Multigrid Research Articles

An Effective Multigrid Method for Solving Problems of High-Frequency Vibrational Convection

An Effective Multigrid Method for Solving Problems of High-Frequency Vibrational Convection

Controlled-source electromagnetic modeling using a versatile secondary electric-field formulation and efficient multigrid-based preconditioner

Frequency-domain controlled-source electromagnetic (CSEM) surveying is a widely used geophysical electromagnetic exploration method. To avoid the singularity near the transmitting sources and improve the accuracy of three-dimensional (3D) CSEM forward modeling, we adopt the vector finite element method and secondary electric field formulation combined with hexahedral grids to solve the frequency-domain Maxwell equations. Primary field computation applies the fast Hankel transform using digital linear filters to quickly calculate the electromagnetic field generated by different electromagnetic dipoles or loops in the layered medium. Therefore, our forward modeling algorithm is highly versatile and compatible with various electromagnetic sources and scenarios of CSEM application (e.g., airborne, offshore, or marine). To improve the convergence rate, we use an algebraic multigrid solver based on the established Hiptmair–Xu decomposition. Using direct solvers requires a vast amount of memory, while conventional iterative solvers usually fail or converge only very slowly. Our new iterative scheme does not require the divergence corrections commonly applied in finite difference or finite volume electromagnetic modeling to accelerate the convergence. We implement complete parallelism in our finite element programming to efficiently solve the CSEM forward problems on massively parallel clusters. We use different numerical methods to obtain high-precision comparison results for one-dimensional and typical 3D models to demonstrate our algorithm's effectiveness and correctness. Finally, a public model of 3 million cells is used to simulate the responses of marine CSEM surveying. The proposed new solver shows robust scalability when using nearly 1000 cores on the cluster, demonstrating that our algorithm is particularly suitable for large-scale 3D forward modeling.

High accuracy compact difference and multigrid methods for two-dimensional time-dependent nonlinear advection-diffusion-reaction problems

We focus on high-accuracy and stable numerical methods for the time-dependent nonlinear advection-diffusion-reaction problems in this work. A novel fourth-order fully implicit compact difference scheme is derived, in which we make use of the fourth-order compact formulas to discretize the diffusion terms, the fourth-order Padé formulas to compute the nonlinear advection terms, and the fourth-order backward differencing formula to approximate the time derivative term. Convergence and stability of the new scheme are analysed by the energy method. On the basis of the novel scheme, a multigrid algorithm is established such that a time advancement algorithm for solving these nonlinear problems are obtained. We provide various numerical experiments to verify the performances of the proposed scheme including the reliability, stability and computational efficiency. And the results obtained by our method show it is more accurate than most same kind schemes reported in the literature.

Overlapping Domain Decomposition Method with Cascadic Multigrid for Image Restoration

In the process of image restoration, it is usually necessary to solve large-scale inverse problems, where the computational cost is very high for large or high-resolution images. The domain decomposition method is one of the most effective algorithms to solve large-scale problems, which can effectively decrease the computational cost. The cascadic multigrid method has a good effect on the linear model of image restoration and can obtain high quality restored images. In this paper, the overlapping domain decomposition method (DDM) with the cascadic multigrid method (CMG) and the DDM with new extrapolation cascadic multigrid method (NECMG) are presented to solve the image restoration problems of denoising and deblurring. We first divide the image problem into some overlapping and independent subproblems. Then, each subproblem is solved independently by CMG or NECMG with the edge-preserving operator. Numerical experiments show that the new method is effective.

Truncated nonsmooth Newton multigrid for phase-field brittle-fracture problems, with analysis

AbstractWe propose the truncated nonsmooth Newton multigrid method (TNNMG) as a solver for the spatial problems of the small-strain brittle-fracture phase-field equations. TNNMG is a nonsmooth multigrid method that can solve biconvex, block-separably nonsmooth minimization problems with linear time complexity. It exploits the variational structure inherent in the problem, and handles the pointwise irreversibility constraint on the damage variable directly, without regularization or the introduction of a local history field. In the paper we introduce the method and show how it can be applied to several established models of phase-field brittle fracture. We then prove convergence of the solver to a solution of the nonsmooth Euler–Lagrange equations of the spatial problem for any load and initial iterate. On the way, we show several crucial convexity and regularity properties of the models considered here. Numerical comparisons to an operator-splitting algorithm show a considerable speed increase, without loss of robustness.

