GAUSS-SEIDEL-TYPE MULTIGRID METHODS

By making use of the Gauss-Seidel-type solution method, the procedure for computing the interpolation operator of multigrid methods is simplified. This leads to a saving of computational time. Three new kinds of interpolation formulae are obtained by adopting different approximate methods, to try to enhance the accuracy of the interpolatory operator. A theoretical study proves the two-level convergence of these Gauss-Seidel-type MG methods. A series of numerical experiments is presented to evaluate the relative performance of the methods with respect to the convergence factor, CPU-time(for one V-cycle and the setup phase) and computational complexity.

MULTIGRID METHODS FOR OBSTACLE PROBLEMS

In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which is closely related to truncated multigrid. The numerical properties of algorithms are carefully assessed by means of a degenerate problem and a problem with a complicated coincidence set.

Multigrid Methods for Elliptic Optimal Control Problems with Pointwise State Constraints

An elliptic optimal control problem with constraints on the state variable is considered.The Lavrentiev-type regularization is used to treat the constraints on the state variable.To solve the problem numerically,the multigrid for optimization(MGOPT)technique and the collective smoothing multigrid(CSMG)are implemented.Numerical results are reported to illustrate and compare the efficiency of both multigrid strategies.

THE CONVERGENCE OF MULTIGRID METHODS FOR SOLVING FINITE ELEMENT EQUATIONS IN THE PRESENCE OF SINGULARITIES

We analyze the convergence of multigrid methods applied to finite elementequations of second order with singularities caused by reentrant angles and abruptchanges in the boundary conditions. Provided much more weaker demand of clas-sical multigrid proofs, it is shown in this paper that, for symmetric and positivedefinite problems in the presence of singularities, multigrid algorithms with evenone smoothing step converge at a rate which is independent of the number of lev-els or unknowns. Furthermore, we extend this result to the nonsymmetric andindefinite problems.

MULTIGRID METHODS FOR THE GENERALIZED STOKES EQUATIONS BASED ON MIXED FINITE ELEMENT METHODS

Examines multigrid methods for the generalized stationary Stokes equations which are discretized by various mixed finite element methods. Information on a multigrid algorithm; preliminaries of convergence; Application of the general multigrid algorithm.

Cascadic multigrid methods for parabolic problems

In this paper,we consider the cascadic multigrid method for a parabolic type equation.Backward Euler approximation in time and linear finite element approximation in space are employed.A stability result is established under some conditions on the smoother.Using new and sharper estimates for the smoothers that reflect the precise dependence on the time step and the spatial mesh parameter,these conditions are verified for a number of popular smoothers.Optimal error bound sare derived for both smooth and non-smooth data.Iteration strategies guaranteeing both the optimal accuracy and the optimal complexity are presented.

Contour Matching Using Wavelet Transform and Multigrid Methods

In this paper, wavelet transform and multigrid method are combined to make the method more practical. It is known that Gaussian filtering causes shrinkage of data. To overcome this disadvantage, Gaussian filtering is replaced with wavelet transform. This method introduces no curve shrinkage. Then, the linearized form of objective equation is proposed. This makes contour matching easier to implement. Finally, the multigrid method is used to speed up the convergence.

Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell’s Equations

We consider the convergence theory of adaptive multigrid methods for secondorder elliptic problems and Maxwell’s equations.The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their“immediate”neighbors.In the context of lowest order conforming finite element approximations,we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms.The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures.The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom.We demonstrate our convergence theory by two numerical experiments.

MULTIGRID METHODS FOR MORLEY ELEMENT ON NONNESTED MESHES

In this paper, we consider some multigrid algorithms for the biharmonic problem discretized by Morley element on nonnested meshes. Through taking the averages of the nodal variables we construct an intergrid transfer operator that satisfies a certain stable approximation property. The so-called regularity-approximation assumption is then established. Optimal convergence properties of the W-cycle and a uniform condition number estimate for the variable V-cycle preconditioner are presented. This technique is applicable to other nonconforming plate elements.

