Multigrid Methods with Newton-Gauss-Seidel Smoothing and Constraint Preserving Interpolation for Obstacle Problems

In this paper, we propose a multigrid algorithm based on the full approximate scheme for solving the membrane constrained obstacle problems and the minimal surface obstacle problems in the formulations of HJB equations. A Newton-Gauss-Seidel (NGS) method is used as smoother. A Galerkin coarse grid operator is proposed for the membrane constrained obstacle problem. Comparing with standard FAS with the direct discretization coarse grid operator, the FAS with the proposed operator converges faster. A special prolongation operator is used to interpolate functions accurately from the coarse grid to the fine grid at the boundary between the active and inactive sets. We will demonstrate the fast convergence of the proposed multigrid method for solving two model obstacle problems and compare the results with other multigrid methods.     

High Accuracy Solution of Three-Dimensional Biharmonic Equations

In this paper, we consider several finite-difference approximations for the three-dimensional biharmonic equation. A symbolic algebra package is utilized to derive a family of finite-difference approximations for the biharmonic equation on a 27 point compact stencil. The unknown solution and its first derivatives are carried as unknowns at selected grid points. This formulation allows us to incorporate the Dirichlet boundary conditions automatically and there is no need to define special formulas near the boundaries, as is the case with the standard discretizations of biharmonic equations. We exhibit the standard second-order, finite-difference approximation that requires 25 grid points. We also exhibit two compact formulations of the 3D biharmonic equations; these compact formulas are defined on a 27 point cubic grid. The fourth-order approximations are used to solve a set of test problems and produce high accuracy numerical solutions. The system of linear equations is solved using a variety of iterative methods. We employ multigrid and preconditioned Krylov iterative methods to solve the system of equations. Test results from two test problems are reported. In these experiments, the multigrid method gives excellent results. The multigrid preconditioning also gives good results using Krylov methods.     

Cascadic multigrid methods combined with sixth order compact scheme for poisson equation

Based on the extrapolation theory and a sixth order compact difference scheme, new extrapolation interpolation operator and extrapolation cascadic multigrid methods for two dimensional Poisson equation are presented. The new extrapolation interpolation operator is used to provide a better initial value on refined grid. The convergence of the new methods are given. Numerical experiments are shown to illustrate that the new methods have higher accuracy and efficiency.     

Efficient numerical methods for pricing American options under stochastic volatility

Five numerical methods for pricing American put options under Heston's stochastic volatility model are described and compared. The option prices are obtained as the solution of a two‐dimensional parabolic partial differential inequality. A finite difference discretization on nonuniform grids leading to linear complementarity problems with M‐matrices is proposed. The projected SOR, a projected multigrid method, an operator splitting method, a penalty method, and a componentwise splitting method are considered. The last one is a direct method while all other methods are iterative. The resulting systems of linear equations in the operator splitting method and in the penalty method are solved using a multigrid method. The projected multigrid method and the componentwise splitting method lead to a sequence of linear complementarity problems with one‐dimensional differential operators that are solved using the Brennan and Schwartz algorithm. The numerical experiments compare the accuracy and speed of the considered methods. The accuracies of all methods appear to be similar. Thus, the additional approximations made in the operator splitting method, in the penalty method, and in the componentwise splitting method do not increase the error essentially. The componentwise splitting method is the fastest one. All multigrid‐based methods have similar rapid grid independent convergence rates. They are about two or three times slower that the componentwise splitting method. On the coarsest grid the speed of the projected SOR is comparable with the multigrid methods while on finer grids it is several times slower. ©John Wiley & Sons, Inc. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Multi-level minimal residual smoothing: a family of general purpose multigrid acceleration techniques

We employ multi-level minimal residual smoothing (MRS) as a pre-optimization technique to accelerate standard multigrid convergence. The MRS method is used to improve the current multigrid iterate by smoothing its corresponding residual before the latter is projected to the coarse grid. We develop different schemes for implementing MRS technique on the finest grid and on the coarse grids, and several versions of the inexact MRS technique. Numerical experiments are conducted to show the efficiency of the multi-level and inexact MRS techniques.     

