Large-scale Systems Of Linear Equations Research Articles

DC3DPAFEM: An efficient and accurate 3-D direct current resistivity anisotropic forward modeling software for complex geological settings

Nowadays, there is a growing trend that direct current (DC) field surveys are shifting towards challenging areas characterized by mountainous topography and electrical anisotropy. Given these complex geological settings, there is an urgent need for 3-D DC forward modeling software capable of effectively addressing large-scale problems and delivering accurate modeling results to interpret field data. However, most open-source software packages face certain limitations, such as the high numerical cost to handle complex surface topography, the lack of consideration for anisotropic conductivity, the absence of mesh refinement techniques to guarantee accuracy in forward modeling, and the lack of parallel computing techniques to solve large-scale problems. In this study, we develop an efficient and highly accurate 3-D DC anisotropic forward modeling software, namely DC3DPAFEM, using the adaptive finite element algorithm based on the unstructured tetrahedral mesh. Firstly, we construct a strong compatible boundary value problem (BVP) for 3-D anisotropic DC problems by adopting a specialized secondary potential approach to handle the surface topography efficiently. Then, we develop a goal-oriented adaptive mesh refinement (AMR) technique to ensure accurate forward modeling results, even with a coarse initial mesh. To ensure time and memory efficiency, we employ a robust conjugate gradient (CG) algorithm preconditioned by the algebraic multigrid (AMG) solver to solve the large-scale linear system of equations resulting from complex geological structures. We aim to investigate the performance of the AMG scheme in anisotropic DC cases. Furthermore, we incorporate the domain decomposition technique into the iterative solution scheme for further efficiency gains. This technique significantly improves computing efficiency for large-scale problems in parallel clusters. Finally, we conduct comprehensive performance tests for DC3DPAFEM using a two-layer anisotropic model and a 3-D complex model with undulating terrain. The results of both examples validate the accuracy of DC3DPAFEM, as they closely align with the analytical solutions and the solutions obtained from the existing 3-D DC forward modeling code. Compared to traditional direct solver MUMPS and ILU-preconditioned iterative solvers, DC3DPAFEM exhibits highly scalable performance for large-scale problems, offering significant advantages in terms of memory and time consumption. Overall, DC3DPAFEM demonstrates substantial advances in efficiency, accuracy, and practicality through a series of numerical examples. This open-source code provides an efficient and available tool for developing a 3-D DC inversion method that can deal with large-scale problems involving intricate topography and anisotropic media.

An Efficient Parallel Iterative Solver for Controlled-source Electromagnetic 3-D Adaptive Forward Modeling in General Anisotropic Media

Controlled-source electromagnetic (CSEM) surveys are moving increasingly toward realistic and complicated scenarios where the survey region may contain undulated topography, complex geometries, and electrical anisotropy media. In this paper, the adaptive finite element numerical method is employed to discrete the total electric field equation, which can provide precise electromagnetic (EM) responses even with a coarse initial mesh. The unstructured tetrahedral grid is employed to effectively address arbitrary irregular geometry and mountain terrain. Then, the flexible generalized minimum residual solver (FGMRES), auxiliary-space Maxwell pre-conditioner, and grid division technique were used to solve the large-scale linear system of equations, which can stably solve ill-conditioned problems using fewer computing resources. Finally, we conducted a numerical experiment via our newly proposed forward modeling scheme on a synthetic model with multi-contrast electrical anisotropy. It validated that accurate EM fields could be obtained against the semi-analytic solutions, and this iterative solver has good robustness for various anisotropic media. As a result, we have developed a state-of-the-art 3D CSEM anisotropic forward modeling engine, which can quickly and accurately deal with large-scale and complex geo-models.

