BiCGStab Algorithm Research Articles

A distributed-memory MPI parallelization scheme for multi-domain incompressible SPH

A parallel scheme for a multi-domain truly incompressible smoothed particle hydrodynamics (SPH) approach is presented. The proposed method is developed for distributed-memory architectures through the Message Passing Interface (MPI) paradigm as communication between partitions.The proposal aims to overcome one of the main drawbacks of the SPH method, which is the high computational cost with respect to mesh-based methods, by coupling a multi-resolution approach with parallel computing techniques.The multi-domain approach aims to employ different resolutions by subdividing the computational domain into non-overlapping blocks separated by block interfaces. The particles belonging to different blocks are efficiently distributed among processors ensuring well balanced loads.The parallelization procedure handles particle exchanges both throughout the blocks and the competence domains of the processors. The matching of the velocity values between neighbouring blocks is obtained solving a system of interpolation equations at each block interfaces through a parallelized BiCGSTAB algorithm. Otherwise, a whole pseudo-pressure system is solved in parallel considering the Pressure Poisson equations of the fluid particles of all the blocks and the interpolation equations of all the block interfaces.The employed test cases show the strong reduction of the computational efforts of the SPH method thanks to the interaction of the employed multi-resolution approach and the proposed parallel algorithms.

Three-Dimensional Microwave Imaging: Fast and Accurate Computations with Block Resolution Algorithms.

This paper considers the microwave imaging reconstruction problem, based on additive penalization and gradient-based optimization. Each evaluation of the cost function and of its gradient requires the resolution of as many high-dimensional linear systems as the number of incident fields, which represents a large amount of computations. Since all such systems involve the same matrix, we propose a block inversion strategy, based on the block-biconjugate gradient stabilized (BiCGStab) algorithm, with efficient implementations specific to the microwave imaging context. Numerical experiments performed on synthetic data and on real measurements show that savings in computing time can reach a factor of two compared to the standard, sequential, BiCGStab implementation. Improvements brought by the block approach are even more important for the most difficult reconstruction problems, that is, with high-frequency illuminations and/or highly contrasted objects. The proposed reconstruction strategy is shown to achieve satisfactory estimates for objects of the Fresnel database, even on the most contrasted ones.

Electrostatic Capacity of a Metallic Cylinder: Effect of the Moment Method Discretization Process on the Performances of the Krylov Subspace Techniques

When a straight cylindrical conductor of finite length is electrostatically charged, its electrostatic potential ϕ depends on the electrostatic charge qe, as expressed by the equation L(qe)=ϕ, where L is an integral operator. Method of moments (MoM) is an excellent candidate for solving L(qe)=ϕ numerically. In fact, considering qe as a piece-wise constant over the length of the conductor, it can be expressed as a finite series of weighted basis functions, qe=∑n=1Nαnfn (with weights αn and N, number of the subsections of the conductor) defined in the L domain so that ϕ becomes a finite sum of integrals from which, considering testing functions suitably combined with the basis functions, one obtains an algebraic system Lmnαn=gm with dense matrix, equivalent to L(qe)=ϕ. Once solved, the linear algebraic system gets αn and therefore qe is obtainable so that the electrostatic capacitance C=qe/V, where V is the external electrical tension applied, can give the corresponding electrostatic capacitance. In this paper, a comparison was made among some Krylov subspace method-based procedures to solve Lmnαn=gm. These methods have, as a basic idea, the projection of a problem related to a matrix A∈Rn×n, having a number of non-null elements of the order of n, in a subspace of lower order. This reduces the computational complexity of the algorithms for solving linear algebraic systems in which the matrix is dense. Five cases were identified to determine Lmn according to the type of basis-testing functions pair used. In particular: (1) pulse function as the basis function and delta function as the testing function; (2) pulse function as the basis function as well as testing function; (3) triangular function as the basis function and delta function as the testing function; (4) triangular function as the basis function and pulse function as the testing function; (5) triangular function as the basis function with the Galerkin Procedure. Therefore, five Lmn and five pair qe and C were computed. For each case, for the resolution of Lmnαn=gm obtained, GMRES, CGS, and BicGStab algorithms (based on Krylov subspaces approach) were implemented in the MatLab® Toolbox to evaluate qe and C as N increases, highlighting asymptotical behaviors of the procedures. Then, a particular value for N is obtained, exploiting both the conditioning number of Lmn and considerations on C, to avoid instability phenomena. The performances of the exploited procedures have been evaluated in terms of convergence speed and CPU-times as the length/diameter and N increase. The results show the superiority of BcGStab, compared to the other procedures used, since even if the number of iterations increases significantly, the CPU-time decreases (more than 50%) when the asymptotic behavior of all the procedures is in place. This superiority is much more evident when the CPU-time of BicGStab is compared with that achieved by exploiting Gauss elimination and Gauss–Seidel approaches.