Numerical Study of Geometric Multigrid Methods on CPU–GPU Heterogeneous Computers

.The geometric multigrid method(GMG)is one of the most efficient solving techniques for discrete algebraic systems arising from elliptic partial differential equations.GMG utilizes a hierarchy of grids or discretizations and reduces the error at a number of frequencies simultaneously.Graphics processing units(GPUs)have recently burst onto the scientific computing scene as a technology that has yielded substantial performance and energy-efficiency improvements.A central challenge in implementing GMG on GPUs,though,is that computational work on coarse levels cannot fully utilize the capacity of a GPU.In this work,we perform numerical studies of GMG on CPU–GPU heterogeneous computers.Furthermore,we compare our implementation with an efficient CPU implementation of GMG and with the most popular fast Poisson solver,Fast Fourier Transform,in the cuFFT library developed by NVIDIA.

A New Extrapolation Economy Cascadic Multigrid Method for Image Restoration Problems

In this paper, a new extrapolation economy cascadic multigrid method is proposed to solve the image restoration model. The new method combines the new extrapolation formula and quadratic interpolation to design a nonlinear prolongation operator, which provides more accurate initial values for the fine grid level. An edge preserving denoising operator is constructed to remove noise and preserve image edges. The local smoothing operator reduces the influence of staircase effect. The experiment results show that the new method not only improves the computational efficiency but also ensures good recovery quality.

A Preconditioned Multigrid Method for Efficient Simulation of Three-dimensional Compressible and Incompressible Flows

To develop an efficient and robust aerodynamic analysis method for numerical optimization designs of wing and complex configuration, a combination of matrix preconditioning and multigrid method is presented and investigated. The time derivatives of three-dimensional Navier-Stokes equations are preconditioned by Choi-Merkle preconditioning matrix that is originally designed for two-dimensional low Mach number viscous flows. An extension to three-dimensional viscous flow is implemented, and a method improving the convergence for transonic flow is proposed. The space discretizaition is performed by employing a finite-volume cell-centered scheme and using a central difference. The time marching is based on an explicit Rtmge-Kutta scheme proposed by Jameson. An efficient FAS multigrid method is used to accelerate the convergence to steady-state solutions. Viscous flows over ONERA M6 wing and M100 wing are numerically simulated with Mach numbers ranging from 0.010 to 0.839. The inviscid flow over the DLR-F4 wing-body configuration is also calculated to preliminarily examine the performance of the presented method for complex configuration. The computed results are compared with the experimental data and good agreement is achieved. It is shown that the presented method is efficient and robust for both compressible and incompressible flows and is very attractive for aerodynamic optimization designs of wing and complex configuration.

An efficient multigrid-DEIM semi-reduced-order model for simulation of single-phase compressible flow in porous media

In this paper,an efcient multigrid-DEIM semi-reduced-order model is developed to accelerate the simulation of unsteady single-phase compressible fow in porous media.The cornerstone of the proposed model is that the full approximate storage multigrid method is used to accelerate the solution of fow equation in original full-order space,and the discrete empirical interpolation method(DEIM)is applied to speed up the solution of Peng-Robinson equation of state in reduced-order subspace.The multigrid-DEIM semi-reduced-order model combines the computation both in full-order space and in reducedorder subspace,which not only preserves good prediction accuracy of full-order model,but also gains dramatic computational acceleration by multigrid and DEIM.Numerical performances including accuracy and acceleration of the proposed model are carefully evaluated by comparing with that of the standard semi-implicit method.In addition,the selection of interpolation points for constructing the low-dimensional subspace for solving the Peng-Robinson equation of state is demonstrated and carried out in detail.Comparison results indicate that the multigrid-DEIM semi-reduced-order model can speed up the simulation substantially at the same time preserve good computational accuracy with negligible errors.The general acceleration is up to 50-60 times faster than that of standard semi-implicit method in two-dimensional simulations,but the average relative errors of numerical results between these two methods only have the order of magnitude 10^(−4)-10^(−6)%.

