bayesian probabilistic estimation framework is deployed for full field estimation. An adaptive, radial basis function based reduced order modeling strategy is implemented to reduce the cost of calculating the posterior. The Bayesian reconstruction approach, accelerated by the proposed model reduction technology, is shown to yield good probabilistic estimates of full error fields, with a computational complexity that is acceptable compared to the evaluation of the goal‐oriented error estimates. The novelty of the work is that a set of computed error estimates are considered as partial observations of an underlying error field, which is to be recovered. Future improvements of the method include the optimal selection of goal‐oriented error measures to be acquired prior to the error field reconstruction.

Efficient Reversible Data Hiding Based on Connected Component Construction and Prediction Error Adjustment

To achieve a good trade-off between the data-embedding payload and the data-embedding distortion, mainstream reversible data hiding (RDH) algorithms perform data embedding on a well-built prediction error histogram. This requires us to design a good predictor to determine the prediction errors of cover elements and find a good strategy to construct an ordered prediction error sequence to be embedded. However, many existing RDH algorithms use a fixed predictor throughout the prediction process, which does not take into account the statistical characteristics of local context. Moreover, during the construction of the prediction error sequence, these algorithms ignore the fact that adjacent cover elements may have the identical priority of data embedding. As a result, there is still room for improving the payload-distortion performance. Motivated by this insight, in this article, we propose a new content prediction and selection strategy for efficient RDH in digital images to provide better payload-distortion performance. The core idea is to construct multiple connected components for a given cover image so that the prediction errors of the cover pixels within a connected component are close to each other. Accordingly, the most suitable connected components can be preferentially used for data embedding. Moreover, the prediction errors of the cover pixels are adaptively adjusted according to their local context, allowing a relatively sharp prediction error histogram to be constructed. Experimental results validate that the proposed method is significantly superior to some advanced works regarding payload-distortion performance, demonstrating the practicality of our method.

Determination of Scandium (Sc), Yttrium (Y), and Rare-Earth Elements (REEs) in Mafic and Ultramafic Rock Powder by a Modified and Validated Digestion Protocol and Inductively Coupled Plasma – Mass Spectrometry (ICP-MS)

The global demand for rare earth elements (REEs) has been associated with the increased use of renewable energy technologies. However, due to the low occurrence of REEs in economic sources, such as geologic samples, it is essential to employ suitable sample preparation methods and measurement techniques that allow for reliable quantification. While acid digestion procedures have been conventionally used to free metals from silicate rocks, few studies have evaluated these methods for quantitative measurement of REEs, including Sc and Y in mafic-ultramafic rocks. This study involves validating a modified digestion procedure to facilitate the determination of Sc, Y, and REEs in mafic-ultramafic matrices. Three reference materials (RMs) (OKUM, BHVO-1, and JB-1a) were used to assess the performance of this modified method. The calculated percent error (< 10%) for most elements was in good agreement with the reference values, except La in OKUM, Er in BHVO-I, and Eu, Gd, Tb, Ho, and Er in JB-1a. The precision of the modified method was satisfactory, with values generally below 5%. The smooth chondrite normalized REE patterns indicate good quality data. The t-values of the studied elements were lower than the t-critical value (± 12.71), suggesting no significant differences between the analytical and Intertek values. Thus, the modified method could be used as a viable alternative in determining REEs in Ni laterite samples by ICP-MS.

A New Algorithm for Fault Tolerance in Redundant Sensor Systems Based on Real-Time Variance Estimation

Low cost and small size integrated sensors bring a significant interest to redundant sensing systems such as sensor array systems. Redundancy may permit to increase both performance and dependability of individual sensors at the system level, however, the emergence of these redundant systems has generated the need for dynamic algorithms able to detect and act against the presence of errors in system elements. Then, a new adaptive algorithm for sensor array systems is presented in this work. This approach is generic and can be implemented for different types of sensor systems, besides it does not require any information about the inputs. The proposed real-time algorithm is based on the following: 1) estimation of sensor individual variances at each time step, 2) identification and removal of sensors affected by errors, and 3) reincorporation of sensors after recovery or replacement. Variance estimation is revisiting MInimum Norm Quadratic Unbiased Estimation (MINQUE) method to allow real-time operation and requires that the number of sensors is strictly greater than two times the number of signals to be measured. Consequently, the proposed method is able to support the presence of m-1 faulty devices, where m is the total number of sensors less two times the number of measured signals. Performance of the proposed algorithm is demonstrated through theoretical demonstrations, simulations, and experimental results.

