Multigrid Solution Research Articles

On the Design of Global-in-Time Newton-Multigrid-Pressure Schur Complement Solvers for Incompressible Flow Problems

In this work, a new global-in-time solution strategy for incompressible flow problems is presented, which highly exploits the pressure Schur complement (PSC) approach for the construction of a space–time multigrid algorithm. For linear problems like the incompressible Stokes equations discretized in space using an inf-sup-stable finite element pair, the fundamental idea is to block the linear systems of equations associated with individual time steps into a single all-at-once saddle point problem for all velocity and pressure unknowns. Then the pressure Schur complement can be used to eliminate the velocity fields and set up an implicitly defined linear system for all pressure variables only. This algebraic manipulation allows the construction of parallel-in-time preconditioners for the corresponding all-at-once Picard iteration by extending frequently used sequential PSC preconditioners in a straightforward manner. For the construction of efficient solution strategies, the so defined preconditioners are employed in a GMRES method and then embedded as a smoother into a space–time multigrid algorithm, where the computational complexity of the coarse grid problem highly depends on the coarsening strategy in space and/or time. While commonly used finite element intergrid transfer operators are used in space, tailor-made prolongation and restriction matrices in time are required due to a special treatment of the pressure variable in the underlying time discretization. The so defined all-at-once multigrid solver is extended to the solution of the nonlinear Navier–Stokes equations by using Newton’s method for linearization of the global-in-time problem. In summary, the presented multigrid solution strategy only requires the efficient solution of time-dependent linear convection–diffusion–reaction equations and several independent Poisson-like problems. In order to demonstrate the potential of the proposed solution strategy for viscous fluid simulations with large time intervals, the convergence behavior is examined for various linear and nonlinear test cases.

A neural network multigrid solver for the Navier-Stokes equations

We present the deep neural network multigrid solver (DNN-MG) that we develop for the instationary Navier-Stokes equations. DNN-MG improves computational efficiency using a judicious combination of a geometric multigrid solver and a recurrent neural network with memory. DNN-MG uses the multi-grid method to classically solve on coarse levels while the neural network corrects interpolated solutions on fine ones, thus avoiding the increasingly expensive computations that would have to be performed there. This results in a reduction in computation time through DNN-MG's highly compact neural network. The compactness results from its design for local patches and the available coarse multigrid solutions that provides a “guide” for the corrections. A compact neural network with a small number of parameters also reduces training time and data. Furthermore, the network's locality facilitates generalizability and allows one to use DNN-MG trained on one mesh domain also on different ones. We demonstrate the efficacy of DNN-MG for variations of the 2D laminar flow around an obstacle. For these, our method significantly improves the solutions as well as lift and drag functionals while requiring only about half the computation time of a full multigrid solution. We also show that DNN-MG trained for the configuration with one obstacle can be generalized to other time dependent problems that can be solved efficiently using a geometric multigrid method.

On a Multigrid Method for Tempered Fractional Diffusion Equations

In this paper, we develop a suitable multigrid iterative solution method for the numerical solution of second- and third-order discrete schemes for the tempered fractional diffusion equation. Our discretizations will be based on tempered weighted and shifted Grünwald difference (tempered-WSGD) operators in space and the Crank–Nicolson scheme in time. We will prove, and show numerically, that a classical multigrid method, based on direct coarse grid discretization and weighted Jacobi relaxation, performs highly satisfactory for this type of equation. We also employ the multigrid method to solve the second- and third-order discrete schemes for the tempered fractional Black–Scholes equation. Some numerical experiments are carried out to confirm accuracy and effectiveness of the proposed method.

