Block Iterative Methods Research Articles

Numerical analysis of the stressed-deformed state of a tubular element under thermal loading

In articles [2-5] solving relations and the block iteration method algorithm for solving linear and nonlinear equations by the semi-analytical finite element method for curvilinear heterogeneous prismatic bodies are implemented. In article [1], a numerical study of the convergence of the solution was performed, a wide range of test problems for bodies with smoothly and abruptly changing physical and geometric characteristics in elastic and elastic-plastic settings were considered. In [6], to confirm the reliability of the results obtained on the basis of the semi-analytical method of finite elements, the effectiveness of the application of this approach for the calculation of curvilinear inhomogeneous prismatic objects is shown. Solving control problems of the theory of elasticity, thermoelasticity, and thermoplasticity, as well as shape change problems, makes it possible to draw a conclusion about the reliability of the results of the research of a selected class of objects based on the developed methodology and implements its application program package.
 In this paper, using the methodology outlined in the above works, the results of the numerical solution of the problem, which have an applied value, were presented. The stress-strain state of a tubular element with a rectangular cutout under conditions of thermoforce loading was studied. As a rule, solving similar problems is carried out in a flat setting without taking into account the bending load. This approach greatly simplifies the formulation of the problem, but leads to very significant errors in the results. At zero value of the bending load, a good agreement of the planar and spatial solutions is observed. The presence of a bending load leads to deviation of the curves from their initial position. The maximum discrepancy of the results is almost 50%.
 The analysis of this structure from the standpoint of the spatial problem of thermoplasticity, which ensures taking into account the dependence of the physical and mechanical characteristics of the material on temperature and taking into account the bending load on the section of the cut, allowed us to reveal the real features of its deformation.

Numerical considerations of block GMRES methods when applied to linear discrete ill-posed problems

Linear systems of equations with a matrix whose singular values decay to zero with increasing index number without a significant gap are commonly referred to as linear discrete ill-posed problems. We are interested in solving large systems of this kind when the right-hand side has k>1 column vectors. The systems may be regarded as one system of equations with a block vector (with k columns) as the right-hand side and then solved by a block iterative method, or as k linear systems of equations (one for each right-hand side vector) that can be solved independently. Thus, the solution is a block vector with k columns. In many applications, including the restoration of color images, the right-hand side represents measurements that are contaminated by errors. Block iterative methods compute all columns of the solution block vector simultaneously. We will illustrate the performance of standard block GMRES methods and global GMRES methods, which also are block methods, and show that they may determine computed solutions of lower quality than when each column of the solution block vector is computed independently by a “standard” iterative method. We introduce a new local block GMRES method that can overcome the problems associated with block GMRES methods applied to linear discrete ill-posed problems.

Randomized average block iterative methods for solving factorised linear systems

Recently, some randomized iterative methods are proposed to solve large-scale factorised linear systems. In this paper, we present two randomized average block iterative methods which still take advantage of the factored form and need not perform the entire matrix. The new methods are pseudoinverse-free and can be implemented for parallel computation. Furthermore, we analyze their convergence behaviors and obtain the exponential convergence rate. Finally, some numerical examples are carried out to show the effectiveness of our new methods.

A fully-coupled framework for solving Cahn-Hilliard Navier-Stokes equations: Second-order, energy-stable numerical methods on adaptive octree based meshes

We present a fully-coupled, implicit-in-time framework for solving a thermodynamically-consistent Cahn-Hilliard Navier-Stokes system that models two-phase flows. In this work, we extend the block iterative method presented in Khanwale et al. [Simulating two-phase flows with thermodynamically consistent energy stable Cahn-Hilliard Navier-Stokes equations on parallel adaptive octree based meshes, J. Comput. Phys. (2020)], to a fully-coupled, provably second-order accurate scheme in time, while maintaining energy-stability. The new method requires fewer matrix assemblies in each Newton iteration resulting in faster solution time. The method is based on a fully-implicit Crank-Nicolson scheme in time and a pressure stabilization for an equal order Galerkin formulation. That is, we use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) procedure to stabilize the pressure. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise, Rayleigh-Taylor instability, and lid-driven cavity flow problems. We analyze in detail the scaling of our numerical implementation.

Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings

<p style='text-indent:20px;'>In this paper, for solving the variational inequality problem over the set of common fixed points of a finite family of demiclosed quasi-nonexpansive mappings in Hilbert spaces, we propose two new strongly convergent methods, constructed by specific combinations between the steepest-descent method and the block-iterative ones. The strong convergence is proved without the boundedly regular assumptions on the family of fixed point sets as well as the approximately shrinking property for each mapping of the family, that are usually assumed in recent literature for similar problems. Applications to the multiple-operator split common fixed point problem (MOSCFPP) and the problem of common minimum points of a finite family of lower semi-continuous convex functions with numerical experiments are given.</p>

Convergence and Semi-Convergence of a Class of Constrained Block Iterative Methods

In this paper, we analyze the convergence properties of projected non-stationary block iterative methods (P-BIM) aiming to find a constrained solution to large linear, usually both noisy and ill-conditioned, systems of equations. We split the error of the kth iterate into noise error and iteration error, and consider each error separately. The iteration error is treated for a more general algorithm, also suited for solving split feasibility problems in Hilbert space. The results for P-BIM come out as a special case. The algorithmic step involves projecting onto closed convex sets. When these sets are polyhedral, and of finite dimension, it is shown that the algorithm converges linearly. We further derive an upper bound for the noise error of P-BIM. Based on this bound, we suggest a new strategy for choosing relaxation parameters, which assist in speeding up the reconstruction process and improving the quality of obtained images. The relaxation parameters may depend on the noise. The performance of the suggested strategy is shown by examples taken from the field of image reconstruction from projections.

Efficient extended-search space full-waveform inversion with unknown source signatures

SUMMARY Full waveform inversion (FWI) requires an accurate estimation of source signatures. Due to the coupling between the source signatures and the subsurface model, small errors in the former can translate into large errors in the latter. When direct methods are used to solve the forward problem, classical frequency-domain FWI efficiently processes multiple sources for source signature and wavefield estimations once a single lower–upper (LU) decomposition of the wave-equation operator has been performed. However, this efficient FWI formulation is based on the exact solution of the wave equation and hence is highly sensitive to the inaccuracy of the velocity model due to the cycle skipping pathology. Recent extended-space FWI variants tackle this sensitivity issue through a relaxation of the wave equation combined with data assimilation, allowing the wavefields to closely match the data from the first inversion iteration. Then, the subsurface parameters are updated by minimizing the wave-equation violations. When the wavefields and the source signatures are jointly estimated with this approach, the extended wave equation operator becomes source dependent, hence making direct methods and, to a lesser extent, block iterative methods ineffective. In this paper, we propose a simple method to bypass this issue and estimate source signatures efficiently during extended FWI. The proposed method replaces each source with a blended source during each data-assimilated wavefield reconstruction to make the extended wave equation operator source independent. Besides computational efficiency, the additional degrees of freedom introduced by spatially distributing the sources allows for a better signature estimation at the physical location when the velocity model is rough. We implement the source signature estimation with a variable projection method in the recently proposed iteratively refined wavefield reconstruction inversion (IR-WRI) method. Numerical tests on the Marmousi II and 2004 BP salt synthetic models confirm the efficiency and the robustness against velocity model errors of the new method compared to the case where source signatures are known.

Semi-analytical method of finished elements in elastic and elastic-plastic position for curviline prismatic objects

In [4, 5, 6] the algorithm of the method of block iterations of solving linear and nonlinear equations by the semivanalytic finite element method for curvilinear inhomogeneous prismatic bodies is realized. This paper presents the results of the effectiveness of the semi-analytical finite element method for the consideration of curvilinear prismatic objects in elastic and elastic-plastic formulation.
 The choice of the optimal in terms of machine time and speed of convergence of the iterative process algorithm for solving systems of linear and nonlinear equations by the semivanalytic finite element method [1, 2, 3] is an important factor influencing the efficiency of the method as a whole. Numerous studies have shown that using the block iteration method to solve systems of equations of the semivanalytic finite element method for prismatic bodies with variable parameters has a number of important advantages over solving systems of the traditional variant of the finite element method.
 The organization of the computational process and its software implementation takes into account the basic requirements for software for calculating strength on modern software packages. The modular structure of the developed system of programs provides its non-closedness concerning new classes of tasks.
 The use of the block iteration method to solve systems of nonlinear equations of SAFEM is approximately an order of magnitude superior to the traditional finite element method.

An Alternative Approach for Finding Newton's Direction in Solving Large-Scale Unconstrained Optimization for Problems with an Arrowhead Hessian Matrix

In this paper, we proposed an alternative way to find the Newton direction in solving large-scale unconstrained optimization problems where the Hessian of the Newton direction is an arrowhead matrix. The alternative approach is a two-point Explicit Group Gauss-Seidel (2EGGS) block iterative method. To check the validity of our proposed Newton’s direction, we combined the Newton method with 2EGGS iteration for solving unconstrained optimization problems and compared it with a combination of the Newton method with Gauss-Seidel (GS) point iteration and the Newton method with Jacobi point iteration. The numerical experiments are carried out using three different artificial test problems with its Hessian in the form of an arrowhead matrix. In conclusion, the numerical results showed that our proposed method is more superior than the reference method in term of the number of inner iterations and the execution time.

