Generalized Minimum Residual Algorithm Research Articles

Least‐squares reverse time migration based on an efficient pure qP‐wave equation

AbstractSeismic waves propagating in anisotropic earth media suffer from waveform distortion, which leads to degraded resolution in seismic data imaging if this adverse effect is not corrected. Additionally, the wavefields simulated by the traditional coupled pseudo‐acoustic wave equations contain undesired shear wave artefacts and are unstable when Thomsen's anisotropy parameter ε is less than δ. To address these issues, many pure qP‐wave equations are proposed to describe wave propagation in anisotropic media. However, these equations are either low precision or high accuracy but computationally expensive. Considering the trade‐off between the computation efficiency and simulation accuracy, we derive a pure qP‐wave equation from the exact dispersion formula in tilted transverse isotropic media. The newly derived equation has only two higher order partial derivatives, which can better balance the accuracy and efficiency than the previous pure qP‐wave equations. The least‐squares reverse time migration method can balance amplitudes, suppress migration artefacts and improve imaging resolution. Then, based on the newly derived pure qP‐wave equation, we derive the Born modelling operator and adjoint migration operator and develop an anisotropic least‐squares reverse time migration method. To achieve the accelerated convergence, we introduce a generalized minimum residual algorithm to implement the anisotropic least‐squares reverse time migration method. Numerical simulation results show that the proposed pure qP‐wave equation outperforms the previous equations in balancing the trade‐off between accuracy and efficiency. In addition, synthetic examples demonstrate the effectiveness and robustness of our proposed anisotropic least‐squares reverse time migration method in correcting the deviations due to anisotropic effects.

Three-dimensional preconditioned FM-IBEM solution to broadband-frequency seismic wave scattering in a layered sedimentary basin

A three-dimensional (3-D) fast multipole indirect boundary element method (FM-IBEM) combined with an inner-outer preconditioned generalized minimal residual (GMRES) method algorithm is proposed to solve the broadband-frequency seismic wave scattering problem in complex sites. This method is established in three steps: first, the scattering wave field is developed with the dynamic Green's function; second, the dynamic Green's functions are expanded to generate a traditional FM-IBEM framework; and finally, the preconditioning matrix and GMRES algorithm are introduced to pre-process the dynamic Green's functions. Using this method, the broadband-frequency scattering of normally incident seismic waves in a hemispherical layered sedimentary basin is investigated in the frequency and time domains. The results show that the proposed method is highly accurate and efficient for solving low- and high-frequency seismic wave problems. Compared with the FM-IBEM without preconditioning, the computational time and storage space required for the FM-IBEM with preconditioning are reduced by 84% and 54%, respectively. The layered sedimentary basin has significant wave amplification (of up to 10 times) for dimensionless frequencies, η, in the range of 0.5–3.0; however, for η > 5, the amplification effect becomes weak.

A novel preconditioned range restricted GMRES algorithm for moving force identification and its experimental validation

Moving force identification (MFI) is a widely concerned inverse problem in structural dynamics and well-known as intrinsically existing ill-posedness. With the help of Arnoldi process and Krylov subspace method, the generalized minimal residual (GMRES) method can be improved to a range restricted generalized minimal residual (RRGMRES) method. Furthermore, by introducing the smoothing-norm preconditioning, a preconditioned range restricted generalized minimal residual (PRRGMRES) method is proposed to provide a stable solution to the ill-posed dynamic force identification problem. Simulations show that the novel method has significant improvement when compared to the classic time domain method and the RRGMRES method. In addition, to show the effectiveness and advantages of the proposed method, the PRRGMRES method is also compared with a newly-proposed regularization method named the preconditioned least square QR-factorization (PLSQR) method. Simulation results show that the PRRGMRES method has much better robustness and higher computational efficiency than the PLSQR method especially in dealing with highly inaccurate measurement cases. Finally, the accuracy and efficiency of the PRRGMRES method is verified by experimental studies. The PRRGMRES method has good performance in both overcoming ill-posed problems and improving computational efficiency, which should be of the highest priority in adoption for MFI.

