Generalized Minimal Residual Research Articles

Preconditioned Multigroup GMRES Algorithms for the Variational Nodal Method

Combinations of three approaches are examined as options to replace the algorithms presently employed in the variational nodal code VARIANT. They are preconditioned Generalized Minimal Residual (GMRES) algorithms, parallelism in energy, and Wielandt acceleration. Together with partitioned matrix and Gauss-Seidel (GS) preconditioners, two GMRES algorithms are formulated to replace the upscattering iteration and facilitate energy parallelism and Wielandt acceleration. The GMRES algorithms are tested on two-dimensional thermal and fast reactor diffusion problems. The two GMRES algorithms yield higher efficiencies in energy group parallelization and Wielandt acceleration than simple parallelization of the existing GS algorithm. With preconditioning the GMRES algorithms reduce the total computing time by a factor of 2 to 4 and in some cases by a factor of >10.A multilevel iteration optimization scheme is investigated that automatically adjusts the relative error tolerance of the inner iterations according to the estimated convergence rate of the corresponding outer iterations and updates the Wielandt shift magnitude as the calculations progress. Numerical results based on large two-dimensional thermal and fast reactor diffusion problems demonstrate that automated optimization of the multilevel iterative processes reduces iteration numbers by as much as an order of magnitude.

A bi-partitioned iterative algorithm for solving linear systems arising from incompressible flow problems

A novel iterative algorithm, called the bi-partitioned method, is introduced for efficiently solving the system of linear equations that arises from the stabilized finite element formulation of the Navier–Stokes equations. The bi-partitioned algorithm is a Krylov subspace method designed for a matrix with separated momentum and continuity blocks. This structure allows for formation of the Schur complement to separately solve for the velocity and pressure unknowns. Hence, the bi-partitioned algorithm can also be applied to problems with similar matrix structure, involving the Schur complement. Two separate spaces are constructed iteratively from the velocity and pressure solution candidates and optimally combined to produce the final solution. The bi-partitioned algorithm calculates the final solution to a given tolerance, regardless of the approximation made in construction of the Schur complement. The proposed algorithm is analyzed and compared to the generalized minimal residual (GMRES) algorithm using two incompressible-flow and one fluid–structure-interaction example, exhibiting up to an order of magnitude improvement is simulation cost while maintaining excellent stability.

Simulation of 3D parachute fluid–structure interaction based on nonlinear finite element method and preconditioning finite volume method

A fluid–structure interaction method combining a nonlinear finite element algorithm with a preconditioning finite volume method is proposed in this paper to simulate parachute transient dynamics. This method uses a three-dimensional membrane–cable fabric model to represent a parachute system at a highly folded configuration. The large shape change during parachute inflation is computed by the nonlinear Newton–Raphson iteration and the linear system equation is solved by the generalized minimal residual (GMRES) method. A membrane wrinkling algorithm is also utilized to evaluate the special uniaxial tension state of membrane elements on the parachute canopy. In order to avoid large time expenses during structural nonlinear iteration, the implicit Hilber–Hughes–Taylor (HHT) time integration method is employed. For the fluid dynamic simulations, the Roe and HLLC (Harten–Lax–van Leer contact) scheme has been modified and extended to compute flow problems at all speeds. The lower–upper symmetric Gauss–Seidel (LU-SGS) approximate factorization is applied to accelerate the numerical convergence speed. Finally, the test model of a highly folded C-9 parachute is simulated at a prescribed speed and the results show similar characteristics compared with experimental results and previous literature.

Performance of Preconditioned Nonstationary Methods for Electromagnetic Scattering From One Dimensional Dielectric Rough Surfaces

For the analysis of electromagnetic scattering from one dimensional dielectric randomly rough surfaces, we apply the right-preconditioning technique to three nonstationary iterative algorithms, namely, the generalized minimal residual (GMRES), conjugate gradient squared (CGS), and biconjugate gradient stabilized (BiCGSTAB). The spectral acceleration technique is also utilized to expedite computation of matrix-vector product. Among the nonstationary methods considered, with the right-preconditioning (RP) applied, GMRES-RP is the most robust, followed by BiCGSTAB-RP, with CGS-RP falling behind. In terms of number of iterations to achieve convergence, for cases when they all converge, GMRES-RP requires the most whereas BiCGSTAB-RP the least, with CGS-RP slightly more demanding than BiCGSTAB-RP. However, in terms of run time, GMRES-RP is the most efficient, followed by BiCGSTAB-RP, and CGS-RP is the least efficient. This relative performance is in contrast to the unpreconditioned case as reported in the literature, which speaks of the impact of the adopted right-preconditioning on the spectral distribution of these nonstationary algorithms.