Multigrid for second‐order ADN elliptic systems

This paper theoretically examines a multigrid strategy for solving systems of elliptic partial differential equations (PDEs) introduced in the work of Lee. Unlike most multigrid solvers that are constructed directly from the whole system operator, this strategy builds the solver using a factorization of the system operator. This factorization is composed of an algebraic coupling term and a diagonal (decoupled) differential operator. Exploiting the factorization, this approach can produce decoupled systems on the coarse levels. The corresponding coarse‐grid operators are in fact the Galerkin variational coarsening of the diagonal differential operator. Thus, rather than performing delicate coarse‐grid selection and interpolation weight procedures on the original strongly coupled system as often done, these procedures are isolated to the diagonal differential operator. To establish the theoretical results, however, we assume that these systems of PDEs are elliptic in the Agmon–Douglis–Nirenberg (ADN) sense and apply the factorization and multigrid only to the principal part of the system of PDEs. Two‐grid error bounds are established for the iteration applied to the complete system of PDEs. Numerical results are presented to illustrate the effectiveness of this strategy and to expose factors that affect the convergence of the methods derived from this strategy.     

Two‐level Fourier analysis of multigrid for higher‐order finite‐element discretizations of the Laplacian

In this paper, we employ local Fourier analysis (LFA) to analyze the convergence properties of multigrid methods for higher‐order finite‐element approximations to the Laplacian problem. We find that the classical LFA smoothing factor, where the coarse‐grid correction is assumed to be an ideal operator that annihilates the low‐frequency error components and leaves the high‐frequency components unchanged, fails to accurately predict the observed multigrid performance and, consequently, cannot be a reliable analysis tool to give good performance estimates of the two‐grid convergence factor. While two‐grid LFA still offers a reliable prediction, it leads to more complex symbols that are cumbersome to use to optimize parameters of the relaxation scheme, as is often needed for complex problems. For the purposes of this analytical optimization as well as to have simple predictive analysis, we propose a modification that is “between” two‐grid LFA and smoothing analysis, which yields reasonable predictions to help choose correct damping parameters for relaxation. This exploration may help us better understand multigrid performance for higher‐order finite element discretizations, including for Q₂‐Q₁ (Taylor‐Hood) elements for the Stokes equations. Finally, we present two‐grid and multigrid experiments, where the corrected parameter choice is shown to yield significant improvements in the resulting two‐grid and multigrid convergence factors.     

Multigrid solution of high order discretisation for three-dimensional biharmonic equation with Dirichlet boundary conditions of second kind

In this paper, we propose two compact finite difference approximations for three-dimensional biharmonic equation with Dirichlet boundary conditions of second kind. In these methods there is no need to define special formulas near the boundaries and boundary conditions are incorporated with these techniques. The unknown solution and its second derivatives are carried as unknowns at grid points. We derive second-order and fourth-order approximations on a 27 point compact stencil. Classical iteration methods such as Gauss–Seidel and SOR for solving the linear system arising from the second-order and fourth-order discretisation suffer from slow convergence. In order to overcome this problem we use multigrid method which exhibit grid-independent convergence and solve the linear system of equations in small amount of computer time. The fourth-order finite difference approximations are used to solve several test problems and produce high accurate numerical solutions.     