A parallel adaptive finite-element approach for 3-D realistic controlled-source electromagnetic problems using hierarchical tetrahedral grids

SUMMARY To effectively and efficiently interpret or invert controlled-source electromagnetic (CSEM) data which are recorded in areas with the kind of complex geological environments and arbitrary topography that are typical, 3-D CSEM forward modelling software that can quickly solve large-scale problems, provide accurate electromagnetic responses for complex geo-electrical models and can be easily incorporated into inversion algorithms are required. We have developed a parallel goal-oriented adaptive mesh refinement finite-element approach for frequency-domain 3-D CSEM forward modelling with hierarchical tetrahedral grids that can offer accurate electromagnetic responses for large-scale complex models and that can efficiently serve for inversion. The approach uses the goal-oriented adaptive vector finite element method to solve the total electric field vector equation. The geo-electrical model is discretized by unstructured tetrahedral grids which can deal with complex underground geological models with arbitrary surface topography. Different from previous adaptive finite element software working on unstructured tetrahedral grids, we have utilized a novel mesh refinement technique named the longest edge bisection method to generate hierarchically refined grids. As the refined grids are nested into the coarse grids, the refinement technique can precisely map the electrical parameters of inversion grids onto the forward modelling grids so that the extra numerical errors generated by the inconsistency of electrical parameters between inversion grids and forward modelling grids are eliminated. In addition, we use the parallel domain-decomposition technique to further accelerate the computations, and the flexible generalized minimum residual solver (FGMRES) with an auxiliary Maxwell solver pre-conditioner to solve the final large-scale system of linear equations. In the end, we validate the performance of the proposed scheme using two synthetic models and one realistic model. We demonstrate that accurate electromagnetic fields can be obtained by comparison with the analytic solutions and that the code is highly scalable for large-scale problems with millions or even hundreds of millions of unknowns. For the synthetic 3-D model and the realistic model with complex geometry, our solutions match well with the results calculated by an existing 3-D CSEM forward modelling code. Both synthetic and realistic examples demonstrate that our newly developed code is an effective, efficient forward modelling engine for interpreting CSEM field data acquired in areas of complex geology and topography.

Solving Large‐Scale Linear Systems of Equations by a Quantum Hybrid Algorithm (Ann. Phys. 7/2022)

Quantum Computing A system to solve large-scale linear systems of equations using a quantum processor is described by V. M. Vinokur and co-workers in article number 2200082. The ability to solve linear systems of record size enables simulation of complex networks, fluid dynamics, financial markets, machine learning applications and much more.

Quantile-Based Iterative Methods for Corrupted Systems of Linear Equations

Often in applications ranging from medical imaging and sensor networks to error correction and data science (and beyond), one needs to solve large-scale linear systems in which a fraction of the measurements have been corrupted. We consider solving such large-scale systems of linear equations $\mathbf{A}\mathbf{x}=\mathbf{b}$ that are inconsistent due to corruptions in the measurement vector $\mathbf{b}$. We develop several variants of iterative methods that converge to the solution of the uncorrupted system of equations, even in the presence of large corruptions. These methods make use of a quantile of the absolute values of the residual vector in determining the iterate update. We present both theoretical and empirical results that demonstrate the promise of these iterative approaches.

Iterative Algorithms for Deblurring of Images in Case of Electrical Capacitance Tomography

Electrical capacitance tomography (ECT) has been used to measure flow by applying gas-solid flow in coal gasification, pharmaceutical, and other industries. ECT is also used for creating images of physically confined objects. The data collected by the acquisition system to produce images undergo blurring because of ambient conditions and the electronic circuitry used. This research includes the principle of ECT techniques for deblurring images that were created during measurement. The data recorded by the said acquisition system ascends a large number of linear equations. This system of equations is sparse and ill-conditioned and hence is ill-posed in nature. A variety of reconstruction algorithms with many pros and cons are available to deal with ill-posed problems. Large-scale systems of linear equations resulting during image deblurring problems are solved using iterative regularization algorithms. The conjugate gradient algorithm for least-squares problems (CGLS), least-squares QR factorization (LSQR), and the modified residual norm steepest descent (MRNSD) algorithm are the famous variations of iterative algorithms. These algorithms exhibit a semiconvergence behavior; that is, the computed solution quality first improves and then reduces as the error norm decreases and later increases with each iteration. In this work, soft thresholding has been used for image deblurring problems to tackle the semiconvergence issues. Numerical test problems were executed to indicate the efficacy of the suggested algorithms with criteria for optimal stopping iterations. Results show marginal improvement compared to the traditional iterative algorithms (CGLS, LSQR, and MRNSD) for resolving semiconvergence behavior and image restoration.

