Error Norm Research Articles

An unstructured finite-volume level set / front tracking method for two-phase flows with large density-ratios

We extend the unstructured LEvel set / froNT tracking (LENT) method [1,2] for handling two-phase flows with strongly different densities (high-density ratios) by providing the theoretical basis for the numerical consistency between the mass and momentum conservation in the collocated Finite Volume discretization of the single-field two-phase Navier-Stokes equations. Our analysis provides the theoretical basis for the mass conservation equation introduced by Ghods and Herrmann [3] and used in [4–8]. We use a mass flux that is consistent with mass conservation in the implicit Finite Volume discretization of the two-phase momentum convection term, and solve the single-field Navier-Stokes equations with our SAAMPLE segregated solution algorithm [2]. The proposed ρLENT method recovers exact numerical stability for the two-phase momentum advection of a spherical droplet with density ratios ρ−/ρ+∈[1,104]. Numerical stability is demonstrated for in terms of the relative L∞ velocity error norm, for density-ratios in the range of [1,104], dynamic viscosity-ratios in the range of [1,104] and very strong surface tension forces, for challenging mercury/air and water/air fluid pairings. In addition, the solver performs well in cases characterized by strong interaction between two phases, i.e., oscillating droplets and rising bubbles. The proposed ρLENT method1 is applicable to any other two-phase flow simulation method that discretizes the single-field two-phase Navier-Stokes Equations using the collocated unstructured Finite Volume Method but does not solve an advection equation for the phase indicator using a flux-based approach, by adding the proposed geometrical approximation of the mass flux and the auxiliary mass conservation equation to the solution algorithm.

A convergent finite element algorithm for mean curvature flow in arbitrary codimension

Optimal-order uniform-in-time H^1 -norm error estimates are given for semi- and full discretizations of mean curvature flow of surfaces in arbitrarily high codimension. The proposed and studied numerical method is based on a parabolic system coupling the surface flow to evolution equations for the mean curvature vector and for the orthogonal projection onto the tangent space. The algorithm uses evolving surface finite elements and linearly implicit backward difference formulas. This numerical method admits a convergence analysis in the case of finite elements of polynomial degree at least 2 and backward difference formulas of orders 2 to 5. Numerical experiments in codimension 2 illustrate and complement our theoretical results.

Application of information theory-based decision support system for high precision modeling of the length-weight relationship (LWR) for five marine shrimps from the northwestern Bay of Bengal

The study was conducted to develop an information theory-based decision support system to understand the variance distribution structure of the data so that a proper modeling approach could be implemented to explain the relationship between the body length and weight of shrimps. Based on biological reasoning, initially, a log-normal multiplicative error structure was assumed and therefore, a log-linearized model was applied. Secondly, the support for normal additive error structure was assessed by fitting a weighted nonlinear model with a power variance structure (wNLM) to address the heteroscedasticity in shrimp weight. The likelihood support for the error structures was ascertained by comparing the AICc of the two competing models. As the general cut-off criterion (ΔAICc>2.0) did not give conclusive evidence from the scrutiny of the probability density diagnostic plot of the residuals, an alternative model scaling criteria, i.e., Akaike weight (Aw) of 0.9 was used for model selection. The corresponding ΔAICc cut-off score of 4.2 was estimated by regressing the ΔAICc score of the competing model against the Aw scores of the best model. The competing models with ΔAICc > 4.2 were rejected and the alternate models with Aw ≥ 0.9 were selected for modeling the length–weight relationship. Both the models were observed to be well founded, as narrow differences in the root mean squared error (RMSE) were observed. A lower RMSE was almost always observed from wNLM despite a higher ΔAICc score, which indicates that RMSE may not be efficient in detecting the model overfitting issue. Contrary to popular belief, only 26.7% of the datasets exhibited a log-normal error structure, whereas, a normal error structure was evident in 33.3% of the datasets. Interestingly, 40.0% of the datasets showed data ambivalence (ΔAICc < 4.2) and therefore, an Akaike weighted model averaging was performed to reduce model uncertainty for the accurate estimation of model parameters and their confidence intervals.