Block low‐rank single precision coarse grid solvers for extreme scale multigrid methods

AbstractExtreme scale simulation requires fast and scalable algorithms, such as multigrid methods. To achieve asymptotically optimal complexity, it is essential to employ a hierarchy of grids. The cost to solve the coarsest grid system can often be neglected in sequential computings, but cannot be ignored in massively parallel executions. In this case, the coarsest grid can be large and its efficient solution becomes a challenging task. We propose solving the coarse grid system using modern, approximate sparse direct methods and investigate the expected gains compared with traditional iterative methods. Since the coarse grid system only requires an approximate solution, we show that we can leverage block low‐rank techniques, combined with the use of single precision arithmetic, to significantly reduce the computational requirements of the direct solver. In the case of extreme scale computing, the coarse grid system is too large for a sequential solution, but too small to permit massively parallel efficiency. We show that the agglomeration of the coarse grid system to a subset of processors is necessary for the sparse direct solver to achieve performance. We demonstrate the efficiency of the proposed method on a Stokes‐type saddle point system solved with a monolithic Uzawa multigrid method. In particular, we show that the use of an approximate sparse direct solver for the coarse grid system can outperform that of a preconditioned minimal residual iterative method. This is demonstrated for the multigrid solution of systems of order up to degrees of freedom on a petascale supercomputer using 43,200 processes.

Impact of mixed precision and storage layout on additive Schwarz smoothers

AbstractThe growing discrepancy between CPU computing power and memory bandwidth drives more and more numerical algorithms into a bandwidth‐bound regime. One example is the overlapping Schwarz smoother, a highly effective building block for iterative multigrid solution of elliptic equations with higher order finite elements. Two options of reducing the required memory bandwidth are sparsity exploiting storage layouts and representing matrix entries with reduced precision in floating point or fixed point format. We investigate the impact of several options on storage demand and contraction rate, both analytically in the context of subspace correction methods and numerically at an example of solid mechanics. Both perspectives agree on the favourite scheme: fixed point representation of Cholesky factors in nested dissection storage.

Fast multigrid solution of high-order accurate multiphase Stokes problems

A fast multigrid solver is presented for high-order accurate Stokes problems discretised by local discontinuous Galerkin (LDG) methods. The multigrid algorithm consists of a simple V-cycle, using an element-wise block Gauss-Seidel smoother. The efficacy of this approach depends on the LDG pressure penalty stabilisation parameter -- provided the parameter is suitably chosen, numerical experiment shows that: (i) for steady-state Stokes problems, the convergence rate of the multigrid solver can match that of classical geometric multigrid methods for Poisson problems; (ii) for unsteady Stokes problems, the convergence rate further accelerates as the effective Reynolds number is increased. An extensive range of two- and three-dimensional test problems demonstrates the solver performance as well as high-order accuracy -- these include cases with periodic, Dirichlet, and stress boundary conditions; variable-viscosity and multi-phase embedded interface problems containing density and viscosity discontinuities several orders in magnitude; and test cases with curved geometries using semi-unstructured meshes.

Efficient multigrid solution of elliptic interface problems using viscosity-upwinded local discontinuous Galerkin methods

Author(s): Saye, RI | Abstract: With an emphasis on achieving ideal multigrid solver performance, this paper explores the design of local discontinuous Galerkin schemes for multiphase elliptic interface problems. In particular, for cases exhibiting coefficient discontinuities several orders in magnitude, the role of viscosity-weighted numerical fluxes on interfacial mesh faces is examined: findings support a known strategy of harmonic weighting, but also show that further improvements can be made via a stronger kind of biasing, denoted herein as viscosity-upwinded weighting. Applying this strategy, multigrid performance is assessed for a variety of elliptic interface problems in 1D, 2D, and 3D, across 16 orders of viscosity ratio. These include constant-and variable-coefficient problems, multiphase checkerboard patterns, implicitly defined interfaces, and 3D problems with intricate geometry. With the exception of a challenging case involving a lattice of vanishingly small droplets, in all demonstrated examples the condition number of the multigrid V-cycle preconditioned system has unit order magnitude, independent of the mesh size h.

