Large Linear Problems Research Articles

New accelerated modulus-based iteration method for solving large and sparse linear complementarity problem

For the large and sparse linear complementarity problem, we provide a family of new accelerated modulus-based iteration methods in this article. We provide some sufficient criteria for the convergence analysis when the system matrix is a \(P\)-matrix or an \(H_+\)-matrix. In addition, we provide some numerical examples of the different parameters to illustrate the efficacy of our proposed methods. These methods help us reduce the number of iterations and the time required by the CPU, which improves convergence performance.

Randomized symmetric Gauss-Seidel method for solving linear least squares problems

<abstract><p>We introduced a random symmetric Gauss-Seidel (RSGS) method, which was designed to handle large scale linear least squares problems involving tall coefficient matrices. This RSGS method projected the approximate residual onto the subspace spanned by two symmetric columns at each iteration. These columns were sampled from the coefficient matrix based on an effective probability criterion. Our theoretical analysis indicated that RSGS converged when the coefficient matrix had full column rank. Furthermore, numerical experiments demonstrated that RSGS outperformed the baseline algorithms in terms of iteration steps and CPU time.</p></abstract>

3D gravity fast inversion based on Krylov subspace methods

Abstract Mapping the density contrast through the 3D gravity inversion can help detect goals under the subsurface. However, it is a challenge to accurately and efficiently solve the 3D gravity inversion. The Krylov subspace method is commonly used for large linear problems due to its high computational efficiency and low storage requirement. In this study, two classical algorithms of Krylov subspace method, namely the generalized minimum residual method and the conjugate gradient method, are applied to 3D gravity inversion. On the basis of the recovered models of the deep mineral and the shallow L-shaped tunnel models, it was found that the generalized minimum residual method provided similar density contrast results to the conjugate gradient method. The obtained inversion results of density contrast corresponded well to the position of the deep mineral resources model and the L-shaped tunnel model. The 3D distribution of Fe content underground was obtained by inverting the measured gravity data from the Olympic Dam in Australia. The recovered results correspond well with the distribution of Fe content in the geological profile collected. The accuracy of inversion using the generalized minimum residual method was similar to that of the conjugate gradient method under the same conditions. However, the generalized minimum residual method had a faster convergence speed and increased inversion efficiency by ∼90%, greatly reducing the inversion time and improves the inversion efficiency.

A note on fixed point method and linear complementarity problem

In this article, we present a general form of the fixed point method for processing the large and sparse linear complementarity problem, as well as a general condition for the method's convergence when the system matrix is a \(P\)-matrix and some sufficient conditions for the proposed method when the system matrix is a \(H_+\)-matrix or symmetric positive definite matrix.

The generalized residual cutting method and its convergence characteristics

AbstractIterative methods and especially Krylov subspace methods (KSM) are a very useful numerical tool in solving for large and sparse linear systems problems arising in science and engineering modeling. More recently, the nested loop KSM have been proposed that improve the convergence of the traditional KSM. In this article, we review the residual cutting (RC) and the generalized residual cutting (GRC) that are nested loop methods for large and sparse linear systems problems. We also show that GRC is a KSM that is equivalent to Orthomin with a variable preconditioning. We use the modified Gram–Schmidt method to derive a stable GRC algorithm. We show that GRC presents a general framework for constructing a class of “hybrid” (nested) KSM based on inner loop method selection. We conduct numerical experiments using nonsymmetric indefinite matrices from a widely used library of sparse matrices that validate the efficiency and the robustness of the proposed methods.

