Robust Multigrid Solver Research Articles

Hp-Robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs

In this work, we formulate and analyze a geometric multigrid method for the iterative solution of the discrete systems arising from the finite element discretization of symmetric second-order linear elliptic diffusion problems. We show that the iterative solver contracts the algebraic error robustly with respect to the polynomial degree p ≥ 1 and the (local) mesh size h. We further prove that the built-in algebraic error estimator which comes with the solver is hp-robustly equivalent to the algebraic error. The application of the solver within the framework of adaptive finite element methods with quasi-optimal computational cost is outlined. Numerical experiments confirm the theoretical findings.

A new semi-smooth Newton multigrid method for control-constrained semi-linear elliptic PDE problems

In this paper a new multigrid algorithm is proposed to accelerate the convergence of the semi-smooth Newton method that is applied to the first order necessary optimality systems arising from a class of semi-linear control-constrained elliptic optimal control problems. Under admissible assumptions on the nonlinearity, the discretized Jacobian matrix is proved to have an uniformly bounded inverse with respect to mesh size. Different from current available approaches, a new numerical implementation that leads to a robust multigrid solver is employed to coarsen the grid operator. Numerical simulations are provided to illustrate the efficiency of the proposed method, which shows to be computationally more efficient than the full-approximation-storage multigrid in current literature. In particular, our proposed approach achieves a mesh-independent convergence and its performance is highly robust with respect to the regularization parameter.

Nonsymmetric Black Box multigrid with coarsening by three

SUMMARYThe classical Petrov–Galerkin approach to Black Box multigrid for nonsymmetric problems due to Dendy is combined with the recent factor‐three‐coarsening Black Box algorithm due to Dendy and Moulton, along with a powerful symmetric line Gauss–Seidel smoother, resulting in an efficient and robust multigrid solver. Focusing on the convection–diffusion operator, the algorithm is tested and shown to achieve fast and reliable convergence with both first‐order and second‐order accurate upstream discretizations of the convection operator. The solver also exhibits robust behavior with respect to discontinuous jumps in the diffusion coefficient and performs well for recirculating flows over a wide range of diffusion coefficients. The efficiency of the solver is supported by results of an analysis for the case of constant coefficients. Copyright © 2012 John Wiley & Sons, Ltd.

Multilevel approximations in sample-based inversion from the Dirichlet-to-Neumann map

In 2005, Christen and Fox introduced a delayed acceptance Metropolis-Hastings (DAMH) algorithm that improved computational efficiency in sample-based imaging of electrical conductivity (EIT). That work used a linear approximation to the forward map in the first step of the algorithm. In this paper, we develop an alternative approximation for use in DAMH, namely a multilevel approximation developed from the hierarchy of coarse-scale models obtained by variational coarsening. This approach builds on two important strengths of robust multigrid solvers. First, the cost of a fine-scale solution of the forward map scales linearly with the degrees of freedom, and hence, it is provides better efficiency for algorithms performing sample-based inference. Second, the homogenization implicit in robust variational multigrid methods gives better solutions at coarse scales than homogenization by averaging of coefficients. We report results from a stylized example in electrical impedance imaging where data is a noisy and incomplete measurement of the Dirichlet-to-Neumann map.

Time-domain modeling of electromagnetic diffusion with a frequency-domain code

We modeled time-domain EM measurements of induction currents for marine and land applications with a frequency-domain code. An analysis of the computational complexity of a number of numerical methods shows that frequency-domain modeling followed by a Fourier transform is an attractive choice if a sufficiently powerful solver is available. A recently developed, robust multigrid solver meets this requirement. An interpolation criterion determined the automatic selection of frequencies. The skin depth controlled the construction of the computational grid at each frequency. Tests of the method against exact solutions for some simple problems and a realistic marine example demonstrate that a limited number of frequencies suffice to provide time-domain solutions after piecewise-cubic Hermite interpolation and a fast Fourier transform.

Multilevel upscaling through variational coarsening

A new efficient multilevel upscaling procedure for single‐phase saturated flow in porous media is presented. While traditional approaches to this problem have focused on the computation of an upscaled hydraulic conductivity, here the coarse‐scale model is created explicitly from the fine‐scale model through the application of operator‐induced variational coarsening. This technique, which originated with robust multigrid solvers, has been shown to accurately capture the influence of fine‐scale heterogeneous structure over the complete hierarchy of coarse‐scale models that it generates. Moreover, implicit in this hierarchy is the construction of interpolation operators that provide a natural and complete multiscale basis for the fine‐scale problem. Thus this new multilevel upscaling methodology is similar to the multiscale finite element method, and indeed, it attains similar accuracy in computations of the fine‐scale hydraulic head and coarse‐scale normal flux on a variety of problems; yet it is an order of magnitude faster on the examples considered here.

A multigrid discrete‐ordinates solution for isotropic transport equation

AbstractWe propose a robust multigrid solver for the isotropic transport equation in three space dimensions. Discrete‐ordinates and Galerkin method are used for angle and space discretizations, respectively. The fully discrete problem is formulated as a compact linear system of algebraic equations with a dense iterate matrix. Using a hierarchy of nested meshes our multigrid algorithm employes the Atkinson‐Brakhage approximate inverse as a smoother while a Krylov subspace method is used to solve the coarse problem. Numerical results and comparisons are shown for a transport problem with thermal source. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A characterization of mapping unstructured grids onto structured grids and using multigrid as a preconditioner

Many problems based on unstructured grids provide a natural multigrid framework due to using an adaptive gridding procedure. When the grids are saved, even starting from just a fine grid problem poses no serious theoretical difficulties in applying multigrid. A more difficult case occurs when a highly unstructured grid problem is to be solved with no hints how the grid was produced. Here, there may be no natural multigrid structure and applying such a solver may be quite difficult to do. Since unstructured grids play a vital role in scientific computing, many modifications have been proposed in order to apply a fast, robust multigrid solver. One suggested solution is to map the unstructured grid onto a structured grid and then apply multigrid to a sequence of structured grids as a preconditioner. In this paper, we derive both general upper and lower bounds on the condition number of this procedure in terms of computable grid parameters. We provide examples to illuminate when this preconditioner is a useful (e. g.,p orh-p formulated finite element problems on semi-structured grids) or should be avoided (e.g., typical computational fluid dynamics (CFD) or boundary layer problems). We show that unless great care is taken, this mapping can lead to a system with a high condition number which eliminates the advantage of the multigrid method.

Using Symmetries and Antisymmetries to Analyze a Parallel Multigrid Algorithm: The Elliptic Boundary Value Problem Case

Symmetry and antisymmetry properties of a class of elliptic partial differential equations are exploited to prove when a particular parallel multilevel algorithm is a direct method rather than the usual iterative method. No smoothing is required for this result. Examples are presented, including variable coefficient ones. A connection between the algorithm in this article and domain decomposition is established, even though this algorithm is more general and different. The parallel algorithm is also analyzed when it is iterative and it is shown how to increase processor utilization. Hackbusch’s robust multigrid algorithm [“A new approach to robust multi-grid solvers,” in ICIAM ’87 : Proceedings of the First International Conference on Industrial and Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1988, pp. 111–126] is analyzed for some model problems and it is shown that the parallel algorithm in this article uses much less computer time with at most the same amount of storage.