Multilevel‐in‐width training for deep neural network regression

AbstractA common challenge in regression is that for many problems, the degrees of freedom required for a high‐quality solution also allows for overfitting. Regularization is a class of strategies that seek to restrict the range of possible solutions so as to discourage overfitting while still enabling good solutions, and different regularization strategies impose different types of restrictions. In this paper, we present a multilevel regularization strategy that constructs and trains a hierarchy of neural networks, each of which has layers that are wider versions of the previous network's layers. We draw intuition and techniques from the field of Algebraic Multigrid (AMG), traditionally used for solving linear and nonlinear systems of equations, and specifically adapt the Full Approximation Scheme (FAS) for nonlinear systems of equations to the problem of deep learning. Training through V‐cycles then encourage the neural networks to build a hierarchical understanding of the problem. We refer to this approach as multilevel‐in‐width to distinguish from prior multilevel works which hierarchically alter the depth of neural networks. The resulting approach is a highly flexible framework that can be applied to a variety of layer types, which we demonstrate with both fully connected and convolutional layers. We experimentally show with PDE regression problems that our multilevel training approach is an effective regularizer, improving the generalize performance of the neural networks studied.

Multigrid Incomplete Factorization Methods in Krylov Subspaces

The paper studies multigrid methods for solving systems of linear algebraic equations resulting from the seven-point discretization of the three-dimensional Dirichlet problem for an elliptic differential equation of the second order in a parallepipedal domain on a regular grid. The algorithms suggested are presented as special iteration processes of incomplete factorization in Krylov subspaces with a hierarchical recursive vector structure that corresponds to a sequence of embedded grids and gives rise to a block tridiagonal recursive representation of the coefficient matrix of the original linear algebraic system. The convergence of iterations is enhanced by using the principle of compensation of the row sums and also the symmetric successive block overrelaxation. An arbitrary m-grid method is defined recursively, based on the two-grid method. For simplicity, the algorithms are considered for linear systems with Stieltjes coefficient matrices. Issues related to generalization of the algorithms to larger classes of problems and, in particular, those with unsymmetric matrices are discussed.

A modified and efficient phase field model for the biological transport network

This paper aims to establish the biological transport network based on the phase field model. In order to ensure that the topological shape is formed under the guidance of electrical conductivity, we generate the biological networks with sufficient information based on the distributions of the venation of the leaf represented by the reaction-diffusion model. We modify the original energy of the network generating model by considering the auxin gradient property. By applying the gradient flow method to minimize the modified energy, we derive the Poisson type equation for pressure, the reaction-diffusion type equation for the network conductance, and the Allen-Cahn type equation for the phase field. The proposed model is significant on the investigation of phase transitions by considering the gradient properties on the boundaries. We have innovatively added conductivity and phase-field coupling terms that inhibit perpendicular transport of nutrients, making it easy to generate thin branches from the trunk through this model. In order to obtain the second-order temporal accuracy, we take the Crank–Nicolson method for the governing system. To obtain the second-order spatial accuracy, we discretize the coupling system with the central finite difference method and linearize the nonlinear terms semi-explicitly to form a linear system at each time step. The discrete energy dissipation is provably preserved and we can use a larger time step. We apply the preconditioned conjugate gradient method with the multigrid method as a preconditioner to implement a practical algorithm with only linear algebraic complexity. The proposed algorithm is easy to implement and achieves a fast convergence. Various numerical tests are demonstrated to verify the efficiency, stability, and robustness of the proposed method.

FFT‐based efficient Poisson solver in nonrectangular domain

AbstractPoisson's equation is one of the most popular partial differential equation (PDE), which is widely used in image processing, computer graphics and other fields. However, solving a large‐scale Poisson's equation always costs huge computational resources. Fast Fourier transform (FFT) is an efficient Poisson solver but it only works in rectangular domain. In this paper, we propose a FFT‐based Poisson solver in nonrectangular domain on regular grids combined with algebraic multigrid (AMG). We extend the original Poisson's equation to a rectangular domain to construct an equivalent equation, so that it can use FFT algorithm to accelerate the solving to Poisson's equation. Experiments show that the FFT‐based Poisson solver can improve the solving speed of large‐scale Poisson's equations in nonrectangular domain. We demonstrate the solver in applications of image processing and fluid simulation.