GMRES Algorithm for Large-Scale Vectoring in DSL Systems

We propose an iterative crosstalk cancellation scheme based on the generalized minimal residual (GMRES) algorithm for large-scale digital subscriber line (DSL) systems. The proposed scheme does not require channel inversion and stores fewer vectors for crosstalk cancellation. We analyze the convergence of the GMRES algorithm and derive computable bounds on the residual error and signal-to-noise ratio in terms of system parameters at each iteration for upstream DSL systems. We show that the GMRES algorithm typically requires a single iteration for very large vectored systems to achieve crosstalk-free performance for the very high-speed DSL (VDSL) frequencies and only a few more in the highest frequency bands of the G.fast spectrum. This yields significant complexity savings and reduction in memory storage, compared to the zero forcing scheme under certain conditions.

The inexact-Newton via GMRES subspace method without line search technique for solving symmetric nonlinear equations

In this paper, we propose an inexact-Newton via GMRES (generalized minimal residual) subspace method without line search technique for solving symmetric nonlinear equations. The iterative direction is obtained by solving the Newton equation of the system of nonlinear equations with the GMRES algorithm. The global convergence and local superlinear convergence rate of the proposed algorithm are established under some reasonable conditions. Finally, the numerical results are reported to show the effectiveness of the proposed algorithm.

An implicit Hermite WENO reconstruction-based discontinuous Galerkin method on tetrahedral grids

An Implicit Reconstructed Discontinuous Galerkin method, IRDG (P1P2), is presented for solving the compressible Euler equations on tetrahedral grids. In this method, a quadratic polynomial (P2) solution is first reconstructed using a least-squares method from the underlying linear polynomial (P1) DG solution. By taking advantage of the derivatives in the DG formulation, the stencils used in the reconstruction involve only von Neumann neighborhood (adjacent face-neighboring cells) and thus are compact and consistent with the underlying DG method. The final P2 solution is then obtained using a WENO reconstruction, which is necessary to ensure stability of the RDG (P1P2) method. A matrix-free GMRES (generalized minimum residual) algorithm is presented to solve the approximate system of linear equations arising from Newton linearization. The LU-SGS (lower–upper symmetric Gauss–Seidel) preconditioner is applied with both the simplified and approximate Jacobian matrices. The numerical experiments on a variety of flow problems demonstrate that the developed IRDG (P1P2) method is able to obtain a speedup of at least two orders of magnitude than its explicit counterpart, maintain the linear stability, and achieve the designed third order of accuracy: one order of accuracy higher than the underlying second-order DG (P1) method without significant increase in computing costs and storage requirements. It is also found that a well approximated Jacobian matrix is essential for the IRDG method to achieve fast converging speed and maintain robustness on large-scale problems.

Imposing symmetry in augmented linear systems

SUMMARY This paper discusses the methods of imposing symmetry in the augmented system formulation (ASF) for least-squares (LS) problems. A particular emphasis is on upper Hessenberg problems, where the challenge lies in leaving all zero-by-definition elements of the LS matrix unperturbed. Analytical solutions for optimal perturbation matrices are given, including upper Hessenberg matrices. Finally, the upper Hessenberg LS problems represented by unsymmetric ASF that indicate a normwise backward stability of the problem (which is not the case in general) are identified. It is observed that such problems normally arise from Arnoldi factorization (for example, in the generalized minimal residual (GMRES) algorithm). The problem is illustrated with a number of practical (arising in the GMRES algorithm) and some ‘purpose-built’ examples. Copyright © 2014 John Wiley & Sons, Ltd.