On an effective multigrid solver for solving a class of variational problems with application to image segmentation

ABSTRACT In this paper we reformulate a class of non-linear variational models for global and selective image segmentation and obtain convergent multigrid solutions. In contrast, non-linear multigrid schemes do not converge for these problems with strong non-linearity and non-smoothness (jumps). Our new approach is to reformulate the non-linear models, using splitting techniques, to generate linear models in a higher dimension which are easier to solve and amenable to the linear multigrid framework. Although splitting techniques are well studied in isolation, direct application of a splitting idea is not sufficient and it is the combination of two splitting approaches and linear multigrid theory approaches which results in a highly effective multigrid algorithm. Numerical results demonstrate the fast convergence of the new multigrid methods.

Multigrid solution of 2D and 3D stress fields in contact mechanics of anisotropic inhomogeneous materials

Increasing demands on performance of machines lead to severer operating conditions of rolling bearings, i.e. higher loads, less lubricant, thinner lubricant films. Under these conditions, the effects of inhomogeneity and anisotropy on the fatigue life become more important. Accurate prediction of such effects requires detailed surface pressure and subsurface stress calculations. For practically relevant 3D cases with realistic grain sizes, this can only be done with very efficient numerical solution methods. In this paper, multigrid techniques are demonstrated to yield the required performance. The influence of inclusions, crystal orientation and roughness on the Von Mises stress distribution is investigated. The algorithm is suited for subsurface material analysis and optimization as well as for computational diagnostics using image analysis.

Efficient and reliable phase‐field simulation of brittle fracture using a nonsmooth multigrid solution scheme

AbstractIn the last decades the phase‐field approach to fracture [1–3] has gained wide popularity due to advantages such as straight‐forward modeling of complex crack patterns and crack branching while allowing standard finite‐element discretizations. Time‐discrete phase‐field models of brittle fracture are typically formulated in terms of biconvex minimization problems for which a standard monolithic Newton‐Raphson scheme usually fails to converge. Solutions can be found by using operator‐splitting methods [2] or predictor‐corrector schemes [4]. Such methods come at the cost of high computational efforts. Furthermore, models incorporating the thermodynamically consistent local irreversibility of the damage phase‐field contain nonsmooth terms [2]. To improve the stability and to reduce the computational costs originating from the biconvexity and the nonsmoothness of the energy functional, we employ a nonsmooth multigrid method that can solve such problems roughly in the time of one equivalent linear problem [5] and which has been shown to be globally convergent [5]. We will demonstrate the computational speed of the proposed solution scheme by means of a classical benchmark problem of brittle fracture.

CLC in AMG solvers for saddle‐point problems

SummaryAn algebraic multigrid (AMG) solution method for saddle‐point problems is described. The indefinite nature of the saddle‐point matrix makes it unsuitable for the simple smoothing methods normally used in AMG. Moreover, even if presented in a stabilised form, as is done here, poorly conditioned matrices will be generated when constructing the coarse‐grid approximation. This is because, with each successive coarsening step, the off‐diagonal matrix blocks (of interfield coupling) reduce in size more slowly than the diagonal blocks (of intrafield coupling). Stabilised smoothing operators are therefore considered. The first is based on an incomplete decomposition of the complete system matrix into the product of lower‐triangular (L) and upper‐triangular (U) matrices, an ILU factorisation. The second is based on an exact block decomposition of an incomplete (simplified) system matrix into lower and upper block‐triangular matrices, a BILU factorisation. However, the degree of stabilisation thus established in the smoothing operators does not guarantee an efficient smoothing at all grid levels. There can still be inefficiency on the least‐stable coarser grids. The breakdown in efficiency begins at a grid level where the ratio of the inter‐ to intrafield coupling strengths exceeds a critical ratio. Provision is thus made for a further conditioning of coarse‐grid operators, a coarse‐level conditioning (CLC). This is another block‐LU factorisation that is only applied at and beyond the critical grid level. It is not applied directly to the operator for any chosen level but to its fine‐grid progenitor. Thus, while ILU and BILU use postcoarsening‐step factorisations, CLC uses precoarsening‐step factorisations. With CLC so deployed, ILU and BILU become efficient at all grid levels, resulting in an AMG convergence that is independent of the total number of grids.