A note on an accelerated high‐accuracy multigrid solution of the convection‐diffusion equation with high Reynolds number

We present a new strategy to accelerate the convergence rate of a high‐accuracy multigrid method for the numerical solution of the convection‐diffusion equation at the high Reynolds number limit. We propose a scaled residual injection operator with a scaling factor proportional to the magnitude of the convection coefficients, an alternating line Gauss‐Seidel relaxation, and a minimal residual smoothing acceleration technique for the multigrid solution method. The new implementation strategy is tested to show an improved convergence rate with three problems, including one with a stagnation point in the computational domain. The effect of residual scaling and the algebraic properties of the coefficient matrix arising from the fourth‐order compact discretization are investigated numerically. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 1–10, 2000     

Conservative solution of the Fokker–Planck equation for stochastic chemical reactions

In this paper we discuss multigrid methods for ill-conditioned symmetric positive definite block Toeplitz matrices. Our block Toeplitz systems are general in the sense that the individual blocks are not necessarily Toeplitz, but we restrict our attention to blocks of small size. We investigate how transfer operators for prolongation and restriction have to be chosen such that our multigrid algorithms converge quickly. We point out why these transfer operators can be understood as block matrices as well and how they relate to the zeroes of the generating matrix function. We explain how our new algorithms can also be combined efficiently with the use of a natural coarse grid operator. We clearly identify a class of ill-conditioned block Toeplitz matrices for which our algorithmic ideas are suitable. In the final section we present an outlook to well-conditioned block Toeplitz systems and to problems of vector Laplace type. In the latter case the small size blocks can be interpreted as degrees of freedom associated with a node. A large number of numerical experiments throughout the article confirms convincingly that our multigrid solvers lead to optimal order convergence. AMS subject classification (2000) 65N55, 65F10     

Monotone multigrid methods for elliptic variational inequalities II

Summary. We derive globally convergent multigrid methods for discrete elliptic variational inequalities of the second kind as obtained from the approximation of related continuous problems by piecewise linear finite elements. The coarse grid corrections are computed from certain obstacle problems. The actual constraints are fixed by the preceding nonlinear fine grid smoothing. This new approach allows the implementation as a classical V-cycle and preserves the usual multigrid efficiency. We give estimates for the asymptotic convergence rates. The numerical results indicate a significant improvement as compared with previous multigrid approaches. Received March 26, 1994 / Revised version received September 22, 1994     

Multigrid transfers for nonsymmetric systems based on Schur complements and Galerkin projections

A framework is proposed for constructing algebraic multigrid transfer operators suitable for nonsymmetric positive definite linear systems. This framework follows a Schur complement perspective as this is suitable for both symmetric and nonsymmetric systems. In particular, a connection between algebraic multigrid and approximate block factorizations is explored. This connection demonstrates that the convergence rate of a two‐level model multigrid iteration is completely governed by how well the coarse discretization approximates a Schur complement operator. The new grid transfer algorithm is then based on computing a Schur complement but restricting the solution space of the corresponding grid transfers in a Galerkin‐style so that a far less expensive approximation is obtained. The final algorithm corresponds to a Richardson‐type iteration that is used to improve a simple initial prolongator or a simple initial restrictor. Numerical results are presented illustrating the performance of the resulting algebraic multigrid method on highly nonsymmetric systems. Copyright © 2013 John Wiley & Sons, Ltd.     

A multigrid upwind strategy for accelerating steady‐state computations of waves propagating with curvature‐dependent speeds

A multigrid strategy using upwind finite differencing is developed for accelerating the steady state computations of waves, [14] propagating with curvature‐dependent speeds. This will allow the rapid computation of a “burn table.” In a high explosive material, a burn table will allow the elimination of solving chemical reaction ODEs by feeding in source terms to the reactive flow equations for solution of the system of ignition of the high explosive material. Standard iterative methods show a quick reduction of the residual followed by a slow final convergence to the solution at high iterations. Such systems, including a nonlinear system such as this, are excellent choices for the use of multigrid methods to speed up convergence. Numerical steady‐state solutions to the eikonal equation on several test grids are conducted. Results are presented for these cases in 2D and a cubic grid in 3D using a Runge‐Kutta time iteration for the smoothing operator until steady state is reached. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 179–192, 2002; DOI 10.1002/num.1002     

Residual scaling techniques in multigrid, II: Practical applications

This paper focuses on the practical applications of the multigrid residual scaling techniques and is the continuation of a companion paper: Residual scaling techniques in multigrid, I: Equivalence proof [Appl. Math. Comput. 86:283–303 (1997)]. We discuss the computational issues of some residual scaling techniques which have been proven mathematically equivalent. A heuristic residual analysis technique, based on the geometry of the grid points and the relaxation pattern, is introduced to estimate the optimal residual scaling factor for a high-order multigrid method. We compare the performance of a typical pre-optimization (pre-acceleration) technique with a typical post-optimization (post-acceleration) technique and show that the pre-optimization is preferable in both convergence and efficiency. Our numerical results support the theoretical conclusions made in the companion paper and demonstrate the full advantage of the pre-optimization technique over the post-optimization technique.     

