Multigrid Procedure Research Articles

On the Regularizing Power of Multigrid-type Algorithms

We consider the deblurring problem of noisy and blurred images in the case of known space invariant point spread functions with four choices of boundary conditions. We combine an algebraic multigrid previously defined ad hoc for structured matrices related to space invariant operators (Toeplitz, circulants, trigonometric matrix algebras, etc.) and the classical geometric multigrid studied in the partial differential equations context. The resulting technique is parameterized in order to have more degrees of freedom: a simple choice of the parameters allows us to devise a quite powerful regularizing method. It defines an iterative regularizing method where the smoother itself has to be an iterative regularizing method (e.g., conjugate gradient, Landweber, conjugate gradient for normal equations, etc.). More precisely, with respect to the smoother, the regularization properties are improved and the total complexity is lower. Furthermore, in several cases, when it is directly applied to the system $A{\bf f}={\bf g}$, the quality of the restored image is comparable with that of all the best known techniques for the normal equations $A^TA{\bf f}=A^T{\bf g}$, but the related convergence is substantially faster. Finally, the associated curves of the relative errors versus the iteration numbers are "flatter" with respect to the smoother (the estimation of the stop iteration is less crucial). Therefore, we can choose multigrid procedures which are much more efficient than classical techniques without losing accuracy in the restored image (as often occurs when using preconditioning). Several numerical experiments show the effectiveness of our proposals.

Spatial interpolation of large climate data sets using bivariate thin plate smoothing splines

Thin plate smoothing splines are widely used to spatially interpolate surface climate, however, their application to large data sets is limited by computational efficiency. Standard analytic calculation of thin plate smoothing splines requires O( n 3) operations, where n is the number of data points, making routine computation infeasible for data sets with more than around 2000 data points. An O( N) iterative procedure for calculating finite element approximations to bivariate minimum generalised cross validation (GCV) thin plate smoothing splines operations was developed, where N is the number of grid points. The key contribution of the method lies in the incorporation of an automatic procedure for optimising smoothness to minimise GCV. The minimum GCV criterion is commonly used to optimise thin plate smoothing spline fits to climate data. The method discretises the bivariate thin plate smoothing spline equations using hierarchical biquadratic B-splines, and uses a nested grid multigrid procedure to solve the system. To optimise smoothness, a double iteration is incorporated, whereby the estimate of the spline solution and the estimate of the optimal smoothing parameter are updated simultaneously. When the method was tested on temperature data from the African and Australian continents, accurate approximations to analytic solutions were obtained.

A dual mesh multigrid preconditioner for the efficient solution of hydraulically driven fracture problems

We present a novel multigrid (MG) procedure for the efficient solution of the large non-symmetric system of algebraic equations used to model the evolution of a hydraulically driven fracture in a multi-layered elastic medium. The governing equations involve a highly non-linear coupled system of integro-partial differential equations along with the fracture front free boundary problem. The conditioning of the algebraic equations typically degrades as O(N3). A number of characteristics of this problem present significant new challenges for designing an effective MG strategy. Large changes in the coefficients of the PDE are dealt with by taking the appropriate harmonic averages of the discrete coefficients. Coarse level Green's functions for multiple elastic layers are constructed using a single dual mesh and superposition. Coarse grids that are sub-sets of the finest grid are used to treat mixed variable problems associated with ‘pinch points.’ Localized approximations to the Jacobian at each MG level are used to devise efficient Gauss–Seidel smoothers and preferential line iterations are used to eliminate grid anisotropy caused by large aspect ratio elements. The performance of the MG preconditioner is demonstrated in a number of numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd.

Spatial direct numerical simulation of transitional boundary layer over compliant surfaces

An iterative implicit fractional step method is developed and employed for the simulation of transitional boundary layer over compliant surfaces. The three-dimensional perturbation Navier–Stokes equations are discretized by curvilinear finite volumes on a collocated grid system. A multigrid procedure is used for computations associated with the pressure-Poisson equation, while simulation is carried out by MPI-based parallel computation with domain decomposition. Results in the literature for oblique linear waves and the non-linear breakdown of a wave triad over a rigid wall are repeated to check the accuracy of the codes that had been developed. Oblique linear TS (Tollmien–Schlichting) waves over finite-length compliant membranes are generally found to coexist with CIFI (compliance induced flow instability) or FISI (flow induced surface instabilities) waves. The latter waves usually possess longer wavelengths and thus propagate at a larger oblique wave angles than the TS waves. Simulation reveals that compliant surfaces may slow down the development of secondary instabilities during the early stages of laminar–turbulent transition. However, during the later stages of fundamental wave breakdown, interactions with CIFI and edge-generated waves may increase the amplitudes of the original 2D and 3D TS waves, leading to an earlier breakdown on compliant surfaces. Linear interaction between the flow and compliant membranes has been assumed.