Time-multipatch discontinuous Galerkin space-time isogeometric analysis of parabolic evolution problems

In this paper, we present a new time-multipatch discontinuous Galerkin Isogeometric Analysis (IgA) technology for solving parabolic initial-boundary problems in space and time simultaneously. We prove coercivity of the IgA variational problem with respect to a suitably chosen norm that together with boundedness, consistency, and approximation results yields a priori discretization error estimates in this norm. Furthermore, we provide efficient parallel generation and parallel multigrid solution technologies. Finally, we present first numerical results on massively parallel computers.

Multigrid solution for the cauchy problem associated with helmholtz type equation on non uniform grids

In this paper, an HOC scheme with multigrid algorithm is developed for solving the Cauchy problem associated with two dimensional Helmholtz type equations. The suggested scheme has up to fourth order accuracy. Lastly, some numerical experiments are given to show the accuracy and performance of the proposed scheme.

A line-by-line hybrid unstructured finite volume/Monte Carlo method for radiation transfer in 3D non-gray medium

A hybrid method combing the unstructured finite volume method and the Monte Carlo method and incorporating the line-by-line model has been developed to simulate the radiative transfer in highly spectral and inhomogeneous medium. In this method, the unstructured finite volume method is adopted to solve the spectral radiative transfer equation at wave numbers or spectral locations determined by the Monte Carlo method. The Monte Carlo method takes effects by firstly defining the monotonic random number relations corresponding to the spectral emitted power density of every discretized element of the concerning medium, and then by reversing the spectral location through comparison of these relations with predefined random numbers. Through this Monte Carlo method, the actual number of spectral locations on which the spectral radiative transfer equations are solved may be reduced: only the spectral locations that have higher spectral emissive powers would be more possibly selected. To increase the performance of the presented method, the total variation diminishing scheme on unstructured grids is adopted in treating the spectral radiative intensity at interface between control volumes. And, the discretized radiative transfer equation is implicitly and iteratively solved by an algebraic multi-grid solution approach to accelerate the convergence of the equation. The presented method was applied to 3D homogeneous and inhomogeneous cases for the validation and performance studies. Results show that for both cases, the presented method agree well with pure Monte Carlo benchmark solutions with acceptable number of spectral locations and computing time.

A multigrid scheme for solving convection–diffusion-integral optimal control problems

The fast multigrid solution of an optimal control problem governed by a convection–diffusion partial-integro differential equation is investigated. This optimization problem considers a cost functional of tracking type and a constrained distributed control. The optimal control sought is characterized by the solution to the corresponding optimality system, which is approximated by a finite volume and quadrature discretization schemes and solved by multigrid techniques. The proposed multigrid approach combines a multigrid method for the governing model with a fast multigrid integration method. The convergence of this solution procedure is analyzed by local Fourier analysis and validated by results of numerical experiments.

Surface Couplings for Subdomain-Wise Isoviscous Gradient Based Stokes Finite Element Discretizations

The Stokes system with constant viscosity can be cast into different formulations by exploiting the incompressibility constraint. For instance, the rate of strain tensor in the weak formulation can be replaced by the velocity-gradient yielding a decoupling of the velocity components in the different coordinate directions. Consequently, the discretization of this partly decoupled formulation leads to fewer nonzero entries in the stiffness matrix. This is of particular interest in large scale simulations where a reduced memory bandwidth requirement can help to significantly accelerate the computations. In the case of a piecewise constant viscosity, as it typically arises in multi-phase flows, or when the boundary conditions involve traction, the situation is more complex, and one has to treat the cross derivatives in the original Stokes system with care. A naive application of the standard vectorial Laplacian results in a physically incorrect solution, while formulations based on the rate of strain tensor increase the computational effort globally. Here, we propose a new approach that is consistent with the stress-divergence formulation and preserves the decoupling advantages of the velocity-gradient-divergence formulation in isoviscous subdomains. The modification is equivalent to locally changing the discretization stencils at interfaces or boundaries. Hence, the more expensive discretization is of lower complexity, making the additional computational cost in large scale simulations negligible. We establish consistency and convergence properties and show that in a massively parallel setup, the multigrid solution of the resulting discrete systems is faster than for the classical stress-divergence formulation.