Uniform Finite Element Error Estimates with Power-Type Asymptotic Constants for Unsteady Navier-Stokes Equations.

Uniform error estimates with power-type asymptotic constants of the finite element method for the unsteady Navier–Stokes equations are deduced in this paper. By introducing an iterative scheme and studying its convergence, we firstly derive that the solution of the Navier–Stokes equations is bounded by power-type constants, where we avoid applying the Gronwall lemma, which generates exponential-type factors. Then, the technique is extended to the error estimate of the long-time finite element approximation. The analyses show that, under some assumptions on the given data, the asymptotic constants in the finite element error estimates for the unsteady Navier–Stokes equations are uniformly power functions with respect to the initial data, the viscosity, and the body force for all time . Finally, some numerical examples are shown to verify the theoretical predictions.

Fast Monte-Carlo Approximation of the Attention Mechanism

We introduce Monte-Carlo Attention (MCA), a randomized approximation method for reducing the computational cost of self-attention mechanisms in Transformer architectures. MCA exploits the fact that the importance of each token in an input sequence vary with respect to their attention scores; thus, some degree of error can be tolerable when encoding tokens with low attention. Using approximate matrix multiplication, MCA applies different error bounds to encode input tokens such that those with low attention scores are computed with relaxed precision, whereas errors of salient elements are minimized. MCA can operate in parallel with other attention optimization schemes and does not require model modification. We study the theoretical error bounds and demonstrate that MCA reduces attention complexity (in FLOPS) for various Transformer models by up to 11 in GLUE benchmarks without compromising model accuracy. Source code and appendix: https://github.com/eis-lab/monte-carlo-attention

Accuracy Improvement of a Miniature Laser Diode Interferometer by Compensating Nonlinear Errors and Active Stabilizing Laser Diode Wavelengths

A miniature laser diode interferometer (MLDI), which can be embedded in precision machines or measurement equipment as an on-line measurement sensor, is designed. A compact laser diode (LD) is adopted as the laser source of the MLDI. The measurement accuracy of the MLDI is affected by the nonlinear errors induced by the installation errors and manufacturing errors of the optical elements and the stability and accuracy of the LD wavelength. An arithmetic is applied to eliminate the nonlinear errors, and an error sensitivity analysis is conducted to enhance the understanding of which error components are more important or influence the measurement accuracy of the MLDI. An active wavelength stabilizer based on a compact laser wavelength meter is proposed to improve the stability of the LD wavelength. A group of experiments are carried out to verify the effectiveness of the proposed methods and the capabilities of the MLDI.

Generalised non-negative matrix factorisation for air pollution source apportionment

Source Apportionment (SA) techniques are widely used for identifying key sources of air pollution, thereby providing critical inputs for policy measures. Positive Matrix Factorisation (PMF) (Paatero and Tapper, 1994) is a widely used SA technique. PMF uses the speciated concentration data (X) collected over several days and factorises it into source contribution (G) and source profile (F) matrices, albeit under positivity constraint. Towards this end, it involves solving an optimisation problem where the elements of X are weighted by the inverse of the standard deviations of the corresponding errors introduced during the sampling and chemical analysis process. Thus, PMF implicitly assumes that the errors in different elements of the X matrix are uncorrelated. This assumption may not hold since the sampling, and chemical analysis steps deployed in any data-collection campaign will inevitably lead to correlated errors. While there are other existing Non-Negative Matrix Factorisation (NMF) methods in literature that can be potentially used for SA, these also make various restrictive assumptions about the error covariance structure. In this work, we propose a new method called Generalised Non-Negative Matrix Factorisation (GNMF) to fill this gap. In particular, the proposed method is able to incorporate any error covariance matrix without making any restrictive assumptions on its structure. Towards this end, we integrate the full error covariance matrix in the objective function to be minimised to obtain F and G matrices. We derive the corresponding update rules for obtaining these matrices iteratively. To ensure non-negativity, we extend the multiplicative and projected gradient-based ideas available in NMF literature to the proposed GNMF approach. The proposed method subsumes various NMF methods available in literature as special cases. The utility of the proposed approach is demonstrated by comparing its performance with other methods on an SA problem using a dataset derived from field measurements.

