An Interpolation Method for Karhunen-Loève Expansion of Random Field Discretization

In the context of global mean square error concerning the number of random variables in the representation,the Karhunen–Loève(KL)expansion is the optimal series expansion method for random field discretization.The computational efficiency and accuracy of the KL expansion are contingent upon the accurate resolution of the Fredholm integral eigenvalue problem(IEVP).The paper proposes an interpolation method based on different interpolation basis functions such as moving least squares(MLS),least squares(LS),and finite element method(FEM)to solve the IEVP.Compared with the Galerkin method based on finite element or Legendre polynomials,the main advantage of the interpolation method is that,in the calculation of eigenvalues and eigenfunctions in one-dimensional random fields,the integral matrix containing covariance function only requires a single integral,which is less than a two-folded integral by the Galerkin method.The effectiveness and computational efficiency of the proposed interpolation method are verified through various one-dimensional examples.Furthermore,based on theKL expansion and polynomial chaos expansion,the stochastic analysis of two-dimensional regular and irregular domains is conducted,and the basis function of the extended finite element method(XFEM)is introduced as the interpolation basis function in two-dimensional irregular domains to solve the IEVP.

Enhancing Feature Discretization in Alarm and Fire Detection Systems Using Probabilistic Inference Models

Sensors for fire alarms require a high level of predictive variables to ensure accurate detection, injury prevention, and loss prevention. Bayesian networks can aid in enhancing early fire detection capabilities and reducing the frequency of erroneous fire alerts, thereby enhancing the effectiveness of numerous safety monitoring systems. This research explores the development of optimized probabilistic graphic models for the discretization thresholds of alarm system predictor variables. The study presents a statistical model framework that increases the efficacy of fire detection by predicting the discretization thresholds of alarm system predictor variable fluctuations used to detect the onset of fire. The work applies the Bayesian networks and probabilistic visual models to reveal the specific characteristics required to cope with fire detection strategies and patterns. The adopted methodology utilizes a combination of prior knowledge and statistical data to draw conclusions from observations. Utilizing domain knowledge to compute conditional dependencies between network variables enabled predictions to be made through the application of specialized analytical and simulation techniques.

A discrete Boltzmann model with symmetric velocity discretization for compressible flow

A discrete Boltzmann model(DBM) with symmetric velocity discretization is constructed for compressible systems with an adjustable specific heat ratio in the external force field. The proposed two-dimensional(2D) nine-velocity scheme has better spatial symmetry and numerical accuracy than the discretized velocity model in literature [Acta Aerodyn. Sin.40 98108(2022)] and owns higher computational efficiency than the one in literature [Phys. Rev. E 99 012142(2019)].In addition, the matrix inversion method is adopted to calculate the discrete equilibrium distribution function and force term, both of which satisfy nine independent kinetic moment relations. Moreover, the DBM could be used to study a few thermodynamic nonequilibrium effects beyond the Euler equations that are recovered from the kinetic model in the hydrodynamic limit via the Chapman–Enskog expansion. Finally, the present method is verified through typical numerical simulations, including the free-falling process, Sod’s shock tube, sound wave, compressible Rayleigh–Taylor instability,and translational motion of a 2D fluid system.

Pseudo-static/dynamic solutions of required reinforcement force for steep slopes using discretization-based kinematic analysis

This paper presents a procedure for assessing the reinforcement force of geosynthetics required for maintaining dynamic stability of a steep soil slope. Such a procedure is achieved with the use of the discretization technique and kinematic analysis of plasticity theory, i.e. discretization-based kinematic analysis. The discretization technique allows discretization of the analyzed slope into various components and generation of a kinematically admissible failure mechanism based on an associated flow rule.Accordingly, variations in soil properties including soil cohesion, internal friction angle and unit weight are accounted for with ease, while the conventional kinematic analysis fails to consider the changes in soil properties. The spatialetemporal effects of dynamic accelerations represented by primary and shear seismic waves are considered using the pseudo-dynamic approach. In the presence of geosynthetic reinforcement, tensile failure is discussed providing that the geosynthetics are installed with sufficient length. Equating the total rates of work done by external forces to the internal rates of work yields the upper bound solution of required reinforcement force, below which slopes fail. The reinforcement force is sought by optimizing the objective function with regard to independent variables, and presented in a normalized form. Pseudo-static analysis is a special case and hence readily transformed from pseudodynamic analysis. Comparisons of the pseudo-static/dynamic solutions calculated in this study are highlighted. Although the pseudo-static approach yields a conservative solution, its ability to give a reasonable result is substantiated for steep slopes. In order to provide a more meaningful solution to a stability analysis, the pseudo-dynamic approach is recommended due to considerations of spatial etemporal effect of earthquake input.