Conformable non-polynomial spline method: A robust and accurate numerical technique

The article introduces a novel and developed a numerical technique that solves nonlinear time fractional differential equations. It combines finite difference and non-polynomial spline methods while using conformable descriptions of fractional derivatives. The strategy has proven to be useful to analyzing intricate datasets across various domains, including finance, science, and engineering. The applicability and validity of the approach are demonstrated through numerical examples. The study assessed the approach's stability using the Fourier method and found it to be unconditionally stable for a specific parameter range. The proposed method has been analyzed for convergence, and the analysis reveals that it exhibits a convergence order of six. The paper also includes graphs comparing the present solution to an analytical one. The method is tested with some examples of the model that have a wide range of applications in fields such as biology, ecology, and physics called the Burgers-Fisher equations and compared to previous approaches to demonstrate its effectiveness and applicability. The research evaluated the approach's accuracy and efficiency using the (L2 and L∞) norm errors. Also, the effects of time and fractional derivatives have been studied, and to the best of our knowledge, it has not been previously published in any academic or industry publication.

One-dimensional heat and advection-diffusion equation based on improvised cubic B-spline collocation, finite element method and Crank-Nicolson technique

In this study, we numerically investigate the solution of one-dimensional heat and advection-diffusion equation by employing improvised quartic order cubic B-spline using Crank-Nicolson method and finite element method (FEM). The numerical technique discretizes the spatial derivatives variables using Crank-Nicolson method and time derivatives using finite element method. It is demonstrated that the method is unconditionally stable with Von-Neumann process and accurate to convergence order Oh4+∆t2. The numerical results are simulated and error bounds are calculated using finite difference scheme on a piecewise uniform mesh. Finally, to support our claim, five test problems are considered and the experimental results also compared to existing methods using MATLAB as well as MATHEMATICA software tools. The error norms L2,L∞,root mean square (RMS) error with computational time and order of convergence are investigated for each example. The method is computationally efficient with less memory storage. The accuracy and efficiency of present scheme is measured in terms of absolute errors and also compared with analytical and published results in literature.

Early warning method of unsafe behavior accidents for offshore drilling operators based on eye-tracking trajectory

Despite the implementation of highly automated and intelligent equipment and facilities at the offshore drilling platform in recent years, drilling operators remain indispensable. During offshore drilling operations, the behavior of operators is susceptible to numerous factors including personal attributes, environmental interferences, and equipment feedback. When the factors that affect the operation of drilling change adversely, unsafe behavior of drilling operators can easily occur, resulting in offshore drilling accidents. As such, the early warning of unsafe behavior accidents of drilling operators is essential to reduce the occurrence of offshore drilling accidents. Firstly, attempts were made to use eye-tracking technology to record the eye movement data of unsafe behavior of drilling operators in real-time. In total, 42 drilling operation trainers had their eye-tracking trajectories recorded in real-time during normal tripping operations throughout the drilling process. A total of 252 eye-tracking trajectory data points were collected for six drilling behaviors, including normal operation, unskilled operation, fatigue operation, action errors, omitted actions and wrong order. Secondly, based on the IETTSM (Improved Eye-tracking Trajectory Sequence Method) algorithm, the eye-tracking trajectory of drilling operators was serialized. Thirdly, an evaluation model of drilling operator operation behavior was constructed based on the DLD (Damerau-Levenshtein Distance) algorithm. Fourthly, the IETTSM-DLD model was used to prompt abnormal behavior among drilling operators, which was accomplished by referencing the appropriate trajectory nodes and issuing early warnings of unsafe behavior based on the coupling of said trajectory nodes. The results reveal that the proposed IETTSM-DLD model can realize real-time prompting of abnormal operation of offshore drilling operators and early warning of accidents caused by unsafe behavior. The average early warning accuracy rate was found to be 91.9%, with an average early warning time of 120.97 s.

Stabilization-free HHO a posteriori error control

The known a posteriori error analysis of hybrid high-order methods treats the stabilization contribution as part of the error and as part of the error estimator for an efficient and reliable error control. This paper circumvents the stabilization contribution on simplicial meshes and arrives at a stabilization-free error analysis with an explicit residual-based a posteriori error estimator for adaptive mesh-refining as well as an equilibrium-based guaranteed upper error bound (GUB). Numerical evidence in a Poisson model problem supports that the GUB leads to realistic upper bounds for the displacement error in the piecewise energy norm. The adaptive mesh-refining algorithm associated to the explicit residual-based a posteriori error estimator recovers the optimal convergence rates in computational benchmarks.