Verification and validation of a high-order strand grid method for two-dimensional turbulent flows

Abstract In this paper, we construct a novel hybrid of two one-dimensional schemes in order to leverage several advantages for solving two-dimensional turbulent flows. Building upon previous work by the authors and others, we combine one-dimensional flux correction along body surfaces along with high-order summation-by-parts finite differences normal to surfaces. A new semi-implicit multigrid solution method is presented that capitalizes on the unique directional properties of each scheme, utilizing an explicit multigrid scheme along the surface direction, and an implicit Gauss-Seidel scheme along the strand direction. Turbulence closure is achieved with a robust version of the Spalart-Allmaras turbulence model that accommodates negative values of the turbulence working variable. The hybrid scheme exhibits fourth-order convergence using the method of manufactured solutions. Fundamental validation studies of the turbulent flux correction method are conducted in two dimensions, using the NASA-Langley turbulence resource as a means for comparison. Results are presented that demonstrate improvements in accuracy with minimal computational and algorithmic overhead over traditional second-order algorithms.

Uzawa Smoother in Multigrid for the Coupled Porous Medium and Stokes Flow System

The multigrid solution of coupled porous media and Stokes flow problems is considered. The Darcy equation as the saturated porous medium model is coupled to the Stokes equations by means of appropriate interface conditions. We focus on an efficient multigrid solution technique for the coupled problem, which is discretized by finite volumes on staggered grids, giving rise to a saddle point linear system. Special treatment is required regarding the discretization at the interface. An Uzawa smoother is employed in multigrid, which is a decoupled procedure based on symmetric Gauss–Seidel smoothing for velocity components and a simple Richardson iteration for the pressure field. Since a relaxation parameter is part of a Richardson iteration, local Fourier analysis is applied to determine the optimal parameters. Highly satisfactory multigrid convergence is reported, and, moreover, the algorithm performs very well for small values of the hydraulic conductivity and fluid viscosity, which are relevant for applications.

A MULTIGRID MULTILEVEL MONTE CARLO METHOD USING HIGH-ORDER FINITE-VOLUME SCHEME FOR LOGNORMAL DIFFUSION PROBLEMS

The aim of this paper is to show that a high-order discretization can be used to improve the convergence of a multilevel Monte Carlo method for elliptic partial differential equations with lognormal random coefficients in combination with the multigrid solution method. To demonstrate this, we consider a fourth-order accurate finite-volume discretization. With the help of the Matern family of covariance functions, we simulate the coefficient field with different degrees of smoothness. The idea behind using a fourth-order scheme is to capture the additional regularity in the solution introduced due to higher smoothness of the random field. Second-order schemes previously utilized for these types of problems are not able to fully exploit this additional regularity. We also propose a practical way of combining a full multigrid solver with the multilevel Monte Carlo estimator constructed on the same mesh hierarchy. Through this integration, one full multigrid solve at any level provides a valid sample for all the preceding Monte Carlo levels. The numerical results show that the fourth-order multilevel estimator consistently outperforms the second-order variant. In addition, we observe an asymptotic gain for the standard Monte Carlo estimator.

A multigrid solution for the Cahn–Hilliard equation on nonuniform grids

We present a nonlinear multigrid method to solve the Cahn–Hilliard (CH) equation on nonuniform grids. The CH equation was originally proposed as a mathematical model to describe phase separation phenomena after the quenching of binary alloys. The model has the characteristics of thin diffusive interfaces. To resolve the sharp interfacial transition, we need a very fine grid, which is computationally expensive. To reduce the cost, we can use a fine grid around the interfacial transition region and a relatively coarser grid in the bulk region. The CH equation is discretized by a conservative finite difference scheme in space and an unconditionally gradient stable type scheme in time. We use a conservative restriction in the nonlinear multigrid method to conserve the total mass in the coarser grid levels. Various numerical results on one-, two-, and three-dimensional spaces are presented to demonstrate the accuracy and effectiveness of the nonuniform grids for the CH equation.