An efficient multigrid method for semilinear interface problems

In this paper, we study an efficient multigrid method to solve the semilinear interface problems. We first give an optimal finite element error estimate for the semilinear interface problems under a weak condition for the nonlinear term compared with the existing conclusions. Then next based on the finite element error estimate, we design a novel multigrid method for semilinear elliptic problems. The proposed multigrid method only requires to solve a linear interface problem in each level of the multilevel space sequence and a small-scale semilinear interface problem in a correction space. The involved linear interface problem can be solved efficiently by the multigrid iteration. The dimension of the correction space is small and fixed, which is independent from the fine spaces. Thus the computational time of the correction step is negligible compared with that of the linear interface problems in the fine spaces. On the whole, the efficiency of the presented multigrid method is nearly the same as that of the multigrid method for linear interface problems. Additionally, unlike the existing finite element error estimates and the multigrid methods for semilinear interface problems, which always require the bounded second order derivatives of the nonlinear terms, all the analysis in our paper only requires a Lipschitz continuous condition.

DETERMINATION OF SECOND, THIRD AND FOURTH COMPONENT ERRORS IN STRUCTURE OF INTER-TRANSITION DIMENSION TOLERANCE FIELD FOR CONVERSION OF GEOMETRICAL ELEMENTS IN STRUCTURE OF INITIAL BLANKS

The structure of the tolerance field of the intertransition size of the form elements to be transformed in the structure of the initial blanks, as well as the intertransition and general allowance the main elements of the regulatory reference information base for solving the tasks of technological preparation of mechanical processing production throughout its cycle. The first component of errors in the structure of the tolerance field of the intertransition time-measure is determined in the information and logical connection of two technological schemes: for basing the initial blanks and blanks into working machines and for basing the mold elements into working machines and their geometric shaping on working machines. The source of the first component of the error is the structure of the technological schemes for basing the initial blanks and blanks into working machines. At the same time, there are conditions for redistribution of structure elements in coordinate systems of slave basing objects by the position of elements in the structure of coordinate systems of leading basing objects. As a result of such redistributions, the coordinate system of the slave home objects: depends on the composition of the “reference” points of the functional elements of the form participating in the conjugation of objects and on the composition of elementary and composite base functions executed by the elements of the form. The first component of the error in the structure of the tolerance field of the intertransition time-measure can be determined only as a result of experimental measurements of dimensional and geometric errors of the shape elements to be transformed directly under the conditions of real working machines for processing. Based on the above, there is a need to take a closer look at other components of the errors in the structure of the tolerance field of the interpersonal dimension. The approaches of different authors to determining the structure of the tolerance field for intertransition dimensions differ in existing knowledge. Unity is that their basis is to operate on the main elements (means) of integration of the structure and control element base provided with design quality parameters in determining parts and assembly units of design products. Other errors to be considered in the structure of the tolerance field of intertransition dimensions in their information and logical connection integral in sin. Their definition has a significant impact on determining the content and sequence of transformations of form elements in their information-on-one and logical connection along the route of technological operations. At the same time, it is necessary to use the technological element base, organic for preparing production, as part of the updated concepts, methods, tools, algorithms, operations associated with the system.

A decoupled and iterative finite element method for generalized Boussinesq equations

In this work, a decoupled and iterative finite element method is developed and analyzed for steady-state generalized Boussinesq equations, in which both the viscosity and thermal conductivity depend on the temperature. By utilizing the solutions obtained in the previous iteration step, the coupled system is reduced to Navier-Stokes equations with temperature-dependent viscosity and a linearized convection-diffusion equation, which can be solved in parallel. The well-posedness and stability of the scheme are proved. Finite element errors and iterative errors are analyzed for the cases of both temperature-dependent and temperature-independent thermal conductivity. Numerical examples are provided to demonstrate the convergence, accuracy and applicability of the proposed method.