Dynamic characteristics of elastic ring squeeze film damper

Purpose The purpose of this paper is to present a numerical model to investigate the dynamic behavior and force coefficients of a compact squeeze film damper with dual film clearances adjusted by an elastic ring, known as elastic ring squeeze film damper (ERSFD). Design/methodology/approach The governing equations of ERSFD as well as the boundary conditions are obtained based on Reynolds equation. A simplified Greenwood–Williamson model is implemented to investigate the contact behavior between the elastic ring and the journal. The interactions between the films and the elastic ring are achieved by block iterative method. Findings The radial deformation as well as velocity of the elastic ring are captured to illustrate the pressure profiles of the inner and outer films. High-order frequency components related to the number of the boss N are observed on the frequency spectrum of the film force. The force coefficients of the ERSFD are constant for a wider range of non-dimensional whirling radius ε compared with conventional squeeze film damper. Originality/value The force coefficients of the ERSFD are obtained by assuming that the journal center moves in a circular centered orbit. High-order frequency components related to the number of bosses N are observed. These findings may provide helpful materials for the application of the ERSFD.

Application of Four-Point EGSOR Iteration with Nonlocal Arithmetic Mean Discretization Scheme for Solving Burger’s Equation

The main objective for this study is to examine the efficiency of block iterative method namely Four-Point Explicit Group Successive Over Relaxation (4EGSOR) iterative method. The nonlinear Burger’s equation is then solved through the application of nonlocal arithmetic mean discretization (AMD) scheme to form a linear system. Next, to scrutinize the efficiency of 4EGSOR with Gauss-Seidel (GS) and Successive Over Relaxation (SOR) iterative methods, the numerical experiments for four proposed problems are being considered. By referring to the numerical results obtained, we concluded that 4EGSOR is more superior than GS and SOR iterative methods in aspects of number of iterations and execution time.

Block Iterative Method for the Solution of Fractional Two-Point Boundary Value Problems

The goal of this paper deals with 4-Point Explicit Group (4-EG) iterative method as solver to solve the linear system which is generated by using Caputo’s finite difference approximation equation. Actually, this approximation equation was been generated from the discretization process of the fractional two-point boundary value problems via finite difference scheme and Caputo’s fractional operator. The formulation and implementation for 4-EG method have been also included. To investigate the efficiency of 4-EG method, this paper considered three examples numerical experiment. Based on numerical results, the finding showed that the 4-EG method is more efficient as compared with GS method in good agreement.

Newton-2EGSOR Method for Unconstrained Optimization Problems with a Block Diagonal Hessian

We find that finding the unconstrained optimizer for large-scale problems with the Newton method can be very expensive. Therefore, in this paper, we developed a new efficient method for solving large-scale unconstrained optimization problems with block diagonal Hessian matrices to reduce the cost of computing Newton’s direction. This method is a combination of the Newton method and the 2-point explicit group successive-over relaxation (2EGSOR) block iterative method. To calculate the performance of the developed method, we used a combination of the Newton method with Gauss–Seidel (Newton-GS) point iteration and the Newton method with successive-over relaxation (Newton-SOR) point iteration as reference methods. The numerical experiment has proven that the developed algorithms generate results that are more efficient compared to the reference methods with less execution time and fewer iterations. Through 90 test cases, we observe that the speedup ratio for our proposed method is up to 659.87 times faster compared to the Newton-SOR method and up to 1963.57 times more rapidly than the Newton-GS method.

A preconditioned modulus-based matrix multisplitting block iteration method for the linear complementarity problems with Toeplitz matrix

A preconditioned modulus-based matrix multisplitting block iteration method is presented for solving the linear complementarity problem with symmetric positive definite Toeplitz matrix. We choose Strang’s preconditioner or T. Chan’s preconditioner as preconditioner in the method. The method has faster convergence rate and less computational work. We also analyze convergence of the method, and show that the new method is effective with some numerical results.

An Adaptive Iterative Thresholding Algorithm for Distributed MIMO Radars

In this paper, a Block Iterative Method with Adaptive Thresholding for Sparse Recovery (BIMATSR) is proposed to recover the received signal in an under-sampled distributed multiple-input multiple-output radar. The BIMATSR scheme induces block sparsity with the aid of a signal-dependent thresholding operator which increases the accuracy of the target parameter estimation task. We have proved that under some sufficient conditions, the suggested scheme converges to a stable solution. Moreover, different simulation scenarios confirm that the BIMATSR algorithm outperforms its counterparts in terms of the target parameter estimation. This superiority is achieved at the expense of slightly more computational complexity. It is also shown that in this application, random sampling scheme offers better performance than random Gaussian matrix, while it has simpler structure.

