Reduction Of Numerical Errors Research Articles

Simplified implicit finite-difference method of spatial derivative using explicit schemes with optimized constant coefficients based on L1 norm

The explicit finite-difference (EFD) method is widely used in numerical simulation of seismic wave propagation to approximate spatial derivatives. However, the traditional and optimized high-order EFD methods suffer from the saturation effect, which seriously restricts the improvement of numerical accuracy. In contrast, the implicit FD (IFD) method approximates the spatial derivatives in the form of rational functions and thus can obtain much higher numerical accuracy with relatively low orders; however, its computational cost is expensive due to the need to invert a multidiagonal matrix. We derive an explicit strategy for the IFD method to reduce the computational cost by constructing the IFD method with the discrete Fourier matrix; then, we transform the inversion of the multidiagonal matrix into an explicit matrix multiplication; next, we construct an objective function based on the [Formula: see text] norm to reduce approximation error of the IFD method. This explicit strategy of the IFD method can avoid inverting the multidiagonal matrix, thus improving the computational efficiency. This constant coefficient optimization method reduces the approximation error in the medium-wavenumber range at the cost of tolerable deviation (smaller than 0.0001) in the low-wavenumber range. For the 2D Marmousi model, the root-mean-square error of the numerical results obtained by this method is one-fifth that of the traditional IFD method with the same order (i.e., 5/3) and one-third that of the traditional EFD method with much higher orders (i.e., 72). The significant reduction of numerical error makes the developed method promising for numerical simulation in large-scale models, especially for long-time simulations.

Accurate Quaternion Polar Harmonic Transform for Color Image Analysis

Polar harmonic transforms (PHTs) have been applied in pattern recognition and image analysis. But the current computational framework of PHTs has two main demerits. First, some significant color information may be lost during color image processing in conventional methods because they are based on RGB decomposition or graying. Second, PHTs are influenced by geometric errors and numerical integration errors, which can be seen from image reconstruction errors. This paper presents a novel computational framework of quaternion polar harmonic transforms (QPHTs), namely, accurate QPHTs (AQPHTs). First, to holistically handle color images, quaternion-based PHTs are introduced by using the algebra of quaternions. Second, the Gaussian numerical integration is adopted for geometric and numerical error reduction. When compared with CNNs (convolutional neural networks)-based methods (i.e., VGG16) on the Oxford5K dataset, our AQPHT achieves better performance of scaling invariant representation. Moreover, when evaluated on standard image retrieval benchmarks, our AQPHT using smaller dimension of feature vector achieves comparable results with CNNs-based methods and outperforms the hand craft-based methods by 9.6% w.r.t mAP on the Holidays dataset.

Mixed finite elements for convection-coupled phase-change in enthalpy form: Open software verified and applied to 2D benchmarks

Melting and solidification processes are often affected by natural convection of the liquid, posing a multi-physics problem involving fluid flow, convective and diffusive heat transfer, and phase-change reactions. Enthalpy methods formulate this convection-coupled phase-change problem on a single computational domain. The governing equations can be solved accurately with a monolithic approach using mixed finite elements and Newton’s method. Previously, the monolithic approach has relied on adaptive mesh refinement to regularize local nonlinearities at phase interfaces. This contribution instead separates mesh refinement from nonlinear problem regularization and provides a continuation procedure which robustly obtains accurate solutions on the tested 2D uniform meshes. A flexible and extensible open source implementation is provided. The code is formally verified to accurately solve the governing equations in time and in 2D space, and convergence rates are shown. Two benchmark simulations are presented in detail with comparison to experimental data sets and corresponding results from the literature, one for the melting of octadecane and another for the freezing of water. Sensitivities to key numerical parameters are presented. For the case of freezing water, effective reduction of numerical errors from these key parameters is successfully demonstrated. Two more simulations are briefly presented, one for melting at a higher Rayleigh number and one for melting gallium.

