scholarly journals Publicly verifiable and efficiency/security-adjustable outsourcing scheme for solving large-scale modular system of linear equations

2019 ◽  
Vol 8 (1) ◽  
Author(s):  
Panpan Meng ◽  
Chengliang Tian ◽  
Xiangguo Cheng
Keyword(s):  
Coding Theory ◽  
Large Scale ◽  
High Efficiency ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Secure Computation ◽  
Modular System ◽  
System Of Linear Equations ◽  
Public Verifiability ◽  
Cloud Server

AbstractSolving large-scale modular system of linear equations ($\mathcal {LMSLE}$ℒℳSℒE) is pervasive in modern computer and communication community, especially in the fields of coding theory and cryptography. However, it is computationally overloaded for lightweight devices arisen in quantity with the dawn of the things of internet (IoT) era. As an important form of cloud computing services, secure computation outsourcing has become a popular topic. In this paper, we design an efficient outsourcing scheme that enables the resource-constrained client to find a solution of the $\mathcal {LMSLE}$ℒℳSℒE with the assistance of a public cloud server. By utilizing affine transformation based on sparse unimodular matrices, our scheme has three merits compared with previous work: 1) Our scheme is efficiency/security-adjustable. Our encryption method is dynamic, and it can balance the security and efficiency to match different application scenarios by skillfully control the number of unimodular matrices. 2) Our scheme is versatile. It is suit for generic m-by-n coefficient matrix A, no matter it is square or not. 3) Our scheme satisfies public verifiability and achieves the optimal verification probability. It enables any verifier which is not necessarily the client to verify the correctness of the results returned from the cloud server with probability 1. Finally, theoretical analysis and comprehensive experimental results confirm our scheme’s security and high efficiency.

Application of Jacobi - Conjugate Gradient Method in the Two-Dimensional Magnetotelluric Forward

2013 ◽  
Vol 416-417 ◽  
pp. 2123-2127
Author(s):  
Chong Li Zhu
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Large Scale ◽  
Linear Equations ◽  
Hessian Matrix ◽  
Simulation Method ◽  
Coefficient Matrix ◽  
Fine Grid ◽  
Solution Efficiency

Using the finite element method and all kinds of numerical simulation method, A large-scale system of linear equations is solved eventually,the solution method of the system of equations largely determines the solution efficiency and precision of numerical calculation. The Jacobi iteration preconditioning conjugate gradient method is adopted, Both overcome the coefficient matrix pathological characteristics and the characteristics of slow convergence speed ,and avoid the disadvantages such as Newton's method to store and Hessian matrix is calculated and inversed,improve forward modeling calculation speed and accuracy. Guarantee for solving numerical stability and efficiency ,of the thick grid combined with verification, the algorithm is feasible and it is verified by coarse grid combine with fine grid.

scholarly journals Numerical solutions to integral equations equivalent to differential equations with fractional time

2010 ◽  
Vol 20 (2) ◽  
pp. 261-269 ◽  
Author(s):  
Bartosz Bandrowski ◽  
Anna Karczewska ◽  
Piotr Rozmej
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Large Scale ◽  
Numerical Solutions ◽  
Symmetric Matrix ◽  
Linear Equations ◽  
Time Variable ◽  
Gmres Method ◽  
System Of Linear Equations ◽  
Variable Equation

Numerical solutions to integral equations equivalent to differential equations with fractional timeThis paper presents an approximate method of solving the fractional (in the time variable) equation which describes the processes lying between heat and wave behavior. The approximation consists in the application of a finite subspace of an infinite basis in the time variable (Galerkin method) and discretization in space variables. In the final step, a large-scale system of linear equations with a non-symmetric matrix is solved with the use of the iterative GMRES method.