Projected fixed point iterative method for large and sparse horizontal linear complementarity problem

Projected fixed point iterative method for large and sparse horizontal linear complementarity problem

Integreted strategic and tactical optimization planning of biomass to bioethanol supply chains coupled with operational plan using vehicle routing: A case study in Ethiopia

The optimization of supply chain, a subject of active research in process system engineering, frequently leads to large Mixed Integer Linear or Non-Linear Problems more particularly when applied to realistic industrial cases. In this domain, one main challenge concerns multi scale modeling to accurately integrate the different decision scales (from strategic to operational) and to study the impact of the decision across these scales, in order to effectively contributes to intelligent and coherent decision making. Unfortunately, very few papers cover the three decision levels - strategic (long-term decisions: supply network design), tactical (medium-term decisions: management of network design) and operational (short- term decisions: routing, inventory management…) to have a holistic perspective of the supply chain. Therefore, the first objective of this paper is to model as finely as possible the three decision levels by focusing more particularly on the transport related decision due to their importance on the supply chain performances. As a progress, the Vehicle Routing Problem, one of the most studied problem in logistic and one of the most complex, is integrated into a strategic and tactical model. But, the resulting large-scale MILP model must be solved in a reasonable computational time. To achieve this second objective, we developed a novel iterative multi scale strategy that optimizes the model at two spatial and two temporal scales. This strategy is based on a deep knowledge of the mathematical models and especially of the variables that strongly influence the resolution. The real biofuel supply chain in the Ethiopian country is used to demonstrate the capabilities of the approach, and the improvement in computing time and solution strategy. In addition to the change in the design of the supply chain network, the use of an operational model that finely describes the transport activities shows that the costs related to these activities are often underestimated by a factor between 2 and 3 in the classical models of the literature. The main perspectives of this work are to address the two main shortcomings of the proposal: writing and solving in a reasonable computational time a single model rather than using a decomposition (especially useful for multi-objective optimization) and integrating other decisions at the operational level to not be limited to those related to transportation.

On multi-step greedy randomized coordinate descent method for solving large linear least-squares problems

On multi-step greedy randomized coordinate descent method for solving large linear least-squares problems

On general fixed point method based on matrix splitting for solving linear complementarity problem

In this article, we introduce a modified fixed point method to process the large and sparse linear complementarity problem (LCP) and formulate an equivalent fixed point equation for the LCP and show the equivalence. Also, we provide convergence conditions when the system matrix is a \(P\)-matrix and two sufficient convergence conditions when the system matrix is an \(H_+\)-matrix. To show the efficiency of our proposed method, we illustrate two numerical examples for different parameters.

Localized meshless approaches based on theta method and BDF2 for nonlinear Sobolev equation arising from fluid dynamics

This paper formulates the second-order BDF and theta schemes based on an efficient localized meshless technique for computing the approximate solution of the Sobolev model. The model is one of the useful description for modeling the flow of fluids through fractured rocks, movement of moisture in soils. This proposed approach discretizes the nonlinear Sobolev equation in two main steps. In the first step, temporal discretization is accomplished with the help of the second-order BDF and theta schemes, respectively. In the second step, the localized radial basis function partition of unity collocation method (L-RBF-PUCM) is used for spatial discretization, thereby constructing a fully-discrete algorithm. For large linear problems, global techniques have the disadvantage of high computational cost. The proposed method overcomes this shortcoming well and reduces the computational burden by sparsifying the linear system. The convergence analysis and unconditional stability of the semi-discrete formulation are examined by means of the energy method. Numerical results are carried out to verify the validity and accuracy of the L-RBF-PUCM.

An Adaptive Generalized Multiscale Finite Element Method Based Two-Grid Preconditioner for Large Scale High-Contrast Linear Elasticity Problems

An Adaptive Generalized Multiscale Finite Element Method Based Two-Grid Preconditioner for Large Scale High-Contrast Linear Elasticity Problems

CMDNet: Learning a Probabilistic Relaxation of Discrete Variables for Soft Detection With Low Complexity