An efficient preconditioned Krylov subspace method for large-scale finite element equations with MPC using Lagrange multiplier method

Purpose– The purpose of this paper is to propose an efficient iterative method for large-scale finite element equations of bad numerical stability arising from deformation analysis with multi-point constraint using Lagrange multiplier method.Design/methodology/approach– In this paper, taking warpage analysis of polymer injection molding based on surface model as an example, the performance of several popular Krylov subspace methods, including conjugate gradient, BiCGSTAB and generalized minimal residual (GMRES), with diffident Incomplete LU (ILU)-type preconditions is investigated and compared. For controlling memory usage, GMRES(m) is also considered. And the ordering technique, commonly used in the direct method, is introduced into the presented iterative method to improve the preconditioner.Findings– It is found that the proposed preconditioned GMRES method is robust and effective for solving problems considered in this paper, and approximate minimum degree (AMD) ordering is most beneficial for the reduction of fill-ins in the ILU preconditioner and acceleration of the convergence, especially for relatively accurate ILU-type preconditioning. And because of concerns about memory usage, GMRES(m) is a good choice if necessary.Originality/value– In this paper, for overcoming difficulties of bad numerical stability resulting from Lagrange multiplier method, together with increasing scale of problems in engineering applications and limited hardware conditions of computer, a stable and efficient preconditioned iterative method is proposed for practical purpose. Before the preconditioning, AMD reordering, commonly used in the direct method, is introduced to improve the preconditioner. The numerical experiments show the good performance of the proposed iterative method for practical cases, which is implemented in in-house and commercial codes on PC.

HIFU scattering by the ribs: constrained optimisation with a complex surface impedance boundary condition

High intensity focused ultrasound (HIFU) enables highly localised, non-invasive tissue ablation and its efficacy has been demonstrated in the treatment of a range of cancers, including those of the kidney, prostate and breast. HIFU offers the ability to treat deep-seated tumours locally, and potentially bears fewer side effects than more established treatment modalities such as resection, chemotherapy and ionising radiation. There remains however a number of significant challenges which currently hinder its widespread clinical application. One of these challenges is the need to transmit sufficient energy through the ribcage to ablate tissue at the required foci whilst minimising the formation of side lobes and sparing healthy tissue. Ribs both absorb and reflect ultrasound strongly. This sometimes results in overheating of bone and overlying tissue during treatment, leading to skin burns. Successful treatment of a patient with tumours in the upper abdomen therefore requires a thorough understanding of the way acoustic and thermal energy is deposited. Previously, a boundary element (BE) approach based on a Generalised Minimal Residual (GMRES) implementation of the Burton-Miller formulation was developed to predict the field of a multi-element HIFU array scattered by human ribs, the topology of which was obtained from CT scan data [1]. Dissipative mechanisms inside the propagating medium have since been implemented, together with a complex surface impedance condition at the surface of the ribs. A reformulation of the boundary element equations as a constrained optimisation problem was carried out to determine the complex surface velocities of a multi-element HIFU array which generated the acoustic pressure field that best fitted a required acoustic pressure distribution in a least-squares sense. This was done whilst ensuring that an acoustic dose rate parameter at the surface of the ribs was kept below a specified threshold. The methodology was tested at an excitation frequency of 1 MHz on a spherical multi-element array in the presence of anatomical ribs.

A fast high-order sinc-based algorithm for pricing options under jump-diffusion processes

An implicit–explicit Euler scheme in temporal direction is employed to discretize a partial integro-differential equation, which arises in pricing options under jump-diffusion process. Then the semi-discretized equation is approximated in space by the Sinc–Galerkin method with exponential accuracy. Meanwhile, the domain decomposition method is incorporated to handle the non-smoothness of the payoff function, and the improved fast Gauss transform is applied to accelerate the evaluation of the jump integral term. An effective preconditioner is proposed for solving the resulting dense Toeplitz-related systems by the preconditioned generalized minimum residual (GMRES) method. Numerical tests are performed to illustrate the efficiency of the proposed algorithm.