A matrix free implicit scheme for solution of resistive magneto-hydrodynamics equations on unstructured grids

The resistive magneto-hydrodynamics (MHD) governing equations represent eight conservation equations for the evolution of density, momentum, energy and induced magnetic fields in an electrically conducting fluid, typically a plasma. A matrix free implicit method is developed to solve the conservation equations within the framework of an unstructured grid finite volume formulation. The analytic form of the convective flux Jacobian is derived on a general unstructured mesh and used in a Lower-Upper Symmetric Gauss Seidel (LU-SGS) technique developed as part of the implicit scheme. A grid coloring technique is also developed to create data parallelism in the algorithm. The computational efficiency of the matrix free method is compared with two common approaches: a global matrix solve technique that uses the GMRES (Generalized minimum residual) algorithm and an explicit method. The matrix-free method is observed to be overall computationally faster than the global matrix solve method and demonstrates excellent parallel scaling on multiple cores. The computational effort and memory requirements for the matrix free approach is comparable to the explicit approach which in turn is much lower than the global solve implicit approach. Both the matrix free and global solve implicit techniques exhibit superior steady state convergence compared to the explicit method.

Preconditioned Krylov Solution of Response Matrix Equations

Multigrid-preconditioned Krylov methods are applied to within-group response matrix equations of the type derived from the variational nodal method for neutron transport with interface conditions represented by orthogonal polynomials in space and spherical harmonics in angle. Since response matrix equations result in nonsymmetric coefficient matrices, the generalized minimal residual (GMRES) Krylov method is employed. Two acceleration methods are employed: response matrix aggregation and multigrid preconditioning. Without approximation, response matrix aggregation combines fine-mesh response matrices into coarse-mesh response matrices with piecewise-orthogonal polynomial interface conditions; this may also be viewed as a form of nonoverlapping domain decomposition on the coarse grid. Two-level multigrid preconditioning is also applied to the GMRES method by performing auxiliary iterations with one degree of freedom per interface that conserve neutron balance for three types of interface conditions: (a) p preconditioning is applied to orthogonal polynomial interface conditions (in conjunction with matrix aggregation), (b) h preconditioning to piecewise-constant interface conditions, and (c) h-p preconditioning to piecewise-orthogonal polynomial interface conditions. Alternately, aggregation is employed outside the GMRES algorithm to coarsen the grid, and multigrid preconditioning is then applied to the coarsened equations. The effectiveness of the combined aggregation and preconditioning techniques is demonstrated in two dimensions on a fixed-source, within-group neutron diffusion problem approximating the fast group of a pressurized water reactor configuration containing six fuel assemblies.

Development of efficient Krylov preconditioning techniques for multi-dimensional method of characteristics

Two techniques are proposed in the preconditioning for the Krylov sub-space method called the Generalized Minimal RESidual (GMRES) method to accelerate inner iterations based on the method of characteristics (MOC). The GMRES method is an iterative method to solve a linear algebraic system byminimizing the norm of the residual vector. The proposed preconditioning technique is based on the first flight collision probability which is efficiently made by the multi-dimensional MOC code. To simplify the preconditioner, slight couplings among regions are ignored by considering the mean free path. And another proposed technique makes simplified preconditioner by the scaling matrix which can homogenize and de-homogenize the fuel region and the cladding region. The scaling technique reduces the size of the matrix and also reduces the calculation time of inverse matrix. Numerical results show that the preconditioner simplified by the mean free path efficiently reduces the number of iterations for the GMRES algorithm. And the scaling technique keeps the efficiency of preconditioner even in the multi-dimensional geometry. The total calculation time is found to be reduced when these techniques are employed.

An efficient implementation of the generalized minimum residual algorithm with a new preconditioner for the boundary element method

Despite the numerical stability and robustness of the generalized minimum residual algorithm, it still suffers from slow convergence rate and unexpected breakdown when applied to the boundary element method, even with the conventional preconditioners. To address these problems, we have devised a new preconditioner by combining the partial pivot method and diagonal scaling preconditioner with use of the selective reorthogonalization criterion. We examine the performance of these implementations through three numerical examples having a simple-domain, a multi-domain and a multiply-connected domain. The results of the numerical analyses confirm that the selective reorthogonalization criterion can retain the orthogonality of the basis vectors with a small number of reorthogonalizations and that the proposed preconditioner improve the computational efficiency.