Causal and Passive Parameterization of S-Parameters Using Neural Networks

Neural networks (NNs) are widely used to create parametric models of S-parameters for various components in electronic systems. The focus of deriving these models has so far been numerical error reduction between the NN-generated S-parameters and the data source. However, this is not sufficient when creating such NNs since it does not guarantee predicted S-parameters to be physically consistent, i.e., passive and causal, which restricts their use cases. This article, therefore, proposes a causality enforcement layer (CEL) and passivity enforcement layer (PEL) that can be used in NNs, which ensures that NN-predicted S-parameters are of a passive and causal system. To achieve this, we utilize Kramers–Kronig relations and singular value properties of S-parameters during the training stage with the purpose of learning a physically consistent representation. This enables end-to-end training where no postprocessing is required to ensure physical consistency. We demonstrate the effectiveness of the presented approach for three different design applications, where the goal is to predict S-parameters from dc to 100 GHz. The results show that when NNs are trained using CEL and PEL, the predicted S-parameters are characterized as 100.0% causal and passive while having the same level of accuracy as NNs that solely focus on error minimization.

A finite volume scheme with improved well modeling in subsurface flow simulation

We present the latest enhancement of the nonlinear monotone finite volume method for the near-well regions. The original nonlinear method is applicable for diffusion, advection-diffusion, and multiphase flow model equations with full anisotropic discontinuous permeability tensors on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which reduces to the conventional two-point flux approximation (TPFA) on cubic meshes but has much better accuracy for the general case of non-orthogonal grids and anisotropic media. The latest modification of the nonlinear method takes into account the nonlinear (e.g., logarithmic) singularity of the pressure in the near-well region and introduces a correction to improve accuracy of the pressure and the flux calculation. In this paper, we consider a linear version of the nonlinear method waiving its monotonicity for sake of better accuracy. The new method is generalized for anisotropic media, polyhedral grids and nontrivial cases such as slanted, partially perforated wells or wells shifted from the cell center. Numerical experiments show noticeable reduction of numerical errors compared to the original monotone nonlinear FV scheme with the conventional Peaceman well model or with the given analytical well rate.

Repeated Richardson extrapolation applied to the two-dimensional Laplace equation using triangular and square grids

The focus of this work is to verify the efficiency of the Repeated Richardson Extrapolation (RRE) to reduce the discretization error in a triangular grid and to compare the result to the one obtained for a square grid for the two-dimensional Laplace equation. Two different geometries were employed: the first one, a unitary square domain, was discretized into a square or triangular grid; and the second, a half square triangle, was discretized into a triangular grid. The methodology employed used the following conditions: the finite volume method, uniform grids, second-order accurate approximations, several variables of interest, Dirichlet boundary conditions, grids with up to 16,777,216 nodes for the square domain and up to 2097,152 nodes for the half square triangle domain, multigrid method, double precision, up to eleven Richardson extrapolations for the first domain and up to ten Richardson extrapolations for the second domain. It was verified that (1) RRE is efficient in reducing the discretization error in a triangular grid, achieving an effective order of approximately 11 for all the variables of interest for the first geometry; (2) for the same number of nodes and with or without RRE, the discretization error is smaller in a square grid than in a triangular grid; and (3) the magnitude of the numerical error reduction depends on, among other factors, the variable of interest and the domain geometry.

On the Error Reduction of a Simple Symplectic Integrator

Abstract An analysis of truncation errors of the simplest symplectic integrator is revisited. We performed an analytical error estimate of this integrator within the framework of a canonical perturbation method, taking the two-body system as an example. Our study was motivated by the desire to remove any confusion in the derivation of this issue in one standard literature along this line of research. We repeated the derivation, and examined how a secular numerical error arises in symplectic integration. We showed that it is possible to substantially eliminate a secular numerical error by choosing appropriate initial conditions, as previous studies have shown. On the other hand, depending on boundary parameter values, such as the eccentricity in the two-body system, we could see that it is sometimes impossible to reduce the secular numerical error by changing the starting conditions. We also numerically demonstrated this reduction, or lack of reduction of numerical errors by taking a restricted three-body system as an example. We confirmed that the so-called “iterative start” of symplectic integrators works in certain systems.

A novel continuum model for the reduction of numerical errors of finite-element analysis

The authors developed a novel numerical formulation that reduces the computational expense of solving linear and non-linear constitutive models. A finite-element constitutive law followed by a novel continuum model is discussed. The proposed approach is employed to reduce numerical errors obtained from the standard finite-element method. The continuum model is based on the Lagrange multipliers strategy along with the finite-difference method and a least-square algorithm. It is crucial to this approach that the components of either the strain or the stress tensor are known at the nodes of the finite-element mesh. The proposed model is independent of element topology which can be strongly and arbitrarily distorted.