The Multiscale Finite-Volume Method on Stratigraphic Grids

Summary Finding a pressure solution for large and highly detailed reservoir models with fine-scale heterogeneities modeled on a meter scale is computationally demanding. One way of making such simulations less compute-intensive is to use multiscale methods that solve coarsened flow problems by use of a set of reusable basis functions to capture flow effects induced by local geological variations. One such method, the multiscale finite-volume (MsFV) method, is well-studied for 2D Cartesian grids but has not been implemented for stratigraphic and unstructured grids with faults in three dimensions. We present an open-source implementation of the MsFV method in three dimensions along with a coarse partitioning algorithm that can handle stratigraphic grids with faults and wells. The resulting solver is an alternative to traditional upscaling methods, but can also be used for accelerating fine-scale simulations. To achieve better precision, the implementation can use the MsFV method as a preconditioner for Arnoldi iterations using the generalized minimal residual (GMRES) method or as a preconditioner in combination with a standard inexpensive smoother. We conduct a series of numerical experiments in which approximate solutions computed by the new MsFV solver are compared with fine-scale solutions computed by a standard two-point scheme for grids with realistic permeabilities and geometries. On the one hand, the results show that the MsFV method can produce accurate approximations for geological models with pinchouts, faults, and nonneighboring connections, but on the other hand, they also show that the method can fail quite spectacularly for highly heterogeneous and anisotropic systems in a way that cannot efficiently be mitigated by iterative approaches. Thus, the MsFV method is, in our opinion, not yet sufficiently robust to be applied as a black-box solver for models with industry-standard complexity. However, extending the method to realistic grids is an important step on the way toward a fast and accurate multiscale solution of large-scale reservoir models. In particular, our open-source implementation provides an efficient framework suitable for further experimentation with partitioning algorithms and MsFV variants.

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.

On $${\varvec{m}}$$ m -step Hermitian and skew-Hermitian splitting preconditioning methods

Preconditioners are often applied in various Krylov subspace iteration methods to improve their computing efficiency. In this paper, we consider solving linear systems with a non-Hermitian positive definite coefficient matrix using preconditioned Krylov subspace iteration methods such as the generalized minimal residual (GMRES) method. An $$m$$ -step polynomial preconditioner is designed based on the Hermitian and skew-Hermitian splitting (HSS) iteration method proposed by Bai et al. (SIAM J Matrix Anal Appl 24:603–626, 2003). The proposed preconditioned system is solved by fully utilizing the HSS iteration method. Theoretical and experimental results show that the proposed $$m$$ -step preconditioner is efficient in accelerating GMRES for solving a non-Hermitian positive definite linear system.

Complete stagnation of GMRES for normal matrices

In this paper we study the problem of complete stagnation of the generalized minimum residual (GMRES) method for normal matrices. We first characterize all n×n nonsingular normal matrices A such that GMRES(A,b) stagnates completely for some vector b. Also we give necessary and sufficient conditions for the non-existence of a real stagnation vector for real normal matrices. The number of real stagnation vectors for normal matrices is studied. Moreover, we characterize all the eigenvalues of nonsingular normal matrices A∈M3(C) such that GMRES(A,b) stagnates completely for some b∈C3. Using the results derived by A. Greenbaum, V. Pták and Z. Strakoš in 1996, we consider the complete stagnation of unitary matrices and derive another characterization for all nonsingular normal matrices A∈M3(R) such that GMRES(A,b) stagnates completely for some vector b∈R3.

Método de elementos de borde jerárquico basado en el árbol de Barnes-Hut aplicado a flujo reptante exterior

In this work, a hierarchical variant of a boundary element method and its use in Stokes flow around three-dimensional rigid bodies in steady regime is presented. The proposal is based on the descending hierarchical low-order and self-adaptive algorithm of Barnes-Hut, and it is used in conjunction with an indirect boundary integral formulation of second class, whose source term is a function of the undisturbed velocity. The solution field is the double layer surface density, which is modified in order to complete the eigenvalue spectrum of the integral operator. In this way, the rigid modes are eliminated and both a non-zero force and a non-null torque on the body could be calculated. The elements are low order flat triangles, and an iterative solution by generalized minimal residual (GMRES) is used. Numerical examples include cases with analytical solutions, bodies with edges and vertices, or with intricate shapes. The main advantage of the presented technique is the possibility of considering a greater number of degrees of freedom regarding traditional collocation methods, due to the decreased demand of main memory and the reduction in the computation times.

A globally optimal tri-vector method to solve an ill-posed linear system

In the present paper, a future cone in the Minkowski space defined in terms of the square-norm of the residual vector for an ill-posed linear system to be solved, is used to derive a nonlinear system of ordinary differential equations. Then the forward Euler scheme is used to generate an iterative algorithm. Two critical values in the critical descent tri-vector are derived, which lead to the largest convergence rate of the resultant iterative algorithm, namely the globally optimal tri-vector method (GOTVM). Some numerical examples are used to reveal the superior performance of the GOTVM than the famous methods of conjugate gradient (CGM) and generalized minimal residual (GMRES). Through the numerical tests we also set forth the rationale by assuming the tri-vector as being a better descent direction.