Implicit time integration of hyperbolic conservation laws via discontinuous Galerkin methods

A high-order, matrix-free implicit method has been developed for the transient solutions of hyperbolic conservation laws. The discontinuous Galerkin method is applied for temporal discretization. This method has the advantage that its discretization error is (Δt2p+1) when a polynomial basis of degree p is used for time discretization. The nonlinear system of equations from the implicit time discretization is solved at each time step using a nonlinear Krylov subspace projection method. The system of linear equations is solved by the generalized minimum residual algorithm with a lower–upper symmetric Gauss–Seidel preconditioner. The numerical results from the inviscid Burgers' equation indicate that the implicit method is several times faster in performance relative to explicit integration by the total variation diminishing Runge–Kutta method of order 3. The forward Euler method requires a time step proportional to the square of the spatial step for stability with equations such as Burgers' equation (J. Sci. Comput. 2001; 16:173–261). It would, hence, be much less efficient than other explicit methods, for example, Cockburn and Shu (Math. Comput. 1989; 52:411–435), which would only require a time step proportional to the spatial step. Copyright © 2009 John Wiley & Sons, Ltd.

Error Control of Iterative Linear Solvers for Integrated Groundwater Models

An open problem that arises when using modern iterative linear solvers, such as the preconditioned conjugate gradient method or Generalized Minimum RESidual (GMRES) method, is how to choose the residual tolerance in the linear solver to be consistent with the tolerance on the solution error. This problem is especially acute for integrated groundwater models, which are implicitly coupled to another model, such as surface water models, and resolve both multiple scales of flow and temporal interaction terms, giving rise to linear systems with variable scaling. This article uses the theory of "forward error bound estimation" to explain the correspondence between the residual error in the preconditioned linear system and the solution error. Using examples of linear systems from models developed by the US Geological Survey and the California State Department of Water Resources, we observe that this error bound guides the choice of a practical measure for controlling the error in linear systems. We implemented a preconditioned GMRES algorithm and benchmarked it against the Successive Over-Relaxation (SOR) method, the most widely known iterative solver for nonsymmetric coefficient matrices. With forward error control, GMRES can easily replace the SOR method in legacy groundwater modeling packages, resulting in the overall simulation speedups as large as 7.74×. This research is expected to broadly impact groundwater modelers through the demonstration of a practical and general approach for setting the residual tolerance in line with the solution error tolerance and presentation of GMRES performance benchmarking results.

Implicit formulation with the boundary element method for nonlinear radiation of water waves

An accurate and efficient numerical method is presented for the two-dimensional nonlinear radiation problem of water waves. The wave motion that occurs on water due to an oscillating body is described under the assumption of ideal fluid flow. The governing Laplace equation is effectively solved by utilizing the GMRES (Generalized Minimal RESidual) algorithm for the boundary element method (BEM) with quadratic approximation. The intersection or corner singularity in the mixed Dirichlet–Neumann problem is resolved by introducing discontinuous elements. The fully implicit trapezoidal rule is used to update solutions at new time-steps, by considering stability and accuracy. Traveling waves generated by the oscillating body are absorbed downstream by the damping zone technique. To avoid the numerical instability caused by the local gathering of grid points, the re-gridding technique is employed, so that all the grids on the free surface may be re-distributed with an equal distance between them. The nonlinear radiation force is evaluated by means of the acceleration potential. For a mixed Dirichlet–Neumann problem in a computational domain with a wavy top boundary, the present BEM yields numerical solutions for the quadratic rate of convergence with respect to the number of boundary elements. It is also demonstrated that the present time-marching and radiation condition work successfully for nonlinear radiation problems of water waves. The results obtained from this study concur reasonably well with other numerical computations.

Improving Iterative Solutions of the Electric-Field Integral Equation Via Transformations Into Normal Equations

We consider the solution of electromagnetics problems involving perfectly conducting objects formulated with the electric-field integral equation (EFIE). Dense matrix equations obtained from the discretization of EFIE are solved iteratively by the generalized minimal residual (GMRES) algorithm accelerated with a parallel multilevel fast multipole algorithm. We show that the number of iterations is halved by transforming the original matrix equations into normal equations. This way, memory required for the GMRES algorithm is reduced by more than 50%, which is significant when the problem size is large.