Efficient and Secure Outsourcing of Large-Scale Linear System of Equations

Solving the large-scale linear system of equations is one of the most fundamental problems both in theory and practice. However this problem requires too much computational resource for most users to solve it. With the rapid development of cloud services, many users tend to outsource the expensive computing to the cloud server, which is regarded as an efficient way of solving such problem. Nevertheless, the cloud server can not protect the data privacy well, especially when the user's linear system of equations contain private and sensitive data. There are many previous research works on secure outsourcing of systems of linear equations. In this paper we first analyze a privacy preserving CGM (conjugate gradient method) algorithm for secure outsourcing of large-scale systems of linear equations proposed in [1] . We find that the cloud server can recover the protected coefficient matrix of the linear system of equations from the message it receives, which makes the security method in this scheme fails. This is a serious problem, which makes the private and sensitive data of the user leak to the cloud server, and privacy preserving does not exist. To overcome this problem, we modified this algorithm to protect the message from leaking, which can protect the users' privacy well. We also show the security of this new scheme and do experiments to show its efficiency.

Randomized double and triple Kaczmarz for solving extended normal equations

The randomized Kaczmarz algorithm has received considerable attention recently because of its simplicity, speed, and the ability to approximately solve large-scale linear systems of equations. In this paper we propose randomized double and triple Kaczmarz algorithms to solve extended normal equations of the form \(\mathbf{A}^\top \mathbf{Ax}=\mathbf{A}^\top \mathbf{b}-\mathbf{c}\). The proposed algorithms avoid forming \(\mathbf{A}^\top \mathbf{A}\) explicitly and work for arbitrary \(\mathbf{A}\in \mathbb {R}^{m\times n}\) (full rank or rank-deficient, \(m\ge n\) or \(m<n\)). Tight upper bounds showing exponential convergence in the mean square sense of the proposed algorithms are presented and numerical experiments are given to illustrate the theoretical results.

New Algorithms for Secure Outsourcing of Large-Scale Systems of Linear Equations

Cloud computing is the on-request accessibility of computer system resources, specially data storage and computing power, without direct dynamic management by the client. In the simplest terms, cloud computing means storing and accessing data and programs over the Internet instead of your computer’s hard drive. Along the improvement of cloud computing, more and more applications are migrated into the cloud. A significant element of distributed computing is pay-more only as costs arise. Distributed computing gives strong computational capacity to the general public at diminished cost that empowers clients with least computational assets to redistribute their huge calculation outstanding burdens to the cloud, and monetarily appreciate the monstrous computational force, transmission capacity, stockpiling, and even reasonable programming that can be partaken in a compensation for each utilization way Tremendous bit of leeway is the essential objective that forestalls the wide scope of registering model for clients when their secret information are expended during the figuring procedure. Critical thinking is a system to arrive at the pragmatic objective of specific instruments that tackles the issues as well as shield from pernicious practices.. In this paper, we examine secure outsourcing for large-scale systems of linear equations, which are the most popular problems in various engineering disciplines. Linear programming is an operation research technique formulates private data by the customer for LP problem as a set of matrices and vectors, to develop a set of efficient privacy-preserving problem transformation techniques, which allow customers to transform original LP problem into some arbitrary one while protecting sensitive input/output information. Identify that LP problem solving in Cloud component is efficient extra cost on cloud server. In this paper we are utilizing Homomorphic encryption system to increase the performance and time efficiency.

