A GENERALIZED SCALAR AUXILIARY VARIABLE METHOD FOR THE TIME-DEPENDENT GINZBURG-LANDAU EQUATIONS

This paper develops a generalized scalar auxiliary variable(SAV)method for the time-dependent Ginzburg-Landau equations.The backward Euler method is used for discretizing the temporal derivative of the time-dependent Ginzburg-Landau equations.In this method,the system is decoupled and linearized to avoid solving the non-linear equation at each step.The theoretical analysis proves that the generalized SAV method can preserve the maximum bound principle and energy stability,and this is confirmed by the numerical result,and also shows that the numerical algorithm is stable.

New Class of Almost Unbiased Modified Ratio Cum Product Estimators with Knownparameters of Auxiliary Variables

This manuscript deals with new class of almost unbiased ratio cum product estimators for the estimation of population mean of the study variable by using the known values auxiliary variable. The bias and mean squared error of proposed estimators are obtained. An empirical study is carried out to assess the efficiency of proposed estimators over the existing estimators with the help of some known natural populations and it shows that the proposed estimators are almost unbiased and it perform better than the existing estimators.

Generalized Ratio-Cum-Product Estimators for Two-Phase Sampling Using Multi-Auxiliary Variables

In this paper, we have proposed estimators of finite population mean using generalized Ratio- cum-product estimator for two-Phase sampling using multi-auxiliary variables under full, partial and no information cases and investigated their finite sample properties. An empirical study is given to compare the performance of the proposed estimators with the existing estimators that utilize auxiliary variable(s) for finite population mean. It has been found that the generalized Ra-tio-cum-product estimator in full information case using multiple auxiliary variables is more efficient than mean per unit, ratio and product estimator using one auxiliary variable, ratio and product estimator using multiple auxiliary variable and ratio-cum-product estimators in both partial and no information case in two phase sampling. A generalized Ratio-cum-product estimator in partial information case is more efficient than Generalized Ratio-cum-product estimator in No information case.

A SCALAR AUXILIARY VARIABLE (SAV) FINITE ELEMENT NUMERICAL SCHEME FOR THE CAHN-HILLIARD-HELE-SHAW SYSTEM WITH DYNAMIC BOUNDARY CONDITIONS

In this paper,we consider the Cahn-Hilliard-Hele-Shaw(CHHS)system with the dynamic boundary conditions,in which both the bulk and surface energy parts play important roles.The scalar auxiliary variable approach is introduced for the physical system;the mass conservation and energy dissipation is proved for the CHHS system.Subsequently,a fully discrete SAV finite element scheme is proposed,with the mass conservation and energy dissipation laws established at a theoretical level.In addition,the convergence analysis and error estimate is provided for the proposed SAV numerical scheme.

Arbitrarily High-Order Energy-Preserving Schemes for the Camassa-Holm Equation Based on the Quadratic Auxiliary Variable Approach

In this paper,we present a quadratic auxiliary variable(QAV)technique to develop a novel class of arbitrarily high-order energy-preserving algorithms for the Camassa-Holm equation.The QAV approach is first utilized to transform the original equation into a reformulated QAV system with a consistent initial condition.Then the reformulated QAV system is discretized by applying the Fourier pseudo-spectral method in space and the symplectic Runge-Kutta methods in time,which arrives at a class of fully discrete schemes.Under the consistent initial condition,they can be rewritten as a new fully discrete system by eliminating the introduced auxiliary variable,which is rigorously proved to be energy-preserving and symmetric.Ample numerical experiments are conducted to confirm the expected order of accuracy,conservative property and efficiency of the proposed methods.The presented numerical strategy makes it possible to directly apply a special class of Runge-Kutta methods to develop energy-preserving algorithms for a general conservative system with any polynomial energy.

