Quantum-Resistant Multi-Feature Attribute-Based Proxy Re-Encryption Scheme for Cloud Services

Cloud-based services have powerful storage functions and can provide accurate computation.However,the question of how to guarantee cloud-based services access control and achieve data sharing security has always been a research highlight.Although the attribute-based proxy re-encryption(ABPRE)schemes based on number theory can solve this problem,it is still difficult to resist quantum attacks and have limited expression capabilities.To address these issues,we present a novel linear secret sharing schemes(LSSS)matrix-based ABPRE scheme with the fine-grained policy on the lattice in the research.Additionally,to detect the activities of illegal proxies,homomorphic signature(HS)technology is introduced to realize the verifiability of re-encryption.Moreover,the non-interactivity,unidirectionality,proxy transparency,multi-use,and anti-quantum attack characteristics of our system are all advantageous.Besides,it can efficiently prevent the loss of processing power brought on by repetitive authorisation and can enable precise and safe data sharing in the cloud.Furthermore,under the standard model,the proposed learning with errors(LWE)-based scheme was proven to be IND-sCPA secure.

A second-order convergent and linearized difference schemefor the initial-boundary value problemof the Korteweg-de Vries equation

To numerically solve the initial-boundary value problem of the Korteweg-de Vries equation,an equivalent coupled system of nonlinear equations is obtained by the method of reduction of order.Then,a difference scheme is constructed for the system.The new variable introduced can be separated from the difference scheme to obtain another difference scheme containing only the original variable.The energy method is applied to the theoretical analysis of the difference scheme.Results show that the difference scheme is uniquely solvable and satisfies the energy conservation law corresponding to the original problem.Moreover,the difference scheme converges when the step ratio satisfies a constraint condition,and the temporal and spatial convergence orders are both two.Numerical examples verify the convergence order and the invariant of the difference scheme.Furthermore,the step ratio constraint is unnecessary for the convergence of the difference scheme.Compared with a known two-level nonlinear difference scheme,the proposed difference scheme has more advantages in numerical calculation.

UNIFORM DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED LINEAR 2ND ORDER HYPERBOLIC PROBLEM WITH ZEROTH ORDER REDUCED EQUATION

In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the asymptotic solution are given. Then an exponentially fitted difference scheme is developed in an equidistant mesh. Finally, uniform convergence in small parameter is proved in the sense of discrete energy norm.

A HIGH ACCURACY DIFFERENCE SCHEME FOR THE SINGULAR PERTURBATION PROBLEM OF THE SECOND-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM

In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h3.

An iterative algorithm for solving ill-conditioned linear least squares problems

Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.

Synchronization of perturbed chaotic systems via nonlinear control

Chaos synchronization of systems with perturbations was investigated.A generic nonlinear control scheme to realize chaos synchronization of systems was proposed.This control scheme is flexible and practicable,and gives more freedom in designing controllers in order to achieve some desired performance.With the aid of Lyapunov stability theorem and partial stability theory,two cases were presented:1) Chaos synchronization of the system without perturbation or with vanishing perturbations;2) The boundness of the error state for the system with nonvanishing perturbations satisfying some conditions.Finally,several simulations for Lorenz system were provided to verify the effectiveness and feasibility of our method.Compared numerically with the existing results of linear feedback control scheme,the results are sharper than the existing ones.

THE NUMERICAL SOLUTION OF A SINGULARLY PERTURBED PROBLEM FOR QUASILINEAR PARABOLIC DIFFERENTIAL EQUATION

We consider the numerical solution of a singularly perturbed problem for the quasilinear parabolic differential equation, and construct a linear three-level finite difference scheme on a nonuniform grid. The uniform convergence in the sense of discrete L2 norm is proved and numerical examples are presented.

Discovering Phase Field Models from Image Data with the Pseudo-Spectral Physics Informed Neural Networks

In this paper,we introduce a new deep learning framework for discovering the phase-field models from existing image data.The new framework embraces the approximation power of physics informed neural networks(PINNs)and the computational efficiency of the pseudo-spectral methods,which we named pseudo-spectral PINN or SPINN.Unlike the baseline PINN,the pseudo-spectral PINN has several advantages.First of all,it requires less training data.A minimum of two temporal snapshots with uniform spatial resolution would be adequate.Secondly,it is computationally efficient,as the pseudo-spectral method is used for spatial discretization.Thirdly,it requires less trainable parameters compared with the baseline PINN,which significantly simplifies the training process and potentially assures fewer local minima or saddle points.We illustrate the effectiveness of pseudo-spectral PINN through several numerical examples.The newly proposed pseudo-spectral PINN is rather general,and it can be readily applied to discover other FDE-based models from image data.