Reduction of Numerical Errors of Time-Stepping Finite Element Analysis for Dynamic Simulation of Electric Machines

The time-stepping finite element method (TS-FEM) can couple the magnetic field, electric circuit and mechanical torque balance equations together and has been widely used to simulate the dynamic characteristics of electric machines. Despite its heavy computational burden, the accuracy of TS-FEM is still limited by a host of practical constraints. Also, it is difficult to accurately model the sliding surface of the stator mesh and the rotor mesh in rotating electric machines. In this paper, a curvilinear element to approximate the curved geometry of sliding surface is presented to increase the computational accuracy. To reduce the numerical error of the derivative quantities, a modified nonlinear iterative formulation is adopted. To reduce the computing time, an adaptive time step size algorithm is also proposed. The proposed strategy and algorithm are verified by the FEM examples as reported in this paper.

Reduction of numerical errors in frequency dependent ADI-FDTD

A new method modifying the frequency dependent alternating direction implicit finite difference time domain (FD-ADI-FDTD) is presented. The method can improve the accuracy of FD-ADI-FDTD without significant increase of computational costs. The proposed method is validated by numerical tests.

준해석적 비선형 설계민감도를 위한 보정변위하중법

Three methods for design sensitivity such as numerical differentiation, analytical method and semianalytical method have been developed for the last three decades. Although analytical design sensitivity analysis is exact, it is hard to implement for practical design problems. Therefore, numerical method such as finite difference method is widely used to simply obtain the design sensitivity in most cases. The numerical differentiation is sufficiently accurate and reliable for most linear problems. However, it turns out that the numerical differentiation is inefficient and inaccurate because its computational cost depends on the number of design variables and large numerical errors can be included especially in nonlinear design sensitivity analysis. Thus semi-analytical method is more suitable for complicated design problems. Moreover semianalytical method is easy to be performed in design procedure, which can be coupled with an analysis solver such as commercial finite element package. In this paper, implementation procedure for the semi-analytical design sensitivity analysis outside of the commercial finite element package is studied and computational technique is proposed, which evaluates the pseudo-load for design sensitivity analysis easily by using the design variation of corresponding internal nodal forces. Errors in semi-analytical design sensitivity analysis are examined and numerical examples are illustrated to confirm the reduction of numerical error considerably.

Combined first-principles calculation and neural-network correction approach for heat of formation

Despite their success, the results of first-principles quantum mechanical calculations contain inherent numerical errors caused by various intrinsic approximations. We propose here a neural-network-based algorithm to greatly reduce these inherent errors. As a demonstration, this combined quantum mechanical calculation and neural-network correction approach is applied to the evaluation of standard heat of formation ΔfH⊖ for 180 small- to medium-sized organic molecules at 298 K. A dramatic reduction of numerical errors is clearly shown with systematic deviation being eliminated. For example, the root-mean-square deviation of the calculated ΔfH⊖ for the 180 molecules is reduced from 21.4 to 3.1 kcal mol−1 for B3LYP/6-311+G(d,p) and from 12.0 to 3.3 kcal mol−1 for B3LYP/6-311+G(3df,2p) before and after the neural-network correction.

Analysis of circular waveguide filter using enhanced FDTD

AbstractThe finite difference time domain (FDTD) analysis of iris‐coupled circular waveguide filters presents several challenges because they contain curved metal surfaces and electrically small slots. In this contribution it is shown that by enhancing the basic FDTD algorithm in three ways, filters of this type may be accurately characterized without the need for a highly dense mesh. The enhancements are: (i) an existing locally conformal mesh algorithm to model the curved metal surface, (ii) novel empirical correction factors to account for the singular field behaviour near the slots and their exact position relative to the FDTD cells and (iii) compensation for the numerical dispersion in the FDTD algorithm. The calculation of the correction factors and the implementation of the technique is discussed and the extent to which each of these three enhancements contributes to the reduction of numerical error is shown. In addition, it is shown that a novel combination (i) and (ii) leads to a robust and effective method of treating metal edges which are offset from the edges of the cells. The improvement obtained for a complete filter is demonstrated by comparison to measured results. Copyright © 2002 John Wiley & Sons, Ltd.