THE MULTIGRID METHOD FOR RESERVOIR SIMULATION

This paper describes a way of solving the reservoir simulation pressure equation using mulligrid technique. The subroutine MG of four-grid method is presented. The result for 2-D two-phase problem is exactly the same as that of the SOR method and the CPU time is much less than that of the latter one.

Geometric Multigrid Method for Isogeometric Analysis

The isogeometric analysis method(IGA)is a new type of numerical method solving partial differential equations.Compared with the traditional finite element method,IGA based on geometric spline can keep the model consistency between geometry and analysis,and provide higher precision with less freedom.However,huge stiffness matrix fromthe subdivision progress still leads to the solution efficiency problems.This paper presents amultigrid method based on geometric multigrid(GMG)to solve the matrix system of IGA.This method extracts the required computational data for multigrid method fromthe IGA process,which also can be used to improve the traditional algebraic multigrid method(AGM).Based on this,a full multigrid method(FMG)based on GMG is proposed.In order to verify the validity and reliability of these methods,this paper did some test on Poisson’s equation and Reynolds’equation and compared the methods on different subdivision methods,different grid degrees of freedom,different cyclic structure degrees,and studied the convergence rate under different subdivision strategies.The results show that the proposed method is superior to the conventional algebraic multigrid method,and for the standard relaxed V-cycle iteration,the method still has a convergence speed independent of the grid size at the same degrees.

The V-cycle Multigrid Method for a Hermite-type Rectangular Element

In this paper, a V-cycle multigrid method is presented for a Hermite rectangular element. By defining proper mesh-dependent inner product and transfer operator, we obtain its convergence property and the uniform convergence rate independent of mesh size and level are established.

Multigrid Method for the Numerical Solution of the Modified Equal Width Wave Equation

Numerical solutions of the modified equal width wave equation are obtained by using the multigrid method and finite difference method. The motion of a single solitary wave, interaction of two solitary waves and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. The numerical solutions are compared with the known analytical solutions. Using error norms and conservative properties of mass, momentum and energy, accuracy and efficiency of the mentioned method will be established through comparison with other methods.

Application of multigrid NLS-4DVar in radar radial velocity data assimilation with WRF-ARW

The nonlinear least-squares four-dimensional variational assimilation(NLS-4DVar)method intro-duced here combines the merits of the ensemble Kalman lter and 4DVar assimilation methods.The multigrid NLS-4DVar method can be implemented without adjoint models and also corrects small-to large-scale errors with greater accuracy.In this paper,the multigrid NLS-4DVar method is used in radar radial velocity data assimilations.Observing system simulation experiments were conducted to determine the capability and efficiency of multigrid NLS-4DVar for assimilating radar radial velocity with WRF-ARW(the Advanced Research Weather Research and Forecasting model).The results show signi cant improvement in 24-h cumulative precipitation prediction due to improved initial conditions after assimilating the radar radial velocity.Additionally,the multigrid NLS-4DVar method reduces computational cost.

A Cascadic Multigrid Algorithm for the Mortar Element Method for Semiliner Ellptic Problems

In this paper a cascadic multigrid algorithm for the mortar finite element approximation of the semilinear elliptic problem is proposed,and corresponding theorems aregiven,which display the error estimate and the computational complexity of the method.

Numericals for total variation-based reconstruction of motion blurred images

In this paper image with horizontal motion blur, vertical motion blur and angled motion blur are considered. We construct several difference schemes to the highly nonlinear term △↓.（△↓u/√｜△↓｜^2＋β） of the total variation-based image motion deblurring problem. The large nonlinear system is linearized by fixed point iteration method. An algebraic multigrid method with Krylov subspace acceleration is used to solve the corresponding linear equations as in [7]. The algorithms can restore the image very well. We give some numerical experiments to demonstrate that our difference schemes are efficient and robust.