Multigrid acceleration for transonic aerodynamic flow simulations based on a hierarchical formulation

AbstractIn a companion paper, the authors used a hierarchical formulation based on a Helmholtz velocity decomposition to simulate transonic flows over airfoils. The potential flow formulation is augmented with entropy and vorticity corrections and the numerical results are compared to standard Euler and Navier–Stokes calculations. For many aerodynamic applications, the corrections are limited to relatively small regions; the flow in the near and far fields is irrotational and isentropic. The entropy and vorticity corrections are governed by convection/diffusion equations while the non‐homogeneous potential equation is of mixed type; elliptic in the subsonic domain and hyperbolic in the supersonic one. The forcing function represents the necessary correction for mass conservation. Upwind schemes are used for the convection terms of the scalar equations of the corrections and for the potential equation in the supersonic region. The formulation can be viewed as an implementation of a viscous/inviscid interaction procedure which is equivalent to Navier–Stokes equations in the inner field.In this paper, convergence acceleration techniques are applied for such a formulation using single and multiple grids. For a single grid, optimal relaxation parameters for subsonic potential flow regions and local artificial time steps for the scalar correction equations have been used. A full multigrid technique is implemented to the augmented potential equation. Only three grids: coarse, intermediate and fine meshes are used. Results for both inviscid and viscous transonic flows are presented. It is noticed that the potential and the viscous flow calculations, based on the present formulation, have comparable convergence histories. The limited applications of multigrid result in an order of magnitude saving of the work units for both calculations. More savings should be achievable with more sophisticated multigrid procedures. Copyright © 2004 John Wiley & Sons, Ltd.

Adaptive multigrid for finite element computations in plasticity

The solution of the system of equilibrium equations is the most time-consuming part in large-scale finite element computations of plasticity problems. The development of efficient solution methods are therefore of utmost importance to the field of computational plasticity. Traditionally, direct solvers have most frequently been used. However, recent developments of iterative solvers and preconditioners may impose a change. In particular, preconditioning by the multigrid technique is especially favorable in FE applications. The multigrid preconditioner uses a number of nested grid levels to improve the convergence of the iterative solver. Prolongation of fine-grid residual forces is done to coarser grids and computed corrections are interpolated to the fine grid such that the fine-grid solution successively is improved. By this technique, large 3D problems, invincible for solvers based on direct methods, can be solved in acceptable time at low memory requirements. By means of a posteriori error estimates the computational grid could successively be refined (adapted) until the solution fulfils a predefined accuracy level. In contrast to procedures where the preceding grids are erased, the previously generated grids are used in the multigrid algorithm to speed up the solution process. The paper presents results using the adaptive multigrid procedure to plasticity problems. In particular, different error indicators are tested.

On the development of an agglomeration multigrid solver for turbulent flows

The paper describes the implementation details and validation results for an agglomeration multigrid procedure developed in the context of hybrid, unstructured grid solutions of aerodynamic flows. The governing equations are discretized using an unstructured grid finite volume method, which is capable of handling hybrid unstructured grids. A centered scheme as well as a second order version of Liou’s AUSM+ upwind scheme are used for the spatial discretization. The time march uses an explicit 5-stage Runge-Kutta time-stepping scheme. Convergence acceleration to steady state is achieved through the implementation of an agglomeration multigrid procedure, which retains all the flexibility previously available in the unstructured grid code. The calculation capability created is validated considering 2-D laminar and turbulent viscous flows over a flat plate. Studies of the various parameters affecting the multigrid acceleration performance are undertaken with the objective of determining optimal numerical parameter combinations.

Multigrid solution of the Navier–Stokes equations at low speeds with large temperature variations

Multigrid methods for the Navier–Stokes equations at low speeds and large temperature variations are investigated. The compressible equations with time-derivative preconditioning and preconditioned flux-difference splitting of the inviscid terms are used. Three implicit smoothers have been incorporated into a common multigrid procedure. Both full coarsening and semi-coarsening with directional fine-grid defect correction have been studied. The resulting methods have been tested on four 2D laminar problems over a range of Reynolds numbers on both uniform and highly stretched grids. Two of the three methods show efficient and robust performance over the entire range of conditions. In addition, none of the methods has any difficulty with the large temperature variations.

From steady to chaotic solutions in a differentially heated cavity of aspect ratio 8

AbstractA splitting method is used for the temperature and the velocity–pressure unknowns to solve the time‐dependent thermal convection problem in an 8:1 enclosure (International Journal for Numerical Methods, this issue). High performance is obtained by means of incomplete Cholesky conjugate gradient for the temperature equation on one hand and a multigrid procedure for Navier–Stokes equations on the other hand. The approximation is achieved by second‐ and third‐order finite differences on staggered grids. The stability of the steady solution is analysed by the computation of the first Lyapunov exponent. The periodic flow at Ra=3.4×105 is widely discussed and some investigations have been done for some higher Rayleigh numbers. Copyright © 2002 John Wiley & Sons, Ltd.