Error Estimate of a Second Order Accurate Scalar Auxiliary Variable (SAV) Numerical Method for the Epitaxial Thin Film Equation

A second order accurate(in time)numerical scheme is analyzed for the slope-selection(SS)equation of the epitaxial thin film growth model,with Fourier pseudo-spectral discretization in space.To make the numerical scheme linear while preserving the nonlinear energy stability,we make use of the scalar auxiliary variable(SAV)approach,in which a modified Crank-Nicolson is applied for the surface diffusion part.The energy stability could be derived a modified form,in comparison with the standard Crank-Nicolson approximation to the surface diffusion term.Such an energy stability leads to an H2 bound for the numerical solution.In addition,this H2 bound is not sufficient for the optimal rate convergence analysis,and we establish a uniform-in-time H3 bound for the numerical solution,based on the higher order Sobolev norm estimate,combined with repeated applications of discrete H¨older inequality and nonlinear embeddings in the Fourier pseudo-spectral space.This discrete H3 bound for the numerical solution enables us to derive the optimal rate error estimate for this alternate SAV method.A few numerical experiments are also presented,which confirm the efficiency and accuracy of the proposed scheme.

An Efficient Scrambled Estimator of Population Mean of Quantitative Sensitive Variable Using General Linear Transformation of Non-sensitive Auxiliary Variable

In the present paper,we propose an efficient scrambled estimator of population mean of quantitative sensitive study variable,using general linear transformation of nonsensitive auxiliary variable.Efficiency comparisons with the existing estimators have been carried out both theoretically and numerically.It has been found that our optimal scrambled estimator is always more efficient than most of the existing scrambled estimators and also it is more efficient than few other scrambled estimators under some conditions.

High-Order Decoupled and Bound Preserving Local Discontinuous Galerkin Methods for a Class of Chemotaxis Models

In this paper,we explore bound preserving and high-order accurate local discontinuous Galerkin(LDG)schemes to solve a class of chemotaxis models,including the classical Keller-Segel(KS)model and two other density-dependent problems.We use the convex splitting method,the variant energy quadratization method,and the scalar auxiliary variable method coupled with the LDG method to construct first-order temporal accurate schemes based on the gradient flow structure of the models.These semi-implicit schemes are decoupled,energy stable,and can be extended to high accuracy schemes using the semi-implicit spectral deferred correction method.Many bound preserving DG discretizations are only worked on explicit time integration methods and are difficult to get high-order accuracy.To overcome these difficulties,we use the Lagrange multipliers to enforce the implicit or semi-implicit LDG schemes to satisfy the bound constraints at each time step.This bound preserving limiter results in the Karush-Kuhn-Tucker condition,which can be solved by an efficient active set semi-smooth Newton method.Various numerical experiments illustrate the high-order accuracy and the effect of bound preserving.

Spatial Interpolation of Soil Texture Using Compositional Kriging and Regression Kriging with Consideration of the Characteristics of Compositional Data and Environment Variables

The spatial interpolation for soil texture does not necessarily satisfy the constant sum and nonnegativity constraints. Meanwhile, although numeric and categorical variables have been used as auxiliary variables to improve prediction accuracy of soil attributes such as soil organic matter, they （especially the categorical variables） are rarely used in spatial prediction of soil texture. The objective of our study was to comparing the performance of the methods for spatial prediction of soil texture with consideration of the characteristics of compositional data and auxiliary variables. These methods include the ordinary kriging with the symmetry logratio transform, regression kriging with the symmetry logratio transform, and compositional kriging （CK） approaches. The root mean squared error （RMSE）, the relative improvement value of RMSE and Aitchison＇s distance （DA） were all utilized to assess the accuracy of prediction and the mean squared deviation ratio was used to evaluate the goodness of fit of the theoretical estimate of error. The results showed that the prediction methods utilized in this paper could enable interpolation results of soil texture to satisfy the constant sum and nonnegativity constraints. Prediction accuracy and model fitting effect of the CK approach were better, suggesting that the CK method was more appropriate for predicting soil texture. The CK method is directly interpolated on soil texture, which ensures that it is optimal unbiased estimator. If the environment variables are appropriately selected as auxiliary variables, spatial variability of soil texture can be predicted reasonably and accordingly the predicted results will be satisfied.

An effective array beamforming scheme based on branch-and-bound algorithm

In this paper, we propose an effective full array and sparse array adaptive beamforming scheme that can be applied for multiple desired signals based on the branch-and-bound algorithm. Adaptive beamforming for the multiple desired signals is realized by the improved Capon method. At the same time,the sidelobe constraint is added to reduce the sidelobe level. To reduce the pointing errors of multiple desired signals, the array response phase of the desired signal is firstly optimized by using auxilary variables while keeping the response amplitude unchanged. The whole design is formulated as a convex optimization problem solved by the branch-and-bound algorithm. In addition,the beamformer weight vector is penalized with the modified reweighted l_(1)-norm to achieve sparsity. Theoretical analysis and simulation results show that the proposed algorithm has lower sidelobe level, higher SINR, and less pointing error than the stateof-the-art methods in the case of a single expected signal and multiple desired signals.