A fast finite-difference algorithm for solving space-fractional filtration equation with a generalised Caputo derivative

The paper considers the problem of finite-difference simulation of filtration processes on the base of a space-fractional-differential equation with a generalised Caputo derivative with respect to a function. To make calculations faster, we propose an algorithm of fractional derivative approximation which reduces the computational complexity of matrix operations that arise while performing simulations on uniform and non-uniform grids. The algorithm is based on Taylor’s series expansions of fractional integral kernel and special representation of the matrices of linear systems that arise during finite-difference approximation. Theoretical estimates of the proposed algorithm’s complexity and accuracy are presented. We describe and experimentally compare computational schemes for matrix-vector multiplication used for numerical solution of linear systems by the BicgStab algorithm and analyse factors that influence its convergence. The results of computational experiments which demonstrate an efficiency of the proposed computational scheme for solving a filtration equation with a generalised Caputo derivative with respect to a space variable on a large-sized grids are given.

Сравнение эффективности решателей разреженных систем линейных алгебраических уравнений на основе методов BiCGStab и FGMRES

The efficiency comparison of solvers for sparse linear algebraic equations systems based on one of the fastest iterative methods, the BiCGStab method (bi-conjugate gradient method with stabilization), and the FGMRES method (flexible method of generalized minimal residuals) is presented in this study. Estimates of computational cost per one iteration are presented for the considered methods. The condition imposed on the Krylov subspace dimensionality in the FGMRES is obtained. When this condition is fulfilled, the computational cost per one iteration of the FGMRES method is less than the computational cost per one iteration of the BiCGStab. In addition, the FGMRES modification, which allows to stop the algorithm before the next restart in case of achieving the specified accuracy, is presented. Solvers on the basis of presented the BiCGStab and FGMRES methods algorithms including ILU and multigrid preconditioning are developed on the C++ language for sparse linear algebraic equations systems. The efficiency comparison of developed solvers was carried out on the difference analogs of the Helmholtz and Poisson equations. The systems were taken from the test problem about simulation of the flow around a circular profile, which makes forced transverse oscillations. The difference scheme for the problem solution is constructed by the LS-STAG method (immersed boundaries method with level-set functions). Computational experiments showed that the FGMRES demonstrates a higher convergence rate on problems of this class in comparison with the BiCGStab. The FGMRES usage allowed to reduce the computation time by more than 6.5 times without preconditioning and more than 3 times with preconditioning. The implementation of the modified FGMRES algorithm was also compared with a similar solver from the Intel® Math Kernel Library. Computational experiments showed that the developed FGMRES implementation allowed to obtain acceleration in comparison with Intel® MKL by 3.4 times without preconditioning and by 1.4 times with ILU-preconditioning.

TÌM NGHIỆM CỦA PHƯƠNG TRÌNH POISSON BA CHIỀU SỬ DỤNG THUẬT TOÁN BiCGstab(l)

This paper presents the use of the BiCGstab(l) algorithm to find the solutions for the three-dimensional Poisson’s equation. This subroutine was incorporated into the self-consistent ensemble Monte-Carlo device simulation program to test its efficiency. Moreover, in order to compare the convergent rate and stability of the BiCGstab(l) algorithm to the corresponding characteristics of the BiCGstab algorithm, we investigated the dependence of the logarithm of the Euclidean norm of the residual vector on the number of the main loops of Poisson subroutine. The results showed that the program for solving Poisson’s equation based on the BiCGstab(l) algorithm gives more accurate simulation results as using the BiCGstab algorithm.