Following the great success of Machine Learning (ML), especially Deep Neural Networks (DNNs), in many research domains in 2010s, several ML-based approaches were proposed for detection in large inverse linear problems, e.g., massive MIMO systems. The main motivation behind is that the complexity of Maximum A-Posteriori (MAP) detection grows exponentially with system dimensions. Instead of using DNNs, essentially being a black-box, we take a slightly different approach and introduce a probabilistic Continuous relaxation of disCrete variables to MAP detection. Enabling close approximation and continuous optimization, we derive an iterative detection algorithm: Concrete MAP Detection (CMD). Furthermore, extending CMD by the idea of deep unfolding into CMDNet, we allow for (online) optimization of a small number of parameters to different working points while limiting complexity. In contrast to recent DNN-based approaches, we select the optimization criterion and output of CMDNet based on information theory and are thus able to learn approximate probabilities of the individual optimal detector. This is crucial for soft decoding in today's communication systems. Numerical simulation results in MIMO systems reveal CMDNet to feature a promising accuracy complexity trade-off compared to State of the Art. Notably, we demonstrate CMDNet's soft outputs to be reliable for decoders.

On two-subspace randomized extended Kaczmarz method for solving large linear least-squares problems

On two-subspace randomized extended Kaczmarz method for solving large linear least-squares problems

Hybrid Projection Methods with Recycling for Inverse Problems

Iterative hybrid projection methods have proven to be very effective for solving large linear inverse problems due to their inherent regularizing properties as well as the added flexibility to sele...

On relaxed greedy randomized coordinate descent methods for solving large linear least-squares problems

The greedy randomized coordinate descent (GRCD) method is an effective iterative method for solving large linear least-squares problems. In this work, we construct a class of relaxed greedy randomized coordinate descent (RGRCD) methods by introducing a relaxation parameter in the probability criterion. Then, we prove the convergence properties of these methods when the coefficient matrix of the linear least-squares problems is of full column rank, with the number of rows being no less than the number of columns. In addition, we propose a max-distance coordinate descent (CD) method, and study its convergence properties and accelerated version. Finally, we provide some numerical experiments to confirm the effectiveness of our new methods.

A joint bidiagonalization based iterative algorithm for large scale general-form Tikhonov regularization

Based on the joint bidiagonalization (JBD) process of the matrix pair {A,L}, an iterative regularization algorithm, called JBDQR, is proposed and developed for large scale linear discrete ill-posed problems in general-form Tikhonov regularization. It is proved that the JBDQR iterates take the form of attractive filtered generalized singular value decomposition (GSVD) expansions, where the filters are given explicitly and insightful. This result and a detailed analysis on it show that JBDQR must have the desired semi-convergence property, where the iteration number k plays the role of the regularization parameter. Embedded with the L-curve criterion or the discrepancy principle that is used to estimate the optimal k⁎ at which the semi-convergence occurs, JBDQR can compute a satisfying good regularized solution. JBDQR is theoretically solid and effective, and it is simple to implement. Numerical experiments confirm our results and the robustness of JBDQR.

Decomposition of arrow type positive semidefinite matrices with application to topology optimization

Decomposition of large matrix inequalities for matrices with chordal sparsity graph has been recently used by Kojima et al. (Math Program 129(1):33–68, 2011) to reduce problem size of large scale semidefinite optimization (SDO) problems and thus increase efficiency of standard SDO software. A by-product of such a decomposition is the introduction of new dense small-size matrix variables. We will show that for arrow type matrices satisfying suitable assumptions, the additional matrix variables have rank one and can thus be replaced by vector variables of the same dimensions. This leads to significant improvement in efficiency of standard SDO software. We will apply this idea to the problem of topology optimization formulated as a large scale linear semidefinite optimization problem. Numerical examples will demonstrate tremendous speed-up in the solution of the decomposed problems, as compared to the original large scale problem. In our numerical example the decomposed problems exhibit linear growth in complexity, compared to the more than cubic growth in the original problem formulation. We will also give a connection of our approach to the standard theory of domain decomposition and show that the additional vector variables are outcomes of the corresponding discrete Steklov–Poincaré operators.