Distributed Solution of Large-Scale Linear Systems via Accelerated Projection-Based Consensus

Solving a large-scale system of linear equations is a key step at the heart of many algorithms in scientific computing, machine learning, and beyond. When the problem dimension is large, computational and/or memory constraints make it desirable, or even necessary, to perform the task in a distributed fashion. In this paper, we consider a common scenario in which a taskmaster intends to solve a large-scale system of linear equations by distributing subsets of the equations among a number of computing machines/cores. We propose a new algorithm called Accelerated Projection-based Consensus , in which at each iteration every machine updates its solution by adding a scaled version of the projection of an error signal onto the nullspace of its system of equations, and the taskmaster conducts an averaging over the solutions with momentum. The convergence behavior of the proposed algorithm is analyzed in detail and analytically shown to compare favorably with the convergence rate of alternative distributed methods, namely distributed gradient descent, distributed versions of Nesterov's accelerated gradient descent and heavy-ball method, the block Cimmino method, and Alternating Direction Method of Multipliers. On randomly chosen linear systems, as well as on real-world data sets, the proposed method offers significant speed-up relative to all the aforementioned methods. Finally, our analysis suggests a novel variation of the distributed heavy-ball method, which employs a particular distributed preconditioning and achieves the same theoretical convergence rate as that in the proposed consensus-based method.

Lanczos-type algorithms with embedded interpolation and extrapolation models for solving large-scale systems of linear equations

The new approach to combating instability in Lanczos-type algorithms for large-scale problems is proposed. It is a modification of so called the embedded interpolation and extrapolation model in Lanczos-type algorithms (EIEMLA), which enables us to interpolate the sequence of vector solutions generated by a Lanczos-type algorithm entirely, without rearranging the position of the entries of the vector solutions. The numerical results show that the new approach performs more effectively than the EIEMLA. In fact, we extend this new approach on the use of a restarting framework to obtain the convergence of Lanczos algorithms accurately. This kind of restarting challenges other existing restarting strategies in Lanczos-type algorithms.

Privacy-preserving large-scale systems of linear equations in outsourcing storage and computation

Along with the prevalence of cloud computing, it can be realised to efficiently outsource costly storage or computations to cloud servers. Recently, secure outsourcing mechanism has received more and more attention. We focus on secure outsourcing storage and computation for large-scale systems of linear equations (LEs) in this paper. Firstly, we construct a new efficient matrix encryption scheme. Then we exploit this encryption scheme to develop a new algorithm which can implement outsourcing storage and computation for large-scale linear equations in the semi-honest setting. Compared with the previous work, the proposed algorithm requires lower storage overhead and is with competitive efficiency.

On Greedy Randomized Kaczmarz Method for Solving Large Sparse Linear Systems

For solving large-scale systems of linear equations by iteration methods, we introduce an effective probability criterion for selecting the working rows from the coefficient matrix and construct a greedy randomized Kaczmarz method. It is proved that this method converges to the unique least-norm solution of the linear system when it is consistent. Theoretical analysis demonstrates that the convergence rate of the greedy randomized Kaczmarz method is much faster than the randomized Kaczmarz method, and numerical results also show that the greedy randomized Kaczmarz method is more efficient than the randomized Kaczmarz method.

Iterative Methods for Solving Factorized Linear Systems

Stochastic iterative algorithms such as the Kaczmarz and Gauss--Seidel methods have gained recent attention because of their speed, simplicity, and the ability to approximately solve large-scale linear systems of equations without needing to access the entire matrix. In this work, we consider the setting where we wish to solve a linear system in a large matrix $X$ that is stored in a factorized form, $X = UV$; this setting either arises naturally in many applications or may be imposed when working with large low-rank datasets for reasons of space required for storage. We propose a variant of the randomized Kaczmarz method for such systems that takes advantage of the factored form and avoids computing $X$. We prove an exponential convergence rate and supplement our theoretical guarantees with experimental evidence demonstrating that the factored variant yields significant acceleration in convergence.