Integrated fast and high‐accuracy computation of convection diffusion equations using multiscale multigrid method

We present an explicit sixth‐order compact finite difference scheme for fast high‐accuracy numerical solutions of the two‐dimensional convection diffusion equation with variable coefficients. The sixth‐order scheme is based on the well‐known fourth‐order compact (FOC) scheme, the Richardson extrapolation technique, and an operator interpolation scheme. For a particular implementation, we use multiscale multigrid method to compute the fourth‐order solutions on both the coarse grid and the fine grid. Then, an operator interpolation scheme combined with the Richardson extrapolation technique is used to compute a sixth‐order accurate fine grid solution. We compare the computed accuracy and the implementation cost of the new scheme with the standard nine‐point FOC scheme and Sun–Zhang's sixth‐order method. Two convection diffusion problems are solved numerically to validate our proposed sixth‐order scheme. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

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.     

Efficient geometric multigrid implementation for triangular grids

This paper deals with a stencil-based implementation of a geometric multigrid method on semi-structured triangular grids (triangulations obtained by regular refinement of an irregular coarse triangulation) for linear finite element methods. An efficient and elegant procedure to construct these stencils using a reference stencil associated to a canonical hexagon is proposed. Local Fourier Analysis (LFA) is applied to obtain asymptotic convergence estimates. Numerical experiments are presented to illustrate the efficiency of this geometric multigrid algorithm, which is based on a three-color smoother.     

NFFT-based extension of a particle simulation method using multigrid

In the calculation of electrostatic problems using multigrid methods, computationally the most expensive part is often the charge assignment to a grid. We show how to improve an existing multigrid method by using an NFFT-based scheme for this task. Numerical tests show that the new method outperforms the direct charge assignment schemes for the case where smooth charge distributions with large support are considered. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

CASCADIC MULTIGRID METHODS FOR MORTAR WILSON FINITE ELEMENT METHODS ON PLANAR LINEAR ELASTICITY

Cascadic multigrid technique for mortar Wilson finite element method of homogeneous boundary value planar linear elasticity is described and analyzed. First the mortar Wilson finite element method for planar linear elasticity will be analyzed, and the error estimate under L2 and H1 norm is optimal. Then a cascadic multigrid method for the mortar finite element discrete problem is described. Suitable grid transfer operator and smoother are developed which lead to an optimal cascadic multigrid method. Finally, the computational results are presented.     

An Iterative Multigrid Regularization Method for Toeplitz Discrete Ill-Posed Problems

Iterative regularization multigrid methods have been successfully applied to signal/image deblurring problems. When zero-Dirichlet boundary conditions are imposed the deblurring matrix has a Toeplitz structure and it is potentially full. A crucial task of a multilevel strategy is to preserve the Toeplitz structure at the coarse levels which can be exploited to obtain fast computations. The smoother has to be an iterative regularization method. The grid transfer operator should preserve the regularization property of the smoother. This paper improves the iterative multigrid method proposed in [11] introducing a wavelet soft-thresholding denoising post-smoother. Such post-smoother avoids the noise amplification that is the cause of the semi-convergence of iterative regularization methods and reduces ringing effects. The resulting iterative multigrid regularization method stabilizes the iterations so that the imprecise (over) estimate of the stopping iteration does not have a deleterious effect on the computed solution. Numerical examples of signal and image deblurring problems confirm the effectiveness of the proposed method.