Novel Method for Improving Motion Accuracy of a Large-Scale Industrial Robot to Perform Offline Teaching Based on Gaussian Process Regression

Industrial robots that can respond to the current needs for variable-type and variable-volume production and that can play a variety of roles such as processing and transporting with a single robot to reduce time and cost. If realized, these robots will help save space in factories and increase production efficiency. However, this requires high positioning accuracy of the robot. In this study, we analyze the motion accuracy of industrial robots and their compensation method to construct this system. Here, we use a laser tracker to measure the coordinates of the hand tip of the robot when the robot is stationary. Subsequently, the error amount in an arbitrary posture is predicted using a Gaussian process. Furthermore, Bayesian optimization is used to efficiently search for points where the positioning error norm is likely to be large, which is then compensated for by a feedback method. This method successfully reduced the time cost of the experiment to approximately one-tenth of that required in the previous study and achieved a correction of approximately 66 %. However, because this method alone does not perform an exhaustive measurement, it is unclear whether all the points predicted to have small errors are so small that they do not require correction. Therefore, future studies, we will aim to verify this issue by considering the time efficiency.

Smooth multi-patch scaled boundary isogeometric analysis for Kirchhoff–Love shells

In this work, a linear Kirchhoff–Love shell formulation in the framework of scaled boundary isogeometric analysis is presented that aims to provide a simple approach to trimming for NURBS-based shell analysis. To obtain a global C^1-regular test function space for the shell discretization, an inter-patch coupling is applied with adjusted basis functions in the vicinity of the scaling center to ensure the approximation ability. Doing so, the scaled boundary geometries are related to the concept of analysis-suitable G^1 parametrizations. This yields a coupling of patch boundaries in a strong sense that is restricted to G^1-smooth surfaces. The proposed approach is advantageous to trimmed geometries due to the incorporation of the trimming curve in the boundary representation that provides an exact representation in the planar domain. Since the coupling of star-shaped blocks is feasible, a mesh generation also for complex domains is implementable. The potential of the approach is demonstrated by several problems of untrimmed and trimmed geometries of Kirchhoff–Love shell analysis evaluated against error norms and displacements. Lastly, the applicability is highlighted in the analysis of a violin structure including arbitrarily shaped patches.

Higher-order error estimates for physics-informed neural networks approximating the primitive equations

Large-scale dynamics of the oceans and the atmosphere are governed by primitive equations (PEs). Due to the nonlinearity and nonlocality, the numerical study of the PEs is generally challenging. Neural networks have been shown to be a promising machine learning tool to tackle this challenge. In this work, we employ physics-informed neural networks (PINNs) to approximate the solutions to the PEs and study the error estimates. We first establish the higher-order regularity for the global solutions to the PEs with either full viscosity and diffusivity, or with only the horizontal ones. Such a result for the case with only the horizontal ones is new and required in the analysis under the PINNs framework. Then we prove the existence of two-layer tanh PINNs of which the corresponding training error can be arbitrarily small by taking the width of PINNs to be sufficiently wide, and the error between the true solution and its approximation can be arbitrarily small provided that the training error is small enough and the sample set is large enough. In particular, all the estimates are a priori, and our analysis includes higher-order (in spatial Sobolev norm) error estimates. Numerical results on prototype systems are presented to further illustrate the advantage of using the H^s norm during the training.

L 2 norm error estimates of BDF methods up to fifth-order for the phase field crystal model

Abstract The well-known backward difference formula (BDF) of the third, the fourth and the fifth orders are investigated for time integration of the phase field crystal model. By building up novel discrete gradient structures of the BDF-$\textrm{k}$ ($\textrm{k}=3,4,5$) formulas, we establish energy dissipation laws at the discrete levels and then obtain a priori solution estimates for the associated numerical schemes; however, we can not build any discrete energy dissipation law for the corresponding BDF-6 scheme because the BDF-6 formula itself does not have any discrete gradient structures. With the help of the discrete orthogonal convolution kernels and Young-type convolution inequalities, some concise $L^{2}$ norm error estimates (with respect to the starting data in the $L^{2}$ norm) are established via the discrete energy technique. To the best of our knowledge, this is the first time such type $L^{2}$ norm error estimates of non-A-stable BDF schemes are obtained for nonlinear parabolic equations. Numerical examples are presented to verify and support the theoretical analysis.