L2-Norm Based a Posteriori Error Estimates of Compressible and Nearly-Incompressible Elastic Finite Element Solutions

The displacement and stress-based error estimates in a posteriori error recovery of compressible and nearly-incompressible elastic finite element solutions is investigated in the present study. The errors in the finite element solutions, i.e., in displacement and stress, at local and global levels are computed in L2-norm of quantity of interest, namely, displacements and gradients. The error estimation techniques are based on the least square fitting of higher order polynomials to stress and displacement in a patch comprising of node/elements surrounding and including the node/elements under consideration. The benchmark examples of compressible and incompressible elastic bodies, with known solutions employing triangular discretization schemes, are implemented to measure the finite element errors in displacements and gradients. The mixed formulation involving displacement and pressure is used for incompressible elastic analysis. The performance of error estimation is measured in terms of convergence properties, effectivity and mesh required for predefined precision. The error convergence rate, in FEM original solution, recovered solution using displacement recovery-based and stress-based error recovery technique for stresses, are obtained as (1.9714, 2.8999, and 2.5018) and (0.9818, 1.7805, and 1.4952) respectively for compressible and incompressible self-loaded elastic plate benchmark example using higher order triangular elements. It is concluded from the study that displacement fitting technique for extracting higher order derivatives shows a very effective technique for recovery of compressible and nearly-incompressible finite element analysis errors.

An Immersed Raviart–Thomas Mixed Finite Element Method for Elliptic Interface Problems on Unfitted Meshes

This paper presents a lowest-order immersed Raviart–Thomas mixed triangular finite element method for solving elliptic interface problems on unfitted meshes independent of the interface. In order to achieve the optimal convergence rates on unfitted meshes, an immersed finite element (IFE) is constructed by modifying the traditional Raviart–Thomas element. Some important properties are derived including the unisolvence of IFE basis functions, the optimal approximation capabilities of the IFE space and the corresponding commuting digram. Optimal finite element error estimates are proved rigorously with the constant independent of the interface location relative to the mesh. Some numerical examples are provided to validate the theoretical analysis.

Comparison between Multi Epipolar Geometry &amp; Affine 2d Transformation-Based Filters for Optical Robot Navigation

Abstract: In previous research papers, Multi Epipolar Geometry-based Filter (M-EGF) has proven its high capability to overcome Single Epipolar Geometry-based Filter (S-EGF) as well as with conformal 2D Transformation-based Filter (C-2DF) in terms of providing precise, trusted, outlier-free, and real time Automatic Image Matching (AIM) results for Optical Robot Navigation (ORN) applications. This paper comes in a series of comparing M-EGF with the most common filters extensively used in ORN. Affine 2D Transformation-based Filter (A-2DF) is another familiar filter used for detecting outliers in AIM results and regards as the advanced version of C-2DF, where it can deal with any changing or distortion in image scales in X and Y directions. In this paper, M-EGF has been compared with A-2DF using the same system, tests, tracks and images and the same evaluation techniques used for comparing M-EGF to S-EGF and C-2DF in the related research papers. Tests show that A-2DF is similar to C-2DF in terms of its disability to filter AIM results in areas with open, narrow, and confused Depth Of Field (DOF). A-2DF is also limited in terms of finding out the 6 correct parameters for its mathematical model when AIM results includes a significant number of mismatching points. Difficult view angles has also a noted effect on the performance of A-2DF, but less than that of C-2DF due to its ability to work with different scales. In tests including limited DOF and low level of outliers, A-2DF has shown comparatively adequate outcomes, that is suitable for non-sensitive applications, in which the error does not entail any serious consequences. A-2DF is independent of the errors in Exterior Orientation Parameters (EOP) and Interior Orientation Elements (IOE) of cameras, where it is image points dependent estimation filter. Tests show that M-EGF is time- effective, efficient with any AIM findings, regardless DOF, capturing angle, outliers rate in observations, type of features and tracks. Unlike, A-2DF, tests show that the performance of M-EGF can affect by the quality of IOEs and EOPs, as a number of correctly matched points might be rejected, and this limitation is not related to the filter mathematical design but the professionalism of cameras calibration and EOP determination. in terms of processing time, A-2DF is considerably slow comparing with M-EGF due to its iterative and repeatability design, which makes it, unlike MEGF, unsuitable for real-time precise ORN applications. Keywords: Optical Robot Navigation, Epipolar Geometry, Automatic Image Matching, Co-planarity Equation, Real Time Applications, Affine 2D Transformation