A numerical algorithm for the solution of a phase-field model of polycrystalline materials

We describe an algorithm for the numerical solution of a phase-field model (PFM) of microstructure evolution in polycrystalline materials. The PFM system of equations includes a local order parameter, a quaternion representation of local orientation and a species composition parameter. The algorithm is based on the implicit integration of a semidiscretization of the PFM system using a backward difference formula (BDF) temporal discretization combined with a Newton–Krylov algorithm to solve the nonlinear system at each time step. The BDF algorithm is combined with a coordinate-projection method to maintain quaternion unit length, which is related to an important solution invariant. A key element of the Newton–Krylov algorithm is the selection of a preconditioner to accelerate the convergence of the Generalized Minimum Residual algorithm used to solve the Jacobian linear system in each Newton step. Results are presented for the application of the algorithm to 2D and 3D examples.

A GMRES iterative solution of FEM‐BEM global systems in skin effect problems

Purpose – This paper aims to extend an efficient method to solve the global system of linear algebraic equations in the hybrid finite element method – boundary element method (FEM‐BEM) solution of open‐boundary skin effect problems. The extension covers the cases in which the skin effect problem is set in a truncated domain in which no homogeneous Dirichlet conditions are imposed.Design/methodology/approach – The extended method is based on use of the generalized minimal residual (GMRES) solver, which is applied virtually to the reduced system of equations in which the unknowns are the nodal values of the normal derivative of the magnetic vector potential on the fictitious truncation boundary. In each step of the GMRES algorithm the FEM equations are solved by means of the standard complex conjugate gradient solver, whereas the BEM equations are not solved but used to perform fast matrix‐by‐vector multiplications. The BEM equations are written in a non‐conventional way, by making the nodes for the potenti...

Efficient Solution of Skin-Effect Problems by Means of the GMRES-Accelerated FEM-BEM Method

This paper shows that the global system of linear equations in the hybrid finite element method/boundary element method (FEM-BEM) solution of open-boundary skin effect problems can be efficiently solved by means of the generalized minimal residual (GMRES) solver. This solver is applied virtually to the reduced system of equations in which the unknowns are the nodal values of the normal derivative of the magnetic vector potential on the fictitious truncation boundary. In each step of the GMRES algorithm, the solution of the FEM equations is performed by means of the standard complex conjugate gradient solver, whereas the BEM equations are not solved but used to perform fast matrix-by-vector multiplications. The BEM equations are written in a nonconventional way, by making the nodes for the potential noncoinciding with the nodes for its normal derivative.

Application of Preconditioned GMRES Algorithm for Analysis of Millimeter Wave Ferrite Sphere Circulators

The generalized minimal residual (GMRES) iterative method is applied to solve such sparse large non-symmetric system of linear equations resulting from the use of edge-based finite element method. In order to speed up the convergence of GMRES, the symmetric successive overrelaxation (SSOR) preconditioning scheme is applied for the analysis of millimeter wave ferrite circulator. Consequently, this preconditioned GMRES (PGMRES) approach can reach convergence ten times faster than GMRES. The reflection and insertion losses of millimeter wave waveguide circulator are compared with those obtained from literature.

An efficient preconditioning technique for numerical simulation of hydrodynamic model semiconductor devices

AbstractWe investigate the application of preconditioned generalized minimal residual (GMRES) algorithm to the equations of hydrodynamic model of semiconductor devices. An introduction to such a model is presented. We use finite‐element method P1‐isoP2 element to discretize the equations. A preconditioning technique is proposed. The CPU times are presented for n+‐n‐n+ diodes and 0.25 μm gate length Si MESFETs by using the preconditioned GMRES algorithm and the GMRES algorithm. The numerical results show that the preconditioning technique accelerates effectively the velocity of convergence. Copyright © 2003 John Wiley & Sons, Ltd.