A fast matrix-free elasto-plastic solver for predicting residual stresses in additive manufacturing

Process planning for additive manufacturing (AM) today relies heavily on multi-physics, multi-scale simulation. The focus of this paper is on one aspect of AM simulation, namely, part-level elasto-plastic simulation for residual stress and distortion predictions. This is one of the crucial steps in AM process optimization, but is computationally expensive, often requiring the use of large computer clusters.The primary bottleneck in elasto-plastic simulation is the repeated solution of large linear systems of equations. While there is a wide range of linear solvers, most cannot exploit the unique structured nature of the mesh underlying AM simulation. Here, we revisit a specific matrix-free solver, namely rigid-body deflated solver that has been successfully deployed for solving large linear elastic problems in such scenarios. The salient feature of this solver is that the stiffness matrix is never assembled, thereby reducing the memory requirements significantly, leading to large computational gains.The objective of this paper is to extend the above solver to elasto-plasticity by efficiently updating the element tangent stiffness matrices, and the corresponding deflation matrix. The performance of the proposed method is evaluated on a benchmark problem using multi-core CPU and GPU architectures, and compared against ANSYS. Then, part-level residual stresses and distortion are predicted using the proposed solver. The present work is restricted to associative plasticity with von-Mises yield criteria, but can be extended to other plasticity models.

Block generalized locally Toeplitz sequences: theory and applications in the unidimensional case

In computational mathematics, when dealing with a large linear discrete problem (e.g., a linear system) arising from the numericaldiscretization of a differential equation (DE), knowledge of the spectral distribution of the associated matrix has proved to be useful information for designing/analyzing appropriate solvers–-especially, preconditioned Krylov and multigrid solvers–-for the considered problem. Actually, this spectral information is of interest also in itself as long as the eigenvalues of the aforementioned matrix represent physical quantities of interest, which is the case for several problems from engineering and applied sciences (e.g., the study of natural vibration frequencies in an elastic material).The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices $A_n$ arising from virtually any kind of numerical discretization of DEs. Indeed, when the mesh-fineness parameter $n$ tends to infinity, these matrices $A_n$ give rise to a sequence $\\{A_n\\}_n$, which often turns out to be a GLT sequence or one of its “relatives”, i.e., a block GLT sequence or a reduced GLT sequence. In particular, block GLT sequences are encountered in the discretization of systems of DEs as well as in the higher-order finite element or discontinuous Galerkin approximation of scalar/vectorial DEs.This work is a review, refinement, extension, and systematic exposition of the theory of block GLT sequences. It also includes several emblematic applications of this theory in the context of DE discretizations.

Block generalized locally Toeplitz sequences: theory and applications in the multidimensional case

In computational mathematics, when dealing with a large linear discrete problem (e.g., a linear system) arising from thenumerical discretization of a partial differential equation (PDE), knowledge of the spectral distribution of the associated matrix has proved to beuseful information for designing/analyzing appropriate solvers–-especially, preconditioned Krylov and multigrid solvers–-for the considered problem.Actually, this spectral information is of interest also in itself as long as the eigenvalues of the aforementioned matrix represent physical quantities of interest,which is the case for several problems from engineering and applied sciences (e.g., the study of natural vibration frequencies in an elastic material).The theory of multilevel generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices $A_n$arising from virtually any kind of numerical discretization of PDEs. Indeed, when the mesh-fineness parameter $n$ tends to infinity, these matrices $A_n$ give rise to asequence $\\{A_n\\}_n$, which often turns out to be a multilevel GLT sequence or one of its “relatives”, i.e., a multilevel block GLT sequence or a (multilevel) reduced GLTsequence. In particular, multilevel block GLT sequences are encountered in the discretization of systems of PDEs as well as in the higher-order finite element or discontinuousGalerkin approximation of scalar/vectorial PDEs.In this work, we systematically develop the theory of multilevel block GLT sequences as an extension of the theories of (unilevel) GLT sequences[Garoni and Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. I., Springer, Cham, 2017],multilevel GLT sequences[Garoni and Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications. Vol. II., Springer, Cham, 2018],and block GLT sequences[Barbarino, Garoni, and Serra-Capizzano, Electron. Trans. Numer. Anal., 53 (2020), pp. 28–112].We also present several emblematic applications of this theory in the context of PDE discretizations.