Half-global discretization algorithm based on rough set theory

It is being widely studied how to extract knowledge from a decision table based on rough set theory. The novel problem is how to discretize a decision table having continuous attribute. In order to obtain more reasonable discretization results, a discretization algorithm is proposed, which arranges half-global discretization based on the correlational coefficient of each continuous attribute while considering the uniqueness of rough set theory. When choosing heuristic information, stability is combined with rough entropy. In terms of stability, the possibility of classifying objects belonging to certain sub-interval of a given attribute into neighbor sub-intervals is minimized. By doing this, rational discrete intervals can be determined. Rough entropy is employed to decide the optimal cut-points while guaranteeing the consistency of the decision table after discretization. Thought of this algorithm is elaborated through Iris data and then some experiments by comparing outcomes of four discritized datasets are also given, which are calculated by the proposed algorithm and four other typical algorithras for discritization respectively. After that, classification rules are deduced and summarized through rough set based classifiers. Results show that the proposed discretization algorithm is able to generate optimal classification accuracy while minimizing the number of discrete intervals. It displays superiority especially when dealing with a decision table having a large attribute number.

A novel index to evaluate discretization methods: A case study of flood susceptibility assessment based on random forest

The selection of a suitable discretization method(DM) to discretize spatially continuous variables(SCVs)is critical in ML-based natural hazard susceptibility assessment. However, few studies start to consider the influence due to the selected DMs and how to efficiently select a suitable DM for each SCV. These issues were well addressed in this study. The information loss rate(ILR), an index based on the information entropy, seems can be used to select optimal DM for each SCV. However, the ILR fails to show the actual influence of discretization because such index only considers the total amount of information of the discretized variables departing from the original SCV. Facing this issue, we propose an index, information change rate(ICR), that focuses on the changed amount of information due to the discretization based on each cell, enabling the identification of the optimal DM. We develop a case study with Random Forest(training/testing ratio of 7 : 3) to assess flood susceptibility in Wanan County, China.The area under the curve-based and susceptibility maps-based approaches were presented to compare the ILR and ICR. The results show the ICR-based optimal DMs are more rational than the ILR-based ones in both cases. Moreover, we observed the ILR values are unnaturally small(<1%), whereas the ICR values are obviously more in line with general recognition(usually 10%–30%). The above results all demonstrate the superiority of the ICR. We consider this study fills up the existing research gaps, improving the MLbased natural hazard susceptibility assessments.

New method for discretization of continuous attributes in rough set theory

A new method for discretization of continuous attributes is put forward to overcome the limitation of the traditional rough sets, which cannot deal with continuous attributes.The method is based on an improved algorithm to produce candidate cut points and an algorithm of reduction based on variable precision rough information entropy. With the guarantee of consistency of decision system, the method can reduce the number of cut points and im- prove efficiency of reduction. Adopting variable precision rough information entropy as measure criterion, it has a good tolerance to noise. Experiments show that the algorithm yields satisfying reduction results.

ON GALERKIN DISCRETIZATION OF AXIALLY MOVING NONLINEAR STRINGS

A computational technique is proposed for the Galerkin discretization of axially moving strings with geometric nonlinearity. The Galerkin discretization is based on the eigenfunctions of stationary strings. The discretized equations are simplified by regrouping nonlinear terms to reduce the computation work. The scheme can be easily implemented in the practical programming. Numerical results show the effectiveness of the technique. The results also highlight the feature of Galerkin＇s discretization of gyroscopic continua that the term number in Galerkin＇s discretization should be even. The technique is generalized from elastic strings to viscoelastic strings.