Meshfree Method in Geophysical Electromagnetic Prospecting: The 2D Magnetotelluric Example

2017 ◽  
Vol 15 (02) ◽  
pp. 1750084 ◽  
Author(s):  
Yanju Ji ◽  
Tingzhe Huang ◽  
Wanyu Huang ◽  
Liangliang Rong
Keyword(s):  
Numerical Simulation ◽  
Large Scale ◽  
High Efficiency ◽  
Interpolation Method ◽  
Rapid Development ◽  
Linear Equations ◽  
Meshfree Method ◽  
Physical Parameters ◽  
Traditional Methods ◽  
Calculation Results

As an important supplement and development of traditional methods, the meshfree method has received a great deal of attention in the field of engineering calculation, and has been successfully used to solve many problems which traditional methods have difficulty in solving. However, the application of meshfree method is relatively less in the area of geophysics. In this paper, we apply the meshfree method to the numerical simulation of geophysical electromagnetic prospecting, taking the 2D magnetotelluric as an example and deduce the corresponding meshfree radial point interpolation method (RPIM) equivalent linear equations in detail. The high-efficiency and accurate solutions of large-scale sparse linear equations are solved by the quasi-minimal residual method based on Krylov subspace. The optimal values of the shape parameters are given by numerical experiments. The correctness of the meshfree method is verified by a layered model. The root mean square error of the calculation results is no more than 0.35%, its accuracy is superior to the finite element method. We also compare the meshfree solution with FEM solution by calculating an inclined vein body model, and the calculation results are in good agreement. A continuously changing fault model and undulating terrain model which traditional methods have difficulty in simulating are respectively calculated, the sectional profiles of the apparent resistivity accurately reflect the trend of the anomalies. The meshfree method does not require the complicated mesh generation, and the physical parameters are loaded at a series of points, thus it is especially suitable for the calculation of the complex geological models. With the rapid development of computational science, the meshfree techniques will certainly become a new robust numerical simulation method in geophysical electromagnetic prospecting.

scholarly journals Parallel Algorithm with Parameters Based on Alternating Direction for Solving Banded Linear Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Xinrong Ma ◽  
Sanyang Liu ◽  
Manyu Xiao ◽  
Gongnan Xie
Keyword(s):  
Parallel Algorithm ◽  
High Efficiency ◽  
Linear Equations ◽  
Positive Definite Matrix ◽  
Coefficient Matrix ◽  
Memory Space ◽  
Alternating Direction ◽  
Banded Linear Systems ◽  
Parallel Iterative ◽  
Better Than

An efficient parallel iterative method with parameters on distributed-memory multicomputer is investigated for solving the banded linear equations in this work. The parallel algorithm at each iterative step is executed using alternating direction by splitting the coefficient matrix and using parameters properly. Only it twice requires the communications of the algorithm between the adjacent processors, so this method has high parallel efficiency. Some convergence theorems for different coefficient matrices are given, such as a Hermite positive definite matrix or anM-matrix. Numerical experiments implemented on HP rx2600 cluster verify that our algorithm has the advantages over the multisplitting one of high efficiency and low memory space, which has a considerable advantage in CPU-times costs over the BSOR one. The efficiency for Example 1 is better than BSOR one significantly. As to Example 2, the acceleration rates and efficiency of our algorithm are better than the PEk inner iterative one.

scholarly journals Convergence of relaxed chaotic parallel iterative methods

1994 ◽  
Vol 50 (1) ◽  
pp. 167-176 ◽  
Author(s):  
Peter E. Kloeden ◽  
Dong-Jin Yuan
Keyword(s):  
Iterative Methods ◽  
Large Scale ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
System Of Linear Equations ◽  
Parallel Iterative Methods ◽  
Parallel Iterative

Sufficient conditions involving uniform multisplittings are established for the convergence of relaxed and AOR versions of asynchronous or chaotic parallel iterative methods for solving a large scale nonsingular system of linear equations Ax = b.