Viscous airfoil computations using adaptive grid redistribution

A quantitative study of the numerical error reduction due to adaptive grid redistribution applied to Navier-Stokes computations of steady airfoil flows is presented. A grid redistribution procedure is described and applied to a range of flow cases including attached and separated flows at subsonic and transonic speeds. The error reduction resulting from grid adaptation is sensitive to the adaptive grid generation parameters used, especially those affecting the smoothness of the adapted grid. A suitable set of adaptive grid generation parameters is determined by comparison of adaptive grid solutions with grid-independent solutions. With the given parameters, adaptive grid redistribution is shown to result in an effective reduction in numerical error and improved resolution of flow features.

Parallel, self-organizing, hierarchical neural networks with forward-backward training

A forward-backward training algorithm for parallel, self-organizing hierarchical neural networks (PSHNNs) is described. Using linear algebra, it is shown that the forward-backward training of ann-stage PSHNN until convergence is equivalent to the pseudo-inverse solution for a single, total network designed in the least-squares sense with the total input vector consisting of the actual input vector and its additional nonlinear transformations. These results are also valid when a single long input vector is partitioned into smaller length vectors. A number of advantages achieved are: small modules for easy and fast learning, parallel implementation of small modules during testing, faster convergence rate, better numerical error-reduction, and suitability for learning input nonlinear transformations by other neural networks. The backpropagation (BP) algorithm is proposed for learning input nonlinearitics. Better performance in terms of deeper minimum of the error function and faster convergence rate is achieved when a single BP network is replaced by a PSHNN of equal complexity in which each stage is a BP network of smaller complexity than the single BP network.

Numerical error reduction by a new skew upwind scheme.

Numerical viscosity is evaluated for a difference scheme of high Reynolds number flow and a new skew upwind scheme, having the following characteristics, is proposed : (1) The value at the center of the calculational mesh is assumed to flow out at the outflow boundary of the mesh, and the volume weighted value of the upwind points is used for the inlet boundary. (2) The net transported quantity varies due to the flow direction, which is obtained by the remainder of the outflow from the inflow quantity. (3) Numerical oscillation is suppressed and the computational resolution is superior to that of the upwind scheme for a steep temperature field with pure convection. (4) The direction of the mainflow agrees with the experimental value which was obtained by Tanaka et al..

The Formulation of Spray Combustion Models: Resolution Compared to Droplet Spacing

Formulations of the governing equations for burning air/fuel spray mixtures are discussed. The subsystem of equations describing liquid properties is formulated in three ways: Eulerian description, Lagrangian description, and probabilistic or distribution function description. The relationships among these approaches and the relative advantages and disadvantages are discussed. The reduction of numerical error, the ability to resolve multivalued solutions, and the ability to achieve resolution a scale smaller than droplet spacing lead to a preference of the Lagrangian method over the Eulerian method. However, when resolution is desired on a scale smaller than the average droplet spacing, the location of each droplet is known only in a statistical manner. The advantages of a probabilistic formulation in this case are evaluated.

An Improvement of the PALLAS Discrete-Ordinates Transport Code

An improvement in the approximation of the spatial distribution of the source term in the PALLAS discrete-ordinates code by a combination of linear and exponential functions results in a reduction of numerical errors compared with those due to the use of only a linear approximation based on a direct integration method. The results obtained by the improved approximation for one-dimensional radiation transport calculations are compared with those by the PALLAS code with the original linear approximation and by the S/sub N/ code ANISN. For further verification, this improved approximation is applied to two-dimensional neutron transport calculations, in which numerical errors due to the spatial mesh sizes used in the calculations are studied. As a result, it is found that the use of the improved approximation gives reasonable solutions with small numerical errors, irrespective of the spatial mesh size in both one- and two-dimensional PALLAS calculations of radiation transport.

Convective difference schemes and Hermite interpolation

AbstractA new method for numerically solving the convection equation is presented. The reduction of numerical errors is achieved by introducing an auxiliary variable and solving an auxiliary equation for the latter. The method is based on a Hermite interpolation of the convected quantity. Three explicit numerical versions of the method are investigated and compared with some of the most widely used schemes, and it is shown that the new method gives better physical results. Arguments are given that the usual mathematical series expansions do not work sufficiently well in that context. Instead of using linear stability analyses the quality of the method is investigated by numerical experiments.