Nonrecursive residual Monte Carlo method for S_(N) transport discretization error estimation

In this paper,we present a nonrecursive residual Monte Carlo method for estimating discretization errors associated with the S_(N) transport solution to radiation transport problems.Although this technique is general,we applied it to the mono-energetic 1-D S_(N) equation with linear-discontinuous finite element method spatial discretization as a demonstration of the theory for the purpose of this study.Two angular flux representations:conforming and simplified representations were considered in this analysis,and the results were compared.It is shown that the simplified representation dramatically reduces the memory footprint and computational complexity of residual source generation and sampling while accurately capturing the error associated with certain types of responses.

Short-duration transient visual evoked potentials and color reflectivity discretization analysis in glaucoma patients and suspects

AIM： To evaluate the use of short-duration transient visual evoked potentials（VEP） and color reflectivity discretization analysis（CORDA） in glaucomatous eyes,eyes suspected of having glaucoma,and healthy eyes.METHODS： The study included 136 eyes from 136 subjects： 49 eyes with glaucoma,45 glaucoma suspect eyes,and 42 healthy eyes.Subjects underwent Humphrey visual field（VF） testing,VEP testing,as well as peripapillary retinal nerve fiber layer optical coherence tomography imaging studies with post-acquisition CORDA applied.Statistical analysis was performed using means and ranges,ANOVA,post-hoc comparisons using Turkey＇s adjustment,Fisher＇s Exact test,area under the curve,and Spearman correlation coefficients.RESULTS： Parameters from VEP and CORDA correlated significantly with VF mean deviation（MD）（P〈0.05）.In distinguishing glaucomatous eyes from controls,VEP demonstrated area under the curve（AUC） values of 0.64-0.75 for amplitude and 0.67-0.81 for latency.The CORDA HR1 parameter was highly discriminative for glaucomatous eyes vs controls（AUC=0.94）.CONCLUSION： Significant correlations are found between MD and parameters of short-duration transient VEP and CORDA,diagnostic modalities which warrant further consideration in identifying glaucoma characteristics.

Discretization of Preisach hysteresis model

In order to reduce the partial derivative errors in Preisach hysteresis model caused by inaccurate experimental data,the concept and correlative method of discretization of Preisach hysteresis model are proposed,the essential of which is to centralize the distribution density of Preisach hysteresis model in local region as an integral,which is defined as the weight of a certain point in that region.For the input composed of an ascending segment and a descending segment,a method to determine the initial weights together with an additional method to determine present weights is given according to the number of input ascending segments.If the number of input ascending segments increases,the weights of the corresponding points in updating rectangle are updated by adding the initial weights of corresponding points.A prominent advantage of discrete Preisach hysteresis model is its memory efficiency.Another advantage of discrete Preisach hysteresis model is that there is no function in the model,and thus,it can be expediently operated using a computer.By generalizing the above updating rectangle method to the continuous Preisach hysteresis model,identification method of distribution density can be given as well.

Lifetime estimation of IGBT module using square-wave loss discretization and power cycling test

The insulated gate bipolar transistor(IGBT)module is one of the most age-affected components in the switch power supply, and its reliability prediction is conducive to timely troubleshooting and reduction in safety risks and unnecessary costs. The pulsed current pattern of the accelerator power supply is different from other converter applications;therefore, this study proposed a lifetime estimation method for IGBT modules in pulsed power supplies for accelerator magnets. The proposed methodology was based on junction temperature calculations using square-wave loss discretization and thermal modeling.Comparison results showed that the junction temperature error between the simulation and IR measurements was less than 3%. An AC power cycling test under real pulsed power supply applications was performed via offline wearout monitoring of the tested power IGBT module. After combining the IGBT4 PC curve and fitting the test results,a simple corrected lifetime model was developed to quantitatively evaluate the lifetime of the IGBT module,which can be employed for the accelerator pulsed power supply in engineering. This method can be applied to other IGBT modules and pulsed power supplies.