On The Numerical Solution of Two Dimensional Model of an Alloy Solidification Problem

In this paper, a linearized three level difference scheme is derived for two-dimensional model of an alloy solidification problem called Sivashinsky equation. Further, it is proved that the scheme is uniquely solvable and convergent with convergence rate of order two in a discrete L∞-norm. At last, numerical experiments are carried out to support the theoretical claims.

Conservative Three-Level Linearized Finite Difference Schemes for the Fisher Equation and Its Maximum Error Estimates

A three-level linearized difference scheme for solving the Fisher equation is firstly proposed in this work.It has the good property of discrete conservative energy.By the discrete energy analysis and mathematical induction method,it is proved to be uniquely solvable and unconditionally convergent with the secondorder accuracy in both time and space.Then another three-level linearized compact difference scheme is derived along with its discrete energy conservation law,unique solvability and unconditional convergence of order two in time and four in space.The resultant schemes preserve the maximum bound principle.The analysis techniques for convergence used in this paper also work for the Euler scheme,the Crank-Nicolson scheme and others.Numerical experiments are carried out to verify the computational efficiency,conservative law and the maximum bound principle of the proposed difference schemes.

TWO-STEP SCHEME FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

In this paper,a stochastic linear two-step scheme has been presented to approximate backward stochastic differential equations(BSDEs).A necessary and sufficient condition is given to judge the L 2-stability of our numerical schemes.This stochastic linear two-step method possesses a family of 3-order convergence schemes in the sense of strong stability.The coefficients in the numerical methods are inferred based on the constraints of strong stability and n-order accuracy(n∈N^(+)).Numerical experiments illustrate that the scheme is an efficient probabilistic numerical method.

ODE-Based Multistep Schemes for Backward Stochastic Differential Equations

In this paper,we explore a new approach to design and analyze numerical schemes for backward stochastic differential equations(BSDEs).By the nonlinear Feynman-Kac formula,we reformulate the BSDE into a pair of reference ordinary differential equations(ODEs),which can be directly discretized by many standard ODE solvers,yielding the corresponding numerical schemes for BSDEs.In particular,by applying strong stability preserving(SSP)time discretizations to the reference ODEs,we can propose new SSP multistep schemes for BSDEs.Theoretical analyses are rigorously performed to prove the consistency,stability and convergency of the proposed SSP multistep schemes.Numerical experiments are further carried out to verify our theoretical results and the capacity of the proposed SSP multistep schemes for solving complex associated problems.

CONVERGENCE ANALYSIS OF NONCONFORMING QUADRILATERAL FINITE ELEMENT METHODS FOR NONLINEAR COUPLED SCHRODINGER-HELMHOLTZ EQUATIONS

The focus of this paper is on two novel linearized Crank-Nicolson schemes with nonconforming quadrilateral finite element methods(FEMs)for the nonlinear coupled Schrodinger-Helmholtz equations.Optimal L^(2) and H^(1) estimates of orders O(h^(2)+τ^(2))and O(h^(2)+τ^(2))are derived respectively without any grid-ratio condition through the following two keys.One is that a time-discrete system is introduced to split the error into the temporal error and the spatial error,which leads to optimal temporal error estimates of order O(τ^(2))in L^(2) and the broken H^(1)-norms,as well as the uniform boundness of numerical solutions in L^(∞) norm.The other is that a novel projection is utilized,which can iron out the difficulty of the existence of the consistency errors.This leads to derive optimal spatial error estimates of orders O(h^(2))in L^(2)-norm and O(h)in the broken H^(1)-norm under the H^(2) regularity of the solutions for the time-discrete system.At last,two numerical examples are provided to confirm the theoretical analysis.Here,h is the subdivision parameter,and τ is the time step.