Balanced and energy norm error bounds for a spatial FEM with Crank-Nicolson and BDF2 time discretisation applied to a singularly perturbed reaction-diffusion problem

Balanced and energy norm error bounds for a spatial FEM with Crank-Nicolson and BDF2 time discretisation applied to a singularly perturbed reaction-diffusion problem

Generalized linear models with structured sparsity estimators

In this paper, we introduce structured sparsity estimators for use in Generalized Linear Models. Structured sparsity estimators in the least squares loss are introduced by Stucky and van de Geer (2018). Their proofs exclusively depend on their use of fixed design and normal errors. We extend their results to debiased structured sparsity estimators with Generalized Linear Model based loss through incorporating random design and non-sub Gaussian data. Structured sparsity estimation means that penalized loss functions with a possible sparsity structure in a norm. These norms include norms generated from convex cones.Our contributions are threefold: (1) We generalize the existing oracle inequality results in penalized Generalized Linear Models; (2) We provide a feasible weighted nodewise regression proof which generalizes the results in the literature; (3) We realize that norms used in feasible nodewise regression proofs should be weaker or equal to the norms in penalized Generalized Linear Model loss.

Study of generalized regularized long wave equation via septic Hermite collocation method with Crank–Nicolson and SSP-RK43 schemes to capture the various solitons

The paper aims to develop a powerful numerical technique for the generalized regularized long wave equation. The proposed technique is based on the orthogonal collocation on finite element method and septic Hermite interpolation polynomials as basis functions called septic Hermite collocation method (SHCM). These basis functions are C3 continuous, ensuring the continuity of dependent variable and its first three derivatives throughout the solution range. Two schemes viz one is implicit Crank–Nicolson scheme and other is Runge–Kutta method of four stages and third-order (SSP-RK43) are used for time discretization and SHCM is used for space discretization. The stability analysis of SHCM with Crank–Nicolson is performed and the scheme is found to be unconditionally stable. The stability of SHCM with SSP-RK43 by CFL conditions is also studied and the method is found to be conditionally stable. The proposed methods are applied to three problems involving single solitary waves, the interaction of two solitary waves, and the evolution of solitons via the Maxwellian initial condition. The formation and behavior of soliton waves are studied numerically. To demonstrate the robustness of proposed technique, error norms and the invariants of motion, i.e., mass, momentum, and energy are calculated. The results are in good agreement with the exact ones and found to be better than the numerical solutions available in the literature. The order of convergence along spatial and temporal directions has been calculated numerically. It is observed that the order of convergence of the explicit SSP-RK43 is higher than the implicit Crank–Nicolson and SHCM with SSP-RK43 provides better accuracy than the SHCM with Crank–Nicolson. The SHCM is capable of displaying solitary wave collision. The proposed technique is simple, fast, efficient, and easy to implement. It does not require unnecessary integration or calculation of weight functions as the case of other numerical methods for instance Galerkin methods, spectral methods, B-spline finite element methods etc.

Sensitivity and robustness of randomization test and F‐test in some experimental designs

AbstractThis work examined the sensitivity of randomization test and F‐test in completely randomized design, randomized complete block design and Latin square design in order to ascertain which is more robust in terms of power (sensitivity) of the test and type I error rate in cases with normal error, skewed error (Chi‐square and Exponential distributions), by varying the sample sizes. Using Monte Carlo simulation, response observations were generated for normal error, skewed error at various sample sizes for each of the experimental designs. The results showed that for the normal error, the power and type I error rate of the F‐test were slightly smaller and greater respectively in each of the designs than that of the randomization test. Furthermore, for the skewed error, the randomization test was more sensitive than the F‐test, and also, the type I error was smaller in each of the designs.

Joint Polarization-Space-Time Processing for Mainlobe Jamming via CP Decomposition

This paper deals with the problem of mainlobe jamming suppression for polarization sensitive phased-array radar system via a joint polarization-space-time processing method resorting to tensor decomposition. Firstly, a radar model accounting for one target and multiple mainlobe jammers in the far field is considered, which is equipped with a polarized transmitted antenna and a uniform linear dual-polarization receiving array. Then, exploiting the polarization-space-time characteristics, tensor CANDECOMP/PARAFAC (CP) decomposition is introduced to recast the received signal as a third order tensor consisting of three factor matrices. Further, the target echo and the jamming signal can be obtained via minimizing the Frobenius norm of the approximation error between the tensor and the outer product of the factor matrices. Finally, the effectiveness of the proposed method by numerical simulation is verified, showing its capability to combat various types of mainlobe jamming as well as their compound forms compared with the art competing method.