A Discretization Method for Nonlinear Delayed Non-affine Systems

The input time delay is always existent in the practical systems. Analysis of the delay phenomenon in a continuous-time domain is sophisticated. It is appropriate to obtain its corresponding discrete-time model for implementation via digital computer. This paper proposes a new discretization method for calculating a sampled-data representation of nonlinear time-delayed non-affine systems. The proposed scheme provides a finite-dimensional representation for nonlinear systems with non-a^ne time-delayed input enabling existing nonlinear controller design techniques to be applied to them. The performance of the proposed discretization procedure is evaluated by using a nonlinear system with non-affine time-delayed input. For this nonlinear system, various time delay values are considered.

Method of Precision Control of 3-D Solid Discretization Based on the Second-Order Osculating Surface

A discretization precision control method based on the second order osculating surface is proposed. The discretization precision of 3 D solid is controlled according to the error between the discrete solid surface and its second order osculating surface. The global maximal error has been gotten after analyzing all the extremums of the error function. It can be used in controlling and optimizing the discretization precision of 3 D solid in computer 3 D modeling and NC milling path generation.

NUMERICAL SOLUTION OF A SINGULARLY PERTURBED ELLIPTIC-HYPERBOLIC PARTIAL DIFFERENTIAL EQUATION ON A NONUNIFORM DISCRETIZATION MESH

In this paper, we consider the upwind difference scheme for singular perturbation problem (1.1). On a special discretization mesh, it is proved that the solution of the upwind difference scheme is first order convergent, uniformly in the small parameter e , to the solution of problem (1.1). Numerical results are finally provided.

SMOOTHNESS OF INERTIAL MANIFOLD UNDER TIME DISCRETIZATION

In this paper we discuss the smoothness of inertial manifolds under time discretization. By the fibre contract principle, see Section 4, we obtain a sufficient condition for C-k(k greater than or equal to 1) inertial manifolds. In view of the numerical computation for dissipative nonlinear evolution equations, it is more important to consider the discretized case than continuous case([3]).

A Principal Theorem of Normal Discretization Schemes for Operator Equations of the First Kind

The authors announce a newly-proved theorem of theirs. This theorem is of principal significance to numerical computation of operator equations of the first kind.

Numerical Solution of a Class of Nonlinear Optimal Control Problems Using Linearization and Discretization

In this paper, a new approach using linear combination property of intervals and discretization is proposed to solve a class of nonlinear optimal control problems, containing a nonlinear system and linear functional, in three phases. In the first phase, using linear combination property of intervals, changes nonlinear system to an equivalent linear system, in the second phase, using discretization method, the attained problem is converted to a linear programming problem, and in the third phase, the latter problem will be solved by linear programming methods. In addition, efficiency of our approach is confirmed by some numerical examples.

High Accuracy Arithmetic Average Discretization for Non-Linear Two Point Boundary Value Problems with a Source Function in Integral Form

In this article, we report the derivation of high accuracy finite difference method based on arithmetic average discretization for the solution of Un=F(x,u,u′)+∫K(x,s)ds , 0 x s < 1 subject to natural boundary conditions on a non-uniform mesh. The proposed variable mesh approximation is directly applicable to the integro-differential equation with singular coefficients. We need not require any special discretization to obtain the solution near the singular point. The convergence analysis of a difference scheme for the diffusion convection equation is briefly discussed. The presented variable mesh strategy is applicable when the internal grid points of the solution space are both even and odd in number as compared to the method discussed by authors in their previous work in which the internal grid points are strictly odd in number. The advantage of using this new variable mesh strategy is highlighted computationally.

Strong-Stability-Preserving, K-Step, 5- to 10-Stage, Hermite-Birkhoff Time-Discretizations of Order 12

We construct optimal k-step, 5- to 10-stage, explicit, strong-stability-preserving Hermite-Birkhoff (SSP HB) methods of order 12 with nonnegative coefficients by combining linear k-step methods of order 9 with 5- to 10-stage Runge-Kutta (RK) methods of order 4. Since these methods maintain the monotonicity property, they are well suited for solving hyperbolic PDEs by the method of lines after a spatial discretization. It is seen that the 8-step 7-stage HB methods have largest effective SSP coefficient among the HB methods of order 12 on hand. On Burgers’ equations, some of the new HB methods have larger maximum effective CFL numbers than Huang’s 7-step hybrid method of order 7, thus allowing larger step size.