A Hessian-free Newton–Raphson method for the configuration of physics systems featured by numerically asymmetric force field

Numerically asymmetric force field is examined in particle-oriented problems such as Quasicontinuum modeling and simulation. To configure the ground state of a large-scale physical system featured by numerically asymmetric force field, we propose a Hessian-free Newton–Raphson method where the Newton equation is solved using central-difference based BiCGstab algorithm (denoted as HFNR-BiCGstab-diff for simplicity). A detailed analytical and experimental investigation on the convergence performance of the HFNR-BiCGstab-diff algorithm is given in this paper. A pure HFNR-BiCGstab-diff algorithm may suffer from unreliable start-up, particularly in the case that the initial guess is far from the minimizer. As a remedy, a hybrid method that couples HFNR-BiCGstab-diff with preconditioned nonlinear conjugate gradient algorithm (PNCG) is developed to achieve optimal computational performance. The algorithms addressed in this paper have been implemented using POSIX-Thread C++. Their performance has been evaluated using three-dimensional Quasicontinuum simulation problems, which are featured by asymmetric force field and large dimensional sizes up to 122,808 degree-of-freedom, as benchmarks. The numerical experiment on IBM SP2 demonstrates that, compared to alternative unconstrained optimization methods such as preconditioned nonlinear conjugate gradient algorithm, the hybrid algorithm saves 20%–60% running time.

Iterative solver approach for turbine interactions: application to wind or marine current turbine farms

This paper presents a numerical investigation for the computation of wind or marine current turbines in a farm. A 3D unsteady Lagrangian vortex method is used together with a panel method in order to take into account for the turbines. In order to enforce the boundary condition onto the panel elements, a linear matrix system is defined. Solving general linear matrix systems is a topic with important scientific literature. But the main concern here is the application to a dedicated matrix which is non-sparse, non-symmetric, neither diagonally dominant nor positive-definite. Several iterative approaches were tested and compared. But after some numerical tests, a Bi-CGSTAB method was finally chosen. The main advantage of the presented method is the use of a specific preconditioner well suited for the desired application. The chosen implementation proved to be very efficient with only 3 iterations of our preconditioned Bi-CGSTAB algorithm whatever the turbine geometrical configuration. Although developed for wind or marine turbines, the proposed algorithm is absolutely not restricted to these cases, and can be applied to many others. At the end of the paper, some applications (specifically, wake computations) in a farm are presented, along with a quantitative assessment of the computational time savings brought by the iterative approach.

A Rapid Online Calculation Method of Three-dimensional Plastic Deformation in Strip Rolling

Three-dimensional strip plastic deformation model is a key part of shape setup model in hot rolling. The numerical efficiency, accuracy and stability of the method are the main factors that may restrict its online application. Considering the theoretical limitations of existing models, basic equations include equilibrium equations with the effect of shear stress, constitutive equations based on incremental theory, and lubricated friction with the respective coefficients of friction at longitudinal and transverse direction. After normalization of parameters and the asymptotic analysis, the strip plastic deformation model is established, and solved by finite volume method and BICGSTAB algorithm to get the transverse distribution of rolling force of each stand. Compared to the finite element model established by ANSYS software, the new model has reliable results and a short computation time of 31 milliseconds, which show its feasibility of online application.

Analysis and Practical Use of Flexible BiCGStab

A flexible version of the BiCGStab algorithm for solving a linear system of equations is analyzed. We show that under variable preconditioning, the perturbation to the outer residual norm is of the same order as that to the application of the preconditioner. Hence, in order to maintain a similar convergence behavior to BiCGStab while reducing the preconditioning cost, the flexible version can be used with a moderate tolerance in the preconditioning Krylov solves. We explored the use of flexible BiCGStab in a large-scale reacting flow application, PFLOTRAN, and showed that the use of a variable multigrid preconditioner significantly accelerates the simulation time on extreme-scale computers using $$O(10^4)$$O(104)---$$O(10^5)$$O(105) processor cores.