Modified Regularized Long Wave Denkleminin Nümerik Çözümü İçin Kuintik Trigonometrik B-spline Algoritması

This study introduces a new numerical algorithm for solving the modified regularized long-wave (MRLW) equation. To discretize the spatial variables and their derivatives, the collocation technique with quintic trigonometric B-spline functions is utilized and for the temporal derivative, the Adam's Moulton scheme is implemented. The performance and efficiency of the computational algorithm is tested on sample problem including the motion of single solitary wave. The error norm and three conservation constants are computed and compared with some of those available in the literature. The computed results verify that the suggested algorithm has the advantage in obtaining a highly accurate approximate solution of the MRLW equation as compared to the existing methods. The advantage of the method is that it is easy to implement and requires the low computational cost.

Accuracy of a Recent Regularized Nuclear Potential.

F. Gygi recently suggested an analytic, norm-conserving, regularized nuclear potential to enable all-electron plane-wave calculations [Gygi J. Chem. Theory Comput. 2023, 19, 1300-1309.]. This potential V(r) is determined by inverting the Schrödinger equation for the wave function Ansatz ϕ(r) = exp[-h(r)]/√π with h(r) = r erf(ar) + b exp(-a2r2), where a and b are parameters. Gygi fixes b by demanding ϕ to be normalized, with the value b(a) depending on the strength of the regularization controlled by a. We begin this work by re-examining the determination of b(a) and find that the original 10-decimal tabulations of Gygi are only correct to 5 decimals, leading to normalization errors in the order of 10-10. In contrast, we show that a simple 100-point radial quadrature scheme not only ensures at least 10 correct decimals of b but also leads to machine-precision level satisfaction of the normalization condition. Moreover, we extend Gygi's plane-wave study by examining the accuracy of V(r) with high-precision finite element calculations with Hartree-Fock and LDA, GGA, and meta-GGA functionals on first- to fifth-period atoms. We find that although the convergence of the total energy appears slow in the regularization parameter a, orbital energies and shapes are indeed reproduced accurately by the regularized potential even with relatively small values of a, as compared to results obtained with a point nucleus. The accuracy of the potential is furthermore studied with s-d excitation energies of Sc-Cu as well as ionization potentials of He-Kr, which are found to converge to sub-meV precision with a = 4. The findings of this work are in full support of Gygi's contribution, indicating that all-electron plane-wave calculations can be accurately performed with the regularized nuclear potential.

NONIC B-SPLINE APPROACH FOR ADVECTION DIFFUSION EQUATION

In this paper, a highly accurate method is introduced to achieve the numerical solution of the advection diffusion equation (ADE). This approach contains collocation technique based on nonic B-spline functions in the spatial-domain discretization and Adams Moulton scheme in the temporal-domain discretization. Two test problems are studied to validate effectiveness of the new presented method and efficiency of the approximate results are tested by calculating rate of temporal-convergence and error norm 𝐿∞ for the suggested method. The obtained numerical results are compared in the tables by the other available studies in literature and it is observed that a better approximate solution is provided than the existing methods.

NUMERICAL SOLUTIONS OF REACTION-DIFFUSION EQUATION SYSTEMS WITH TRIGONOMETRIC QUINTIC B-SPLINE COLLOCATION ALGORITHM

In this study, trigonometric quintic B-spline collocation method is constructed for computing numerical solutions of the reaction-diffusion system (RDS). Schnakenberg, Gray-Scott and Brusselator models are special cases of reaction-diffusion systems considered as examples in this paper. Crank-Nicolson formulae is used for the time discretization of the generalized RDS and the nonlinear terms in time-discretized form of RDS are linearized using the Taylor expansion. The fully integration of the generalized system is carried out using the collocation method based on the trigonometric quintic B-splines. The method is tested on different problems to illustrate the accuracy. The error norms are calculated for the linear problem whereas the relative error is given for nonlinear problems. Both simple and easy B-spline algorithms are illustrated to give the solutions of RDS and also the graphical representation of the efficient solutions are presented for the nonlinear RDSs. Combination of the quintic B-splines and the collocation method is shown to present numerical solutions of the RDS successfully. With the presented method, it is possible to get approximate solutions as well as their derivatives up to an order of four on the problem domain.