Low-Order Preconditioning for the High-Order Finite Element de Rham Complex

In this paper we present a unified framework for constructing spectrally equivalent low-order-refined discretizations for the high-order finite element de Rham complex. This theory covers diffusion problems in $H^1$, $H({\rm curl})$, and $H({\rm div})$, and is based on combining a low-order discretization posed on a refined mesh with a high-order basis for N\'ed\'elec and Raviart-Thomas elements that makes use of the concept of polynomial histopolation (polynomial fitting using prescribed mean values over certain regions). This spectral equivalence, coupled with algebraic multigrid methods constructed using the low-order discretization, results in highly scalable matrix-free preconditioners for high-order finite element problems in the full de Rham complex. Additionally, a new lowest-order (piecewise constant) preconditioner is developed for high-order interior penalty discontinuous Galerkin (DG) discretizations, for which spectral equivalence results and convergence proofs for algebraic multigrid methods are provided. In all cases, the spectral equivalence results are independent of polynomial degree and mesh size; for DG methods, they are also independent of the penalty parameter. These new solvers are flexible and easy to use; any "black-box" preconditioner for low-order problems can be used to create an effective and efficient preconditioner for the corresponding high-order problem. A number of numerical experiments are presented, based on an implmentation in the finite element library MFEM. The theoretical properties of these preconditioners are corroborated, and the flexibility and scalability of the method are demonstrated on a range of challenging three-dimensional problems.

Symbol based convergence analysis in multigrid methods for saddle point problems

Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations. They can be formulated such that the system matrix is symmetric, but indefinite, so the variational convergence theory that is usually used to prove multigrid convergence cannot be applied. In a 2016 paper in Numerische Mathematik Notay has presented a different algebraic approach that analyzes properly preconditioned saddle point problems, proving convergence of the two-grid method. The present paper analyzes saddle point problems where the blocks are circulant within this framework. It contains sufficient conditions for convergence and optimal parameters for the preconditioning of the unilevel and multilevel saddle point problem and for the point smoother that is used. The analysis is based on the generating symbols of the circulant blocks. Further, it is shown that the structure can be kept on the coarse level, allowing for a recursive application of the approach in a W- or V-cycle and studying the “level independency” property. Numerical results demonstrate the efficiency of the proposed method in the circulant and the Toeplitz case.

A coupled multigrid solver with wall functions forthree‐dimensionalturbulent flows over urban‐like obstacles

AbstractOne of the objectives for rapid operational tools for urban atmospheric events isfastcalculation of the computational fluid dynamics (CFD) models for multiphase flows to respond to deliberate or accidental chemical, biological, and radiological (CBR) releases. This article addresses the implementation of a coupled multigrid (CMG) method in an in‐house pressure‐based low‐speed CFD solver called STREAM in detail, including its validation against several 2D/3D benchmark test problems pertinent to urban flows. It was identified that solving the advection–diffusion equation of concentration for the prediction of dispersion of CBR agents is computationally very efficient, provided that the input data of turbulence viscosity and flow fields can be supplied in a timely fashion. Since most pressure‐based solvers adopt the segregated SIMPLE algorithm, and the coupling among different (linearized) equations is established via outer iterations on a single grid, its convergence rate is generally poor, particularly for turbulent flows in urban environments involving massive flow separation. The proposed approach and main contribution here is to adopt CMG to solve turbulent flows over urban‐like obstacles efficiently in 2D and particularly 3D, in which the standard Reynolds‐averaged Navier–Stokes turbulence model in conjunction with wall functions is employed. The results presented in this paper demonstrate a speedup ratio based on work unit (WU) for 3D laminar cavity flows of roughly 100, and 25 for 3D turbulent urban flow, depending on grid sizes.