A Note on the Block and Seed BiCGSTAB Algorithms for Nonsymmetric Multiple Linear Systems

In this paper we ϐirst give a new derivation of the Bl-BiCGSTAB algorithm for solving block linear systems. This derivation is based only on linear algebra and didn't needs matrix orthogonal polynomials. We also propose a new method for solving linear systems with the same coefϐicient matrix and different right-hand sides. The proposed method is based on the BiCGSTAB algorithm and is especially efϐicient when the right-hand sides vector-columns are closely related. Some numerical examples are reported to illustrate the effectiveness of the proposed methods.

Recycling BiCGSTAB with an Application to Parametric Model Order Reduction

Krylov subspace recycling is a process for accelerating the convergence of sequences of linear systems. Based on this technique, the recycling BiCG algorithm has been developed recently. Here, we now generalize and extend this recycling theory to BiCGSTAB. Recycling BiCG focuses on efficiently solving sequences of dual linear systems, while the focus here is on efficiently solving sequences of single linear systems (assuming non-symmetric matrices for both recycling BiCG and recycling BiCGSTAB). As compared with other methods for solving sequences of single linear systems with non-symmetric matrices (e.g., recycling variants of GMRES), BiCG based recycling algorithms, like recycling BiCGSTAB, have the advantage that they involve a short-term recurrence, and hence, do not suffer from storage issues and are also cheaper with respect to the orthogonalizations. We modify the BiCGSTAB algorithm to use a recycle space, which is built from left and right approximate invariant subspaces. Using our algorithm for a parametric model order reduction example gives good results. We show about 40% savings in the number of matrix-vector products and about 35% savings in runtime.

Investigation of the Finite Planar Frequency Selective Surface with Defect Patterns

In this paper, RCS characteristics on defect pattern of crossed dipole slot FSS having a finite size have been analyzed. To analyze RCS, we applied the electric field integral equation analysis which applies BiCGSTab algorithm with iterative method and uses RWG basis function. To verify the validity of this paper, RCS of PEC sphere has been compared to the theoretical results and FSSs with defect patterns are fabricated and measured. As defect patterns in FSS, missing one column, missing some elements, and discontinuity in surfaces are simulated and compared with the measurement results. Resonant frequency shifts in pass band and changes in bandwidth are observed. From the results, precisely predicting and designing frequency characteristics over defect patterns are essential when applying FSS structures such as FSS radomes.

ML([formula omitted])BiCGStabt: A ML([formula omitted])BiCGStab variant with [formula omitted]-transpose

The 1980 IDR method (Wesseling and Sonneveld, 1980 [12]) plays an important role in the history of Krylov subspace methods. It started the research of transpose-free Krylov subspace methods. The ML(n)BiCGStab method (Yeung, 2012) is one of such methods. In this paper, we present a new ML(n)BiCGStab variant that involves A-transpose in its implementation. Comparison of this new algorithm with the existing ML(n)BiCGStab algorithms and some other Krylov subspace algorithms will be presented.

Iterative methods for 3D implicit finite-difference migration using the complex Padé approximation

Conventional implementations of 3D finite-difference (FD) migration use splitting techniques to accelerate performance and save computational cost. However, such techniques are plagued with numerical anisotropy that jeopardises the correct positioning of dipping reflectors in the directions not used for the operator splitting. We implement 3D downward continuation FD migration without splitting using a complex Pade approximation. In this way, the numerical anisotropy is eliminated at the expense of a computationally more intensive solution of a large-band linear system. We compare the performance of the iterative stabilized biconjugate gradient (BICGSTAB) and that of the multifrontal massively parallel direct solver (MUMPS). It turns out that the use of the complex Pade approximation not only stabilizes the solution, but also acts as an effective preconditioner for the BICGSTAB algorithm, reducing the number of iterations as compared to the implementation using the real Pade expansion. As a consequence, the iterative BICGSTAB method is more efficient than the direct MUMPS method when solving a single term in the Pade expansion. The results of both algorithms, here evaluated by computing the migration impulse response in the SEG/EAGE salt model, are of comparable quality.