A Solution Strategy Based on Segmented Domain Decomposition Multigrid for Turbomachinery Flows

A Segmented Domain Decomposition Multigrid (SDDMG) procedure was developed for viscous flow problems as they apply to turbomachinery flows. The procedure divides the computational domain into a coarse mesh comprised of uniformly spaced cells. To resolve smaller length scales such as the viscous layer near a surface, segments of the coarse mesh are subdivided into a finer mesh. This is repeated until adequate resolution of the smallest relevant length scale is obtained. Multigrid is used to communicate information between the different grid levels [1]. To test the procedure, simulation results will be presented for a compressor and turbine cascade. These simulations are intended to show the ability of the present method to generate grid independent solutions. Comparisons with data will also be presented. These comparisons will further demonstrate the usefulness of the present work for they allow an estimate of the accuracy of the flow modeling equations independent of error attributed to numerical discretization.

Adaptive and fixed segmented domain decomposition multigrid procedures for internal viscous flows

A segmented domain decomposition multigrid procedure is reviewed. The methodology is outlined for both an adaptive formulation applied primarily to inlet, nozzle and duct flows, and for a fixed, predetermined, subdomain formulation applied for turbomachinery geometries. The procedure has been considered for both low and high speed flows in two- and three-dimensions, and for laminar and turbulent-model applications. With these procedures, the grids remain uniform in each subdomain; although the global grid appears to be highly stretched. This formulation allows for computations with a high degree of accuracy and grid independence at reasonable computer cost. This is generally not possible to replicate with full domain multigrid procedures on stretched grids with large cell aspect ratio.

Iterative modified approximate factorization

An implicit procedure for minimizing the error introduced by approximate factorization in solving scalar or block multidiagonal matrix equations is presented with applications to fluid dynamics. High convergence rates of approximately 0.8 are achieved upon application to a wide range of problems. The implicit procedure can be combined with a multigrid procedure for further convergence acceleration.

On the Representation of Functions and Finite Difference Operators on Adaptive Dyadic Grids

Abstract In this paper we describe methods to approximate functions and dif- ferential operators on adaptive sparse (dyadic) grids. We distinguish between several representations of a function on the sparse grid and we describe how finite difference (FD) operators can be applied to these representations. For general variable coefficient equations on sparse grids, genuine finite element (FE) discretizations are not feasible and FD operators allow an easier operator evaluation than the adapted FE operators. However, the structure of the FD operators is complex. With the aim to construct an efficient multigrid procedure, we analyze the structure of the discrete Laplacian in its hierarchical representation and show the relation between the full and the sparse grid case. The rather complex relations, that are expressed by scaling matrices for each separate coordinate direction, make us doubt about the pos- sibility of constructing efficient preconditioners that show spectral equivalence. Hence, we question the possibility of constructing a natural multigrid algorithm with optimal O(N) efficiency. We conjecture that for the efficient solution of a general class of adaptive grid problems it is better to accept an additional condition for the dyadic grids (condition L) and to apply adaptive hp-discretization.

A Multigrid-Preconditioned Newton--Krylov Method for the Incompressible Navier--Stokes Equations

Globalized inexact Newton methods are well suited for solving large-scale systems of nonlinear equations. When combined with a Krylov iterative method, an explicit Jacobian is never needed, and the resulting matrix-free Newton--Krylov method greatly simplifies application of the method to complex problems. Despite asymptotically superlinear rates of convergence, the overall efficiency of a Newton--Krylov solver is determined by the preconditioner. High-quality preconditioners can be constructed from methods that incorporate problem-specific information, and for the incompressible Navier--Stokes equations, classical pressure-correction methods such as SIMPLE and SIMPLER fulfill this requirement. A preconditioner is constructed by using these pressure-correction methods as smoothers in a linear multigrid procedure. The effectiveness of the resulting Newton--Krylov-multigrid method is demonstrated on benchmark incompressible flow problems.

Acceleration of Anisotropic Scattering Computations Using Coupled Ordinates Method (COMET)

Traditional sequential solution procedures for solving coupled heat transfer and radiation problems are known to perform very poorly when the optical thickness is high. This poor performance can be traced to the fact that sequential procedures cannot handle strong inter-equation coupling resulting from large absorption and scattering coefficients. This paper extends the use of the coupled ordinates method (COMET) to problems involving anisotropic scattering, where inter-equation coupling results from the in-scattering terms. A finite volume formulation for arbitrary unstructured meshes is used to discretize the radiative transfer equation. An efficient point-solution procedure is devised which is suitable for different scattering phase functions, and eliminates the cubic dependence of the operation count on the angular discretization. This solution scheme is used as a relaxation sweep in a multigrid procedure designed to eliminate long-wavelength errors. The method is applied to scattering problems and shown to substantially accelerate convergence for moderate to large optical thicknesses.