Fast Stokesian dynamics

2019 ◽  
Vol 878 ◽  
pp. 544-597 ◽  
Author(s):  
Andrew M. Fiore ◽  
James W. Swan
Keyword(s):  
Saddle Point ◽  
Large Scale ◽  
Near Field ◽  
Linear Equations ◽  
Far Field ◽  
System Of Linear Equations ◽  
Stokesian Dynamics ◽  
Brownian Force ◽  
Saddle Point Matrix ◽  
Stochastic Forces

We present a new method for large scale dynamic simulation of colloidal particles with hydrodynamic interactions and Brownian forces, which we call fast Stokesian dynamics (FSD). The approach for modelling the hydrodynamic interactions between particles is based on the Stokesian dynamics (SD) algorithm (J. Fluid Mech., vol. 448, 2001, pp. 115–146), which decomposes the interactions into near-field (short-ranged, pairwise additive and diverging) and far-field (long-ranged many-body) contributions. In FSD, the standard system of linear equations for SD is reformulated using a single saddle point matrix. We show that this reformulation is generalizable to a host of particular simulation methods enabling the self-consistent inclusion of a wide range of constraints, geometries and physics in the SD simulation scheme. Importantly for fast, large scale simulations, we show that the saddle point equation is solved very efficiently by iterative methods for which novel preconditioners are derived. In contrast to existing approaches to accelerating SD algorithms, the FSD algorithm avoids explicit inversion of ill-conditioned hydrodynamic operators without adequate preconditioning, which drastically reduces computation time. Furthermore, the FSD formulation is combined with advanced sampling techniques in order to rapidly generate the stochastic forces required for Brownian motion. Specifically, we adopt the standard approach of decomposing the stochastic forces into near-field and far-field parts. The near-field Brownian force is readily computed using an iterative Krylov subspace method, for which a novel preconditioner is developed, while the far-field Brownian force is efficiently computed by linearly transforming those forces into a fluctuating velocity field, computed easily using the positively split Ewald approach (J. Chem. Phys., vol. 146, 2017, 124116). The resultant effect of this field on the particle motion is determined through solution of a system of linear equations using the same saddle point matrix used for deterministic calculations. Thus, this calculation is also very efficient. Additionally, application of the saddle point formulation to develop high-resolution hydrodynamic models from constrained collections of particles (similar to the immersed boundary method) is demonstrated and the convergence of such models is discussed in detail. Finally, an optimized graphics processing unit implementation of FSD for mono-disperse spherical particles is used to demonstrated performance and accuracy of dynamic simulations of $O(10^{5})$ particles, and an open source plugin for the HOOMD-blue suite of molecular dynamics software is included in the supplementary material.

scholarly journals Assure deletion supporting dynamic insertion for outsourced data in cloud computing

2020 ◽  
Vol 16 (9) ◽  
pp. 155014772095829
Author(s):  
Changsong Yang ◽  
Yueling Liu ◽  
Xiaoling Tao
Keyword(s):  
Cloud Computing ◽  
Cloud Storage ◽  
Large Scale ◽  
High Efficiency ◽  
Rapid Development ◽  
Data Set ◽  
Outsourced Data ◽  
Data Owner ◽  
Cloud Storage Service ◽  
Cloud Server

With the rapid development of cloud computing, an increasing number of data owners are willing to employ cloud storage service. In cloud storage, the resource-constraint data owners can outsource their large-scale data to the remote cloud server, by which they can greatly reduce local storage overhead and computation cost. Despite plenty of attractive advantages, cloud storage inevitably suffers from some new security challenges due to the separation of outsourced data ownership and its management, such as secure data insertion and deletion. The cloud server may maliciously reserve some data copies and return a wrong deletion result to cheat the data owner. Moreover, it is very difficult for the data owner to securely insert some new data blocks into the outsourced data set. To solve the above two problems, we adopt the primitive of Merkle sum hash tree to design a novel publicly verifiable cloud data deletion scheme, which can also simultaneously achieve provable data storage and dynamic data insertion. Moreover, an interesting property of our proposed scheme is that it can satisfy private and public verifiability without requiring any trusted third party. Furthermore, we formally prove that our proposed scheme not only can achieve the desired security properties, but also can realize the high efficiency and practicality.

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