The generalized QMRCGSTAB algorithm for solving Sylvester-transpose matrix equations

In this work we generalize the Quasi-Minimal Residual Variant of the Bi-CGSTAB algorithm (QMRCGSTAB) to obtain a matrix iterative method for solving the Sylvester-transpose matrix equation ∑ i = 1 l ( A i X B i + C i X T D i ) = E . In order to compare the generalized algorithm with some existing methods, we present two numerical examples.

유한크기 FSS의 곡률효과에 따른 주파수 특성 해석

In this paper, we have analyzed the curvature effects of the finite FSS(frequency selective surface) using the 3-D method of moment using the BiCGSTab algorithm as an iterative method. To validate the analysis method in this paper, we compared the RCS(radar cross section) of PEC(perfect electric conductor) sphere with theoretical results and it shows well agreements. The tripole slot FSSs which have many application in military were selected for the investigation of curvature effect and RCS as a frequency characteristics were observed with the variation of the curvature rate. Simulation results shows that curvature effect can significant effect the passband frequency and bandwidth of FSS. We suggest that the curvature effect must be considered at the stage of design FSS application like FSS radome.

A second-order sharp numerical method for solving the linear elasticity equations on irregular domains and adaptive grids – Application to shape optimization

We present a numerical method for solving the equations of linear elasticity on irregular domains in two and three spatial dimensions. We combine a finite volume and a finite difference approaches to derive discretizations that produce second-order accurate solutions in the L∞-norm. Our discretization is ‘sharp’ in the sense that the physical boundary conditions (mixed Dirichlet/Neumann-type) are imposed at the interface and the solution is computed inside the irregular domain only, without the need of smearing the solution across the interface. The irregular domain is represented implicitly using a level-set function so that this approach is applicable to free moving boundary problems; we provide a simple example of shape optimization to illustrate this capability. In addition, we provide an extension of our method to the case of adaptive meshes in both two and three spatial dimensions: we use non-graded quadtree (2D) and octree (3D) data structures to represent the grid that is automatically refined near the irregular domain’s boundary. This extension to quadtree/octree grids produces second-order accurate solutions albeit non-symmetric linear systems, due to the node-based sampling nature of the approach. However, the linear system can be solved with simple linear solvers; in this work we use the BICGSTAB algorithm.

ML(n)BiCGStab: Reformulation, Analysis and Implementation

With the aid of index functions, we re-derive the ML(n)BiCGStab algorithm in [39] more systematically. There are n ways to define the ML(n)BiCGStab residual vector. Each definition leads to a different ML(n)BiCGStab algorithm. We demonstrate this by presenting a second algorithm which requires less storage. In theory, this second algorithm serves as a bridge that connects the Lanczos-based BiCGStab and the Arnoldi-based FOM while ML(n)BiCG is a bridge connecting BiCG and FOM. We also analyze the breakdown situation from the probabilistic point of view and summarize some useful properties of ML(n)BiCGStab. Implementation issues are also addressed.

ML(n)BiCGStab: Reformulation, Analysis and Implementation

With the aid of index functions, we re-derive the ML(n)BiCGStab algorithm in a paper by Yeung and Chan in 1999 in a more systematic way. It turns out that there are n ways to define the ML(n)BiCGStab residual vector. Each definition will lead to a different ML(n)BiCGStab algorithm. We demonstrate this by presenting a second algorithm which requires less storage. In theory, this second algorithm serves as a bridge that connects the Lanczos-based BiCGStab and the Arnoldi-based FOM while ML(n)BiCG a bridge connecting BiCG and FOM. We also analyze the breakdown situations from the probabilistic point of view and summarize some useful properties of ML(n)BiCGStab. Implementation issues are also addressed.