Multigrid for American option pricing with stochastic volatility

The paper describes an implicit finite difference approach to the pricing of American options on assets with a stochastic volatility. A multigrid procedure is described for the fast iterative solution of the discrete linear complementarity problems that result. The accuracy and performance of this approach is improved considerably by a strike-price related analytic transformation of asset prices and adaptive time-stepping.

Implicit and multigrid procedures for steady-state computations with upwind algorithms

A computational study for the convergence acceleration of Euler and Navier–Stokes computations with upwind schemes has been conducted in a unified framework. It involves the flux-vector splitting algorithms due to Steger–Warming and Van Leer, the flux-difference splitting algorithms due to Roe and Osher and the hybrid algorithms, AUSM (Advection Upstream Splitting Method) and HUS (Hybrid Upwind Splitting). Implicit time integration with line Gauss–Seidel relaxation and multigrid are among the procedures which have been systematically investigated on an individual as well as cumulative basis. The upwind schemes have been tested in various implicit–explicit operator combinations such that the optimal among them can be determined based on extensive computations for two-dimensional flows in subsonic, transonic, supersonic and hypersonic flow regimes. In this study, the performance of these implicit time-integration procedures has been systematically compared with those corresponding to a multigrid accelerated explicit Runge–Kutta method. It has been demonstrated that a multigrid method employed in conjunction with an implicit time-integration scheme yields distinctly superior convergence as compared to those associated with either of the acceleration procedures provided that effective smoothers, which have been identified in this investigation, are prescribed in the implicit operator.

Dense estimation and object-based segmentation of the optical flow with robust techniques

In this paper, we address the issue of recovering and segmenting the apparent velocity field in sequences of images. As for motion estimation, we minimize an objective function involving two robust terms. The first one cautiously captures the optical flow constraint, while the second (a priori) term incorporates a discontinuity-preserving smoothness constraint. To cope with the nonconvex minimization problem thus defined, we design an efficient deterministic multigrid procedure. It converges fast toward estimates of good quality, while revealing the large discontinuity structures of flow fields. We then propose an extension of the model by attaching to it a flexible object-based segmentation device based on deformable closed curves (different families of curve equipped with different kinds of prior can be easily supported). Experimental results on synthetic and natural sequences are presented, including an analysis of sensitivity to parameter tuning.

Multigrid Methods for Nearly Singular Linear Equations and Eigenvalue Problems

The purpose of this paper is to develop a convergence theory for multigrid methods applied to nearly singular linear elliptic partial differential equations of the type produced from a positive definite system by a shift with the identity. One of the important aspects of this theory is that it allows such shifts to vary anywhere in the multigrid scheme, enabling its application to a wider class of eigenproblem solvers. The theory is first applied to a method for computing eigenvalues and eigenvectors that consists of multigrid iterations with zero right-hand side and updating the shift from the Rayleigh quotient before every cycle. It is then applied to the Rayleigh quotient multigrid (RQMG) method, which is a more direct multigrid procedure for solving eigenproblems. Local convergence of the multigrid V-cycle and global convergence for a full multigrid version of both methods is obtained.

Calculation of two-phase flow in gas turbine combustors

A method is presented for computing steady two-phase turbulent combusting flow in a gas turbine combustor. The gas phase equations are solved in an Eulerian frame of reference. The two-phase calculations are performed by using a liquid droplet spray combustion a model and treating the motion of the evaporating fuel droplets in a Lagrangian frame of reference. The numerical algorithm employs nonorthogonal curvilinear coordinates, a multigrid iterative solution procedure, the standard k-{epsilon} turbulence model, and a combustion model comprising an assumed shape probability density function and the conserved scalar formulation. The trajectory computation of the fuel provides the source terms for all the gas phase equations. This two-phase model was applied to a real piece of combustion hardware in the form of a modern GE/SNECMA single annular CFM56 turbofan engine combustor. For the purposes of comparison, calculations were also performed by treating the fuel as a single gaseous phase. The effect on the solution of two extreme situations of the fuel as a gas and initially as a liquid was examined. The distribution of the velocity field and the conserved scalar within the combustor, as well as the distribution of the temperature field in the reaction zone and in themore » exhaust, were all predicted with the combustor operating both at high-power and low-power (ground idle) conditions. The calculated exit gas temperature was compared with test rig measurements. Under both low and high-power conditions, the temperature appeared to show an improved agreement with the measured data when the calculations were performed with the spray model as compared to a single-phase calculation.« less