An Efficient Class of Estimators for the Finite Population Mean in Ranked Set Sampling

In this paper, we propose a class of estimators for estimating the finite population mean of the study variable under Ranked Set Sampling (RSS) when population mean of the auxiliary variable is known. The bias and Mean Squared Error (MSE) of the proposed class of estimators are obtained to first degree of approximation. It is identified that the proposed class of estimators is more efficient as compared to [1] estimator and several other estimators. A simulation study is carried out to judge the performances of the estimators.

Novel Partitioned Time-Stepping Algorithms for Fast Computation of Random Interface-Coupled Problems with Uncertain Parameters

The simulation of multi-domain,multi-physics mathematical models with uncertain parameters can be quite demanding in terms of algorithm design and com-putation costs.Our main objective in this paper is to examine a physical interface coupling between two random dissipative systems with uncertain parameters.Due to the complexity and uncertainty inherent in such interface-coupled problems,un-certain diffusion coefficients or friction parameters often arise,leading to consid-ering random systems.We employ Monte Carlo methods to produce independent and identically distributed deterministic heat-heat model samples to address ran-dom systems,and adroitly integrate the ensemble idea to facilitate the fast calcu-lation of these samples.To achieve unconditional stability,we introduce the scalar auxiliary variable(SAV)method to overcome the time constraints of the ensemble implicit-explicit algorithm.Furthermore,for a more accurate and stable scheme,the ensemble data-passing algorithm is raised,which is unconditionally stable and convergent without any auxiliary variables.These algorithms employ the same co-efficient matrix for multiple linear systems and enable easy parallelization,which can significantly reduce the computational cost.Finally,numerical experiments are conducted to support the theoretical results and showcase the unique features of the proposed algorithms.

A Stable Arbitrarily High Order Time-Stepping Method for Thermal Phase Change Problems

Thermal phase change problems are widespread in mathematics,nature,and science.They are particularly useful in simulating the phenomena of melting and solidification in materials science.In this paper we propose a novel class of arbitrarily high-order and unconditionally energy stable schemes for a thermal phase changemodel,which is the coupling of a heat transfer equation and a phase field equation.The unconditional energy stability and consistency error estimates are rigorously proved for the proposed schemes.A detailed implementation demonstrates that the proposed method requires only the solution of a system of linear elliptic equations at each time step,with an efficient scheme of sufficient accuracy to calculate the solution at the first step.It is observed from the comparison with the classical explicit Runge-Kutta method that the new schemes allow to use larger time steps.Adaptive time step size strategies can be applied to further benefit from this unconditional stability.Numerical experiments are presented to verify the theoretical claims and to illustrate the accuracy and effectiveness of our method.

An SAV Method for Imaginary Time Gradient Flow Model in Density Functional Theory

In this paper,based on the imaginary time gradient flow model in the density functional theory,a scalar auxiliary variable(SAV)method is developed for the ground state calculation of a given electronic structure system.To handle the orthonormality constraint on those wave functions,two kinds of penalty terms are introduced in designing the modified energy functional in SAV,i.e.,one for the norm preserving of each wave function,another for the orthogonality between each pair of different wave functions.A numerical method consisting of a designed scheme and a linear finite element method is used for the discretization.Theoretically,the desired unconditional decay of modified energy can be obtained from our method,while computationally,both the original energy and modified energy decay behaviors can be observed successfully from a number of numerical experiments.More importantly,numerical results show that the orthonormality among those wave functions can be automatically preserved,without explicitly preserving orthogonalization operations.This implies the potential of our method in large-scale simulations in density functional theory.