Coarse-grid selection using simulated annealing

Multilevel techniques are efficient approaches for solving the large linear systems that arise from discretized partial differential equations and other problems. While geometric multigrid requires detailed knowledge about the underlying problem and its discretization, algebraic multigrid aims to be less intrusive, requiring less knowledge about the origin of the linear system. A key step in algebraic multigrid is the choice of the coarse/fine partitioning, aiming to balance the convergence of the iteration with its cost. In work by MacLachlan and Saad (2007), a constrained combinatorial optimization problem is used to define the “best” coarse grid within the setting of a two-level reduction-based algebraic multigrid method and is shown to be NP-complete. Here, we develop a new coarsening algorithm based on simulated annealing to approximate solutions to this problem, which yields improved results over the greedy algorithm developed previously. We present numerical results for test problems on both structured and unstructured meshes, demonstrating the ability to exploit knowledge about the underlying grid structure if it is available.

Exploiting spatial symmetries for solving Poisson's equation

This paper presents a strategy to accelerate virtually any Poisson solver by taking advantage of s spatial reflection symmetries. More precisely, we have proved the existence of an inexpensive block diagonalisation that transforms the original Poisson equation into a set of 2s fully decoupled subsystems then solved concurrently. This block diagonalisation is identical regardless of the mesh connectivity (structured or unstructured) and the geometric complexity of the problem, therefore applying to a wide range of academic and industrial configurations. In fact, it simplifies the task of discretising complex geometries since it only requires meshing a portion of the domain that is then mirrored implicitly by the symmetries' hyperplanes. Thus, the resulting meshes naturally inherit the exploited symmetries, and their memory footprint becomes 2s times smaller. Thanks to the subsystems' better spectral properties, iterative solvers converge significantly faster. Additionally, imposing an adequate grid points' ordering allows reducing the operators' footprint and replacing the standard sparse matrix-vector products with the sparse matrix-matrix product, a higher arithmetic intensity kernel. As a result, matrix multiplications are accelerated, and massive simulations become more affordable. Finally, we include numerical experiments based on a turbulent flow simulation and making state-of-the-art solvers exploit a varying number of symmetries. On the one hand, algebraic multigrid and preconditioned Krylov subspace methods require up to 23% and 72% fewer iterations, resulting in up to 1.7x and 5.6x overall speedups, respectively. On the other, sparse direct solvers' memory footprint, setup and solution costs are reduced by up to 48%, 58% and 46%, respectively.

Adaptive multigrid strategy for geometry optimization of large-scale three dimensional molecular mechanics

In this paper, we present an efficient adaptive multigrid strategy for the geometry optimization of large-scale three dimensional molecular mechanics. The resulting method can achieve significantly reduced complexity by exploiting the intrinsic low-rank property of the material configurations and by combining the state-of-the-art adaptive techniques with the hierarchical structure of multigrid algorithms. To be more precise, we develop a oneway multigrid method with adaptive atomistic/continuum (a/c) coupling, e.g., blended ghost force correction (BGFC) [1] approximations with gradient-based a posteriori error estimators on the coarse levels. We utilize state-of-the-art 3D mesh generation techniques to effectively implement the method. For 3D crystalline defects, such as vacancies, micro-cracks and dislocations, compared with brute-force optimization, complexity with superior rates can be observed numerically, and the strategy has a five-fold acceleration in terms of CPU time for systems with 108 atoms.

Effect of the Curvature Angle in a Conduit with an Adiabatic Cylinder over a Backward Facing Step on the Magnetohydrodynamic Behaviour in the Presence of a Nanofluid

A backward facing duct are present in various industrial applications especially those focused on heat transfer. The flow through a curved backward facing duct especially in the presence of nanofluid presents complexities compared to a straight backward facing step (BFS) duct. Therefore, the present numerical study deals a nanofluid flow (Fe3O4-H2O) forced convection in a curved backward facing duct. The objective of this investigation is to visualize at different curvature angles g (0°, 30°, 45°, 60°, 90°) imposed on the top wall of the duct, the effect of Hartmann number Ha (0, 50, 100), magnetic field inclination angle γ (0°, 60°, 90°), Reynolds number Re (10, 100, 200) and nanoparticle volume fraction φ (0 %, 0.05 %, 0.1 %). The dimensionless governing equations are solved using the multigrid finite element method. The results showed that the heat transfer was enhanced at the curved angle g = 90° for large Hartman numbers, thus, the average Nusselt number increased with a ratio of 240.74 % in the case of Hartmann number (Ha = 100).