Towards the optimisation of acoustic fields for ablative therapies of tumours in the upper abdomen

The efficacy of high intensity focused ultrasound (HIFU) for the non-invasive treatment of cancer has been demonstrated for a range of different cancers including those of the liver, kidney, prostate and breast. As a non-invasive focused therapy, HIFU offers considerable advantages over other techniques such as chemotherapy and surgical resection, in terms of its non-invasiveness and low risk of harmful side effects. There is, however, a number of significant challenges which currently hinder its widespread clinical application. One of these challenges is the need to transmit sufficient energy through the ribcage to induce tissue necrosis at the required foci whilst minimising the formation of side lobes and sparing healthy tissue. Ribs both absorb and reflect ultrasound strongly. As such, a common side effect of focusing ultrasound in regions located behind the rib cage is the overheating of bone and surrounding tissue, which can lead to skin burns. Successful treatment of a patient with tumours in the upper abdomen therefore requires a thorough understanding of the way acoustic and thermal energy are deposited. This is likely to rely on a treatment planning procedure in which optimal source velocity distributions are obtained so as to maximise a dose quantity at the treatment sites, whilst ensuring that this quantity does not exceed a specified threshold at other field locations, particularly on the surface of the ribs. Previously, a boundary element approach based on a Generalised Minimal Residual (GMRES) implementation of the Burton-Miller formulation was developed to predict the field of a multi-element HIFU array scattered by human ribs, the topology of which was obtained from CT scan data [1]. This work describes the reformulation of the boundary element equations as a least-squares minimisation problem with non-linear constraints. The methodology was subsequently tested at an excitation frequency of 100 kHz on a spherical multi-element array in the presence of a perfectly rigid cylindrical scatterer with hemi-spherical end-caps. Reduction of side lobes in the exterior domain and of acoustic pressure magnitudes on the surface of the cylinder were achieved whilst preserving a local maximum in the focal region.

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.

Two-level additive preconditioners for edge element discretizations of time-harmonic Maxwell equations

Two-level additive preconditioners are presented for edge element discretizations of time-harmonic Maxwell equations. The key is to construct a special “coarse mesh” space, which adds the kernel of the curl-operator in a fine space to a coarse mesh space, to solve the original problem, and then uses the fine mesh space to solve the H(curl)-elliptic problem. It is shown that the generalized minimal residual (GMRES) method applied to the preconditioned system converges uniformly provided that the coarsest mesh size is reasonably small (but independent of the fine mesh size) and the parameter for the “coarse mesh” space solver is sufficiently large. Numerical experiments show the efficiency of the proposed approach.

Application of PETSc in soil-water coupled geotechnical problems

In this paper, parallel library, portable extensible toolkit for scientific computation (PETSc), is used to solve linear systems in soil-water coupled finite element method (FEM) for geotechnical problems. The parallel environment is integrated into GLEAVES, which is a geotechnical software package used for the finite element simulation. The linear system ·· = · which is a fundamental and the most time-consuming part of the FEM is solved with iterative solvers in PETSc. In order to find a robust and effective combination of iterative solvers and corresponding preconditioners for the soil-water coupled problems, performance evaluations on Krylov subspace methods and four preconditioners are carried out. The results indicate that general minimal residual (GMRES) method coupled with preconditioners can provide an effective solution. The application to a construction project is presented to illustrate the potential of the proposed solution.

Numerical simulations of microwave heating of liquids: enhancements using Krylov subspace methods

In this paper, we compare the performances of three iterative solvers for large sparse linear systems arising in the numerical computations of incompressible Navier-Stokes (NS) equations. These equations are employed mainly in the simulation of microwave heating of liquids. The emphasis of this work is on the application of Krylov projection techniques such as Generalized Minimal Residual (GMRES) to solve the Pressure Poisson Equations that result from discretisation of the NS equations. The performance of the GMRES method is compared with the traditional Gauss-Seidel (GS) and point successive over relaxation (PSOR) techniques through their application to simulate the dynamics of water housed inside a vertical cylindrical vessel which is subjected to microwave radiation. It is found that as the mesh size increases, GMRES gives the fastest convergence rate in terms of computational times and number of iterations.

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.