A Conservative SAV-RRK Finite Element Method for the Nonlinear Schrodinger Equation

Abstract.In this paper,we propose,analyze and numerically validate a conservative finite element method for the nonlinear Schrodinger equation.A scalar auxiliary variable(SAV)is introduced to reformulate the nonlinear Schrodinger equation into an equivalent system and to transform the energy into a quadratic form.We use the standard continuous finite element method for the spatial discretization,and the relaxation Runge-Kutta method for the time discretization.Both mass and energy conservation laws are shown for the semi-discrete finite element scheme,and also preserved for the full-discrete scheme with suitable relaxation coefficient in the relaxation Runge-Kutta method.Numerical examples are presented to demonstrate the accuracy of the proposed method,and the conservation of mass and energy in long time simulations.

Active label distribution learning via kernel maximum mean discrepancy

Label distribution learning(LDL)is a new learning paradigm to deal with label ambiguity and many researches have achieved the prominent performances.Compared with traditional supervised learning scenarios,the annotation with label distribution is more expensive.Direct use of existing active learning(AL)approaches,which aim to reduce the annotation cost in traditional learning,may lead to the degradation of their performance.To deal with the problem of high annotation cost in LDL,we propose the active label distribution learning via kernel maximum mean discrepancy(ALDL-kMMD)method to tackle this crucial but rarely studied problem.ALDL-kMMD captures the structural information of both data and label,extracts the most representative instances from the unlabeled ones by incorporating the nonlinear model and marginal probability distribution matching.Besides,it is also able to markedly decrease the amount of queried unlabeled instances.Meanwhile,an effective solution is proposed for the original optimization problem of ALDL-kMMD by constructing auxiliary variables.The effectiveness of our method is validated with experiments on the real-world datasets.

A LINEARLY-IMPLICIT ENERGY-PRESERVING ALGORITHM FOR THE TWO-DIMENSIONAL SPACE-FRACTIONAL NONLINEAR SCHRÖDINGER EQUATION BASED ON THE SAV APPROACH

The main objective of this paper is to present an efficient structure-preserving scheme,which is based on the idea of the scalar auxiliary variable approach,for solving the twodimensional space-fractional nonlinear Schrodinger equation.First,we reformulate the equation as an canonical Hamiltonian system,and obtain a new equivalent system via introducing a scalar variable.Then,we construct a semi-discrete energy-preserving scheme by using the Fourier pseudo-spectral method to discretize the equivalent system in space direction.After that,applying the Crank-Nicolson method on the temporal direction gives a linearly-implicit scheme in the fully-discrete version.As expected,the proposed scheme can preserve the energy exactly and more efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time step.Finally,numerical experiments are provided to demonstrate the efficiency and conservation of the scheme.

SAV Finite Element Method for the Peng-Robinson Equation of State with Dynamic Boundary Conditions

In this paper,the Peng-Robinson equation of state with dynamic boundary conditions is discussed,which considers the interactions with solid walls.At first,the model is introduced and the regularization method on the nonlinear term is adopted.Next,The scalar auxiliary variable(SAV)method in temporal and finite element method in spatial are used to handle the Peng-Robinson equation of state.Then,the energy dissipation law of the numerical method is obtained.Also,we acquire the convergence of the discrete SAV finite element method(FEM).Finally,a numerical example is provided to confirm the theoretical result.

Efficient linear and unconditionally energy stable schemes for the modified phase field crystal equation

In this paper,we construct efficient schemes based on the scalar auxiliary variable block-centered finite difference method for the modified phase field crystal equation,which is a sixth-order nonlinear damped wave equation.The schemes are linear,conserve mass and unconditionally dissipate a pseudo energy.We prove rigorously second-order error estimates in both time and space for the phase field variable in discrete norms.We also present some numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy.

Sample rotation theory with missing data

This paper studies how the sample rotation method is applied to the case where item nonresponse occurs in surveys. The two cases where the response to the first occasion is complete or incomplete are considered. Using ratio imputation method, the estimators of the current population mean are proposed, which are valid under uniform response regardless of the model and under the ratio model regardless of the response mechanism. Under uniform response, the variances of the proposed estimators are derived. Interestingly, although their expressions are similar, the estimator for the case of incomplete response on the first occasion can have smaller variance than the one for the case of complete response on the first occasion under uniform response. The linearized jackknife variance estimators are also given. These variance estimators prove to be approximately design-unbiased under uniform response. It should be noted that similar property on variance estimators has not been discussed in literature.