Efficient Distributed Matrix-free Multigrid Methods on Locally Refined Meshes for FEM Computations

This work studies three multigrid variants for matrix-free finite-element computations on locally refined meshes: geometric local smoothing, geometric global coarsening (both h -multigrid), and polynomial global coarsening (a variant of p -multigrid). We have integrated the algorithms into the same framework—the open source finite-element library deal.II —, which allows us to make fair comparisons regarding their implementation complexity, computational efficiency, and parallel scalability as well as to compare the measurements with theoretically derived performance metrics. Serial simulations and parallel weak and strong scaling on up to 147,456 CPU cores on 3,072 compute nodes are presented. The results obtained indicate that global-coarsening algorithms show a better parallel behavior for comparable smoothers due to the better load balance, particularly on the expensive fine levels. In the serial case, the costs of applying hanging-node constraints might be significant, leading to advantages of local smoothing, even though the number of solver iterations needed is slightly higher. When using p - and h -multigrid in sequence ( hp -multigrid), the results indicate that it makes sense to decrease the degree of the elements first from a performance point of view due to the cheaper transfer.

MGM: A Meshfree Geometric Multilevel Method for Systems Arising from Elliptic Equations on Point Cloud Surfaces

We develop a new meshfree geometric multilevel (MGM) method for solving linear systems that arise from discretizing elliptic PDEs on surfaces represented by point clouds. The method uses a Poisson disk sampling-type technique for coarsening the point clouds and new meshfree restriction/interpolation operators based on polyharmonic splines for transferring information between the coarsened point clouds. These are then combined with standard smoothing and operator coarsening methods in a V-cycle iteration. MGM is applicable to discretizations of elliptic PDEs based on various localized meshfree methods, including RBF finite differences (RBF-FD) and generalized finite differences (GFD). We test MGM both as a standalone solver and preconditioner for Krylov subspace methods on several test problems using RBF-FD and GFD and numerically analyze convergence rates, efficiency, and scaling with increasing point cloud sizes. We also perform a side-by-side comparison to algebraic multigrid methods for solving the same systems. Finally, we further demonstrate the effectiveness of MGM by applying it to three challenging applications on complicated surfaces: pattern formation, surface harmonics, and geodesic distance.

FV-MgNet: Fully connected V-cycle MgNet for interpretable time series forecasting

By investigating iterative methods for a constrained linear model, we propose a new class of fully connected V-cycle MgNet for long-term time series forecasting, which is one of the most difficult tasks in forecasting. MgNet is a CNN model that was proposed for image classification based on the multigrid (MG) methods for solving discretized partial differential equations (PDEs). We replace the convolutional operations with fully connected operations in the existing MgNet and then apply them to forecasting problems. Motivated by the V-cycle structure in MG, we further propose the FV-MgNet, a V-cycle version of the fully connected MgNet, to extract features hierarchically. By evaluating the performance of FV-MgNet on popular datasets and comparing it with state-of-the-art models, we show that the FV-MgNet achieves better results with less memory usage and faster inference speed. In addition, we develop ablation experiments to demonstrate that the structure of FV-MgNet is the best choice among the many variants.

A Geometric Multigrid Method for Space-Time Finite Element Discretizations of the Navier–Stokes Equations and its Application to 3D Flow Simulation

We present a parallelized geometric multigrid (GMG) method, based on the cell-based Vanka smoother, for higher order space-time finite element methods (STFEM) to the incompressible Navier–Stokes equations. The STFEM is implemented as a time marching scheme. The GMG solver is applied as a preconditioner for generalized minimal residual iterations. Its performance properties are demonstrated for 2D and 3D benchmarks of flow around a cylinder. The key ingredients of the GMG approach are the construction of the local Vanka smoother over all degrees of freedom in time of the respective subinterval and its efficient application. For this, data structures that store pre-computed cell inverses of the Jacobian for all hierarchical levels and require only a reasonable amount of memory overhead are generated. The GMG method is built for the deal.II finite element library. The concepts are flexible and can be transferred to similar software platforms.