Linear multi-secret sharing schemes

In this paper the linear multi-secret sharing schemes are studied by using monotone span programs. A relation between computing monotone Boolean functions by using monotone span programs and realizing multi-access structures by using linear multi-secret sharing schemes is shown. Furthermore, the concept of optimal linear multi-secret sharing scheme is presented and the several schemes are proved to be optimal.

Linear Secret Sharing Schemes and Rearrangements of Access Structures

In this paper we study linear secret sharing schemes by monotone span programs, according to the relation between realizing access structures by linear secret sharing schemes and computing monotone Boolean functions by monotone span programs. We construct some linear secret sharing schemes. Furthermore, we study the rearrangements of access structures that is very important in practice.

Convergence of a Linearized and Conservative Difference Scheme for the Klein-Gordon-Zakharov Equation

A linearized and conservative finite difference scheme is presented for the initial-boundary value problem of the Klein-Gordon-Zakharov （KGZ） equation. The new scheme is also decoupled in computation, which means that no iteration is needed and parallel computation can be used, so it is expected to be more efficient in imple- mentation. The existence of the difference solution is proved by Browder fixed point theorem. Besides the standard energy method, in order to overcome the difficulty in obtaining a priori estimate, an induction argument is used to prove that the new scheme is uniquely solvable and second order convergent for U in the discrete L∞- norm, and for N in the discrete L2-norm, respectively, where U and N are the numeri- cal solutions of the KGZ equation. Numerical results verify the theoretical analysis.

A Convergence Analysis of a Structure-Preserving Gradient Flow Method for the All-Electron Kohn-Sham Model

In[Dai et al.,Multi.Model.Simul.18(4)(2020)],a structure-preserving gradient flow method was proposed for the ground state calculation in Kohn-Sham density functional theory,based on which a linearized method was developed in[Hu et al.,EAJAM.13(2)(2023)]for further improving the numerical efficiency.In this paper,a complete convergence analysis is delivered for such a linearized method for the all-electron Kohn-Sham model.Temporally,the convergence,the asymptotic stability,as well as the structure-preserving property of the linearized numerical scheme in the method is discussed following previous works,while spatially,the convergence of the h-adaptive mesh method is demonstrated following[Chen et al.,Multi.Model.Simul.12(2014)],with a key study on the boundedness of the Kohn-Sham potential for the all-electron Kohn-Sham model.Numerical examples confirm the theoretical results very well.

Analysis of L1-Galerkin FEMs for Time-Fractional Nonlinear Parabolic Problems

This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality.In this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative.In terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems.The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.

Semi-Implicit Spectral Deferred Correction Method Based on the Invariant Energy Quadratization Approach for Phase Field Problems

This paper presents a high order time discretization method by combining the semi-implicit spectral deferred correction method with energy stable linear schemes to simulate a series of phase field problems.We start with the linear scheme,which is based on the invariant energy quadratization approach and is proved to be linear unconditionally energy stable.The scheme also takes advantage of avoiding nonlinear iteration and the restriction of time step to guarantee the nonlinear system uniquely solvable.Moreover,the scheme leads to linear algebraic system to solve at each iteration,and we employ the multigrid solver to solve it efficiently.Numerical re-sults are given to illustrate that the combination of local discontinuous Galerkin(LDG)spatial discretization and the high order temporal scheme is a practical,accurate and efficient simulation tool when solving phase field problems.Namely,we can obtain high order accuracy in both time and space by solving some simple linear algebraic equations.

Efficient Numerical Solution of Dynamical Ginzburg-Landau Equations under the Lorentz Gauge

In this paper,a new numerical scheme for the time dependent Ginzburg-Landau(GL)equations under the Lorentz gauge is proposed.We first rewrite the original GL equations into a new mixed formulation,which consists of three parabolic equations for the order parameterψ,the magnetic fieldσ=curlA,the electric potentialθ=divA and a vector ordinary differential equation for the magnetic potential A,respectively.Then,an efficient fully linearized backward Euler finite element method(FEM)is proposed for the mixed GL system,where conventional Lagrange element method is used in spatial discretization.The new approach offers many advantages on both accuracy and efficiency over existing methods for the GL equations under the Lorentz gauge.Three physical variablesψ,σandθcan be solved accurately and directly.More importantly,the new approach is well suitable for non-convex superconductors.We present a set of numerical examples to confirm these advantages.