Function-based block multigrid strategy for a two-dimensional linear elasticity-type problem

We consider the solution of block-coupled large-scale linear systems of equations, arising from the finite element approximation of the linear elasticity problem. Due to the large scale of the problems we use properly preconditioned iterative methods, where the preconditioners utilize the underlying block matrix structures, involving inner block solvers and, when suited, broadly established tools such as the algebraic Multigrid method (AMG).For the considered problem, despite of its optimal rate of convergence, AMG, as implemented in some publicly available scientific libraries, imposes unacceptably high demands for computer resources. In this paper we propose and analyze an efficient multilevel preconditioner, based on the Generalized Locally Toeplitz framework, with a specialized transfer operator. We prove and numerically illustrate the optimal convergence rate of the proposed preconditioner, and experimentally report memory and CPU time savings. We also provide comparisons with respect to another aggregation-based algebraic multigrid algorithm.

A decomposition method for estimating recursive logit based route choice models

Fosgerau et al. (2013) recently proposed the recursive logit (RL) model for route choice problems, that can be consistently estimated and easily used for prediction without any sampling of choice sets. Its estimation however requires solving many large-scale systems of linear equations, which can be computationally costly for real data sets. We design a decomposition (DeC) method in order to reduce the number of linear systems to be solved, opening the possibility to estimate more complex RL based models, for instance mixed RL models. We test the performance of the DeC method by estimating the RL model on two networks of more than 7000 and 40,000 links, and we show that the DeC method significantly reduces the estimation time. We also use the DeC method to estimate two mixed RL specifications, one using random coefficients and one incorporating error components associated with subnetworks (Frejinger and Bierlaire 2007). The models are estimated on a real network and a cross-validation study is performed. The results suggest that the mixed RL models can be estimated in a reasonable time with the DeC method. These models yield sensible parameter estimates and the in-sample and out-of sample fits are significantly better than the RL model.

Comment on “Harnessing the Cloud for Securely Outsourcing Large-Scale Systems of Linear Equations”

We remark that the Wang et al.'s scheme [ IEEE Transanction Parallel Distributed Systems , 24 (6), 1172-1181, 2013] fails because the involved homomorphic encryption system is invalid in the context of the scheme. This is due to that the general arithmetic over the field $\mathbb {R}$ (the outsourced linear equations are constrained to the field) is not compatible with the modular arithmetic over any finite domain (the homomorphic encryption is constrained to the domain).

New Algorithms for Secure Outsourcing of Large-Scale Systems of Linear Equations

With the rapid development in availability of cloud services, the techniques for securely outsourcing the prohibitively expensive computations to untrusted servers are getting more and more attentions in the scientific community. In this paper, we investigate secure outsourcing for large-scale systems of linear equations, which are the most popular problems in various engineering disciplines. For the first time, we utilize the sparse matrix to propose a new secure outsourcing algorithm of large-scale linear equations in the fully malicious model. Compared with the state-of-the-art algorithm, the proposed algorithm only requires ( optimal ) one round communication (while the algorithm requires $L$ rounds of interactions between the client and cloud server, where $L$ denotes the number of iteration in iterative methods). Furthermore, the client in our algorithm can detect the misbehavior of cloud server with the ( optimal ) probability 1. Therefore, our proposed algorithm is superior in both efficiency and checkability. We also provide the experimental evaluation that demonstrates the efficiency and effectiveness of our algorithm.

Inverse subspace problems with applications

SUMMARY Given a square matrix A, the inverse subspace problem is concerned with determining a closest matrix to A with a prescribed invariant subspace. When A is Hermitian, the closest matrix may be required to be Hermitian. We measure distance in the Frobenius norm and discuss applications to Krylov subspace methods for the solution of large-scale linear systems of equations and eigenvalue problems as well as to the construction of blurring matrices. Extensions that allow the matrix A to be rectangular and applications to Lanczos bidiagonalization, as well as to the recently proposed subspace-restricted SVD method for the solution of linear discrete ill-posed problems, also are considered.Copyright © 2013 John Wiley & Sons, Ltd.

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

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.