A Block Iteration with Parallelization Method for the Greedy Selection in Radial Basis Functions Based Mesh Deformation

Greedy algorithm is one of the important point selection methods in the radial basis function based mesh deformation. However, in large-scale mesh, the conventional greedy selection will generate expensive time consumption and result in performance penalties. To accelerate the computational procedure of the point selection, a block iteration with parallelization method is proposed in this paper. By the block iteration method, the computational complexities of three steps in the greedy selection are all reduced from O ( n 3 ) to O ( n 2 ) . In addition, the parallelization of two steps in the greedy selection separates boundary points into sub-cores, efficiently accelerating the procedure. Specifically, three typical models of three-dimensional undulating fish, ONERA M6 wing and three-dimensional Super-cavitating Hydrofoil are taken as the test cases to validate the proposed method and the results show that it improves 17.41 times performance compared to the conventional method.

PERFORMANCE ANALYSIS OF FOUR-POINT EGAOR ITERATIVE METHOD APPLIED TO POISSON IMAGE BLENDING PROBLEM

Poisson image blending is one of the useful editing tools in image processing to generate a desirable image which is impossible to acquire. The key to this solution is to obtain the unique solution of Poisson equation. Thus, the motivation of this paper is to examine the effectiveness of 4-EGAOR iterative method to solve the linear system generated from the Poisson image blending problem. To evaluate its effectiveness, the formulation and implementation of 4-EGAOR, SOR and AOR iterative methods are demonstrated. The numerical results revealed that 4-EGAOR iterative method improved the computational time taken and reduced the number of iterations used. In fact, the new images generated by the proposed block iterative method give a satisfactory visual effect.

Mathematical modelling of filtration processes in drainage systems using conformal mapping

AbstractThe situation when groundwater considerably rises above the “normal” level, water intake, lowering of groundwater levels and other relevant practical tasks require the drainage facilities. The most effective techniques of numerical studies of the corresponding boundary problems at present time are methods of dealing with inverse boundary value problems (conformal and quasi-conformal mappings). As basis of this research we used the case of combining the fictitious domain methods with quasi-conformal mappings of the solution of nonlinear boundary value problems for the calculation of filtration regimes in environments with free boundary areas (depression curves) and zones of “mountainous” areas.This paper reviews the stationary issue of flat-vertical stationary non-pressure liquid filtration to horizontal symmetric drainage. In the paper a practical methodology for solving boundary value problems on conformal mappings is suggested for the calculation of the filtration process in the horizontal symmetrical drainage.The idea of block iterative methods was used during the creation of the corresponding algorithm which is based on the alternating “freeze” of the anticipated conformance parameter, the internal and boundary connections of the curvilinear area.The results of the conducted numerical calculations confirmed the effectiveness of the suggested problem formulations and algorithms of their numerical solution and the possibility of their use in the modelling of nonlinear filtration processes occurring in horizontal drainage systems, as well as in the design of drainage facilities and optimizing other hydrosystems. Therefore these results are of great importance.

Iterative Method for Simultaneous Sparse Approximation

This paper studies the problem of Simultaneous Sparse Approximation (SSA). This problem arises in many applications which work with multiple signals maintaining some degree of dependency such as radar and sensor networks. In this paper, we introduce a new method towards joint recovery of several independent sparse signals with the same support. We provide an analytical discussion on the convergence of our method called Simultaneous Iterative Method with Adaptive Thresholding (SIMAT). Additionally, we compare our method with other group-sparse reconstruction techniques, i.e., Simultaneous Orthogonal Matching Pursuit (SOMP), and Block Iterative Method with Adaptive Thresholding (BIMAT) through numerical experiments. The simulation results demonstrate that SIMAT outperforms these algorithms in terms of the metrics Signal to Noise Ratio (SNR) and Success Rate (SR). Moreover, SIMAT is considerably less complicated than BIMAT, which makes it feasible for practical applications such as implementation in MIMO radar systems.

Block hybrid multilevel method to compute the dominant [formula omitted]-modes of the neutron diffusion equation

The dominant λ-modes associated with a nuclear reactor configuration describe the neutron steady-state distribution and its criticality. Furthermore, they are useful to develop modal methods to study reactor instabilities. Different eigenvalues solvers have been successfully used to obtain such modes, most of them are implemented reducing the original generalized eigenvalue problem to an ordinary one. Thus, it is necessary to solve many linear systems making these methods not very efficient, especially for large problems. In this work, the original generalized eigenvalue problem is considered and two block iterative methods to solve it are studied: the block inverse-free preconditioned Arnoldi method and the modified block Newton method. All of these iterative solvers are initialized using a block multilevel technique. A hybrid multilevel method is also proposed based on the combination of the methods proposed. Two benchmark problems are studied illustrating the convergence and the competitiveness of the methods proposed. A comparison with the Krylov-Schur method and the Generalized Davidson is also included.