Fast Iterative Solvers Research Articles

Analysis of preconditioned iterative solvers for incompressible flow problems

AbstractSolving efficiently the incompressible Navier–Stokes equations is a major challenge, especially in the three‐dimensional case. The approach investigated by Elman et al. (Finite Elements and Fast Iterative Solvers. Oxford University Press: Oxford, 2005) consists in applying a preconditioned GMRES method to the linearized problem at each iteration of a nonlinear scheme. The preconditioner is built as an approximation of an ideal block‐preconditioner that guarantees convergence in 2 or 3 iterations. In this paper, we investigate the numerical behavior for the three‐dimensional lid‐driven cavity problem with wedge elements; the ultimate motivation of this analysis is indeed the development of a preconditioned Krylov solver for stratified oceanic flows which can be efficiently tackled using such meshes. Numerical results for steady‐state solutions of both the Stokes and the Navier–Stokes problems are presented. Theoretical bounds on the spectrum and the rate of convergence appear to be in agreement with the numerical experiments. Sensitivity analysis on different aspects of the structure of the preconditioner and the block decomposition strategies are also discussed. Copyright © 2011 John Wiley & Sons, Ltd.

Recycling Krylov subspaces for efficient large-scale electrical impedance tomography

Electrical impedance tomography (EIT) captures images of internal features of a body. Electrodes are attached to the boundary of the body, low intensity alternating currents are applied, and the resulting electric potentials are measured. Then, based on the measurements, an estimation algorithm obtains the three-dimensional internal admittivity distribution that corresponds to the image. One of the main goals of medical EIT is to achieve high resolution and an accurate result at low computational cost. However, when the finite element method (FEM) is employed and the corresponding mesh is refined to increase resolution and accuracy, the computational cost increases substantially, especially in the estimation of absolute admittivity distributions. Therefore, we consider in this work a fast iterative solver for the forward problem, which was previously reported in the context of structural optimization. We propose several improvements to this solver to increase its performance in the EIT context. The solver is based on the recycling of approximate invariant subspaces, and it is applied to reduce the EIT computation time for a constant and high resolution finite element mesh. In addition, we consider a powerful preconditioner and provide a detailed pseudocode for the improved iterative solver. The numerical results show the effectiveness of our approach: the proposed algorithm is faster than the preconditioned conjugate gradient (CG) algorithm. The results also show that even on a standard PC without parallelization, a high mesh resolution (more than 150,000 degrees of freedom) can be used for image estimation at a relatively low computational cost.

Efficient monolithic simulation techniques for the stationary Lattice Boltzmann equation on general meshes

In this paper, we present special discretization and solution techniques for the numerical simulation of the Lattice Boltzmann equation (LBE). In Hübner and Turek (Computing, 81:281–296, 2007), the concept of the generalized mean intensity had been proposed for radiative transfer equations which we adapt here to the LBE, treating it as an analogous (semi-discretized) integro-differential equation with constant characteristics. Thus, we combine an efficient finite difference-like discretization based on short-characteristic upwinding techniques on unstructured, locally adapted grids with fast iterative solvers. The fully implicit treatment of the LBE leads to nonlinear systems which can be efficiently solved with the Newton method, even for a direct solution of the stationary LBE. With special exact preconditioning by the transport part due to the short-characteristic upwinding, we obtain an efficient linear solver for transport dominated configurations (macroscopic Stokes regime), while collision dominated cases (Navier-Stokes regime for larger Re numbers) are treated with a special block-diagonal preconditioning. Due to the new generalized equilibrium formulation (GEF) we can combine the advantages of both preconditioners, i.e. independence of the number of unknowns for convection-dominated cases with robustness for stiff configurations. We further improve the GEF approach by using hierarchical multigrid algorithms to obtain grid-independent convergence rates for a wide range of problem parameters, and provide representative results for various benchmark problems. Finally, we present quantitative comparisons between a highly optimized CFD-solver based on the Navier-Stokes equation (FeatFlow) and our new LBE solver (FeatLBE).

3D multidomain BEM for a Poisson equation

This paper deals with the efficient 3D multidomain boundary element method (BEM) for solving a Poisson equation. The integral boundary equation is discretized using linear mixed boundary elements. Sparse system matrices similar to the finite element method are obtained, using a multidomain approach, also known as the ‘subdomain technique’. Interface boundary conditions between subdomains lead to an overdetermined system matrix, which is solved using a fast iterative linear least square solver. The accuracy, efficiency and robustness of the developed numerical algorithm are presented using cube and sphere geometry, where the comparison with the competitive BEM is performed. The efficiency is demonstrated using a mesh with over 200,000 hexahedral volume elements on a personal computer with 1 GB memory.

Improvement of Unstructured Quadrilateral Mesh Quality for Multigrid Analysis

Due to more complex and severe design restrictions, more effective and faster finite element analyses are demanded. There are several ways to compute FE analysis efficiently: parallel computing, fast iterative or direct solvers, adaptive analysis and so on. One of the most effective analysis ways is the combination of adaptive analysis and multigrid iterative solver, because an adaptive analysis requires several meshes with difference resolutions and multigrid solver utilizes such meshes to accelerate its computation. However, convergence of multigrid solver is largely affected by initial shape of each element. An effective mesh improvement method is proposed here. It is the combination of mesh coarsening and refinement. A good mesh can be obtained by the method to be applied to an initial mesh, and better convergence is achieved by the improved initial mesh.

An Algorithm for Total Variation Inpainting Based on Nonlinear Multi-Grid Methods

Image inpainting refers to restoring a damaged image with missing information. The total variation (TV) inpainting model is one such method that simultaneously fills in the regions with available information from their surroundings and eliminates noises. The method works well with small narrow inpainting domains. However there remains an urgent need to develop fast iterative solvers, as the underlying problem sizes are large. In addition one needs to tackle the imbalance of results between inpainting and denoising. When the inpainting regions are thick and large, the procedure of inpainting works quite slowly and usually requires a significant number of iterations and leads inevitably to oversmoothing in the outside of the inpainting domain. To overcome these difficulties, we propose a solution for TV inpainting method based on the nonlinear multi-grid algorithm.

Advances in Iterative Methods and Preconditioners for the Helmholtz Equation

In this paper we survey the development of fast iterative solvers aimed at solving 2D/3D Helmholtz problems. In the first half of the paper, a survey on some recently developed methods is given. The second half of the paper focuses on the development of the shifted Laplacian preconditioner used to accelerate the convergence of Krylov subspace methods applied to the Helmholtz equation. Numerical examples are given for some difficult problems, which had not been solved iteratively before.

3D multidomain BEM for solving the Laplace equation

An efficient 3D multidomain boundary element method (BEM) for solving problems governed by the Laplace equation is presented. Integral boundary equations are discretized using mixed boundary elements. The field function is interpolated using a continuous linear function while its derivative in a normal direction is interpolated using a discontinuous constant function over surface boundary elements. Using a multidomain approach, also known as the subdomain technique, sparse system matrices similar to the finite element method (FEM) are obtained. Interface boundary conditions between subdomains leads to an over-determined system matrix, which is solved using a fast iterative linear least square solver. The accuracy and robustness of the developed numerical algorithm is presented on a scalar diffusion problem using simple cube geometry and various types of meshes. Efficiency is demonstrated with potential flow around the complex geometry of a fighter airplane using tetrahedral mesh with over 100,000 subdomains on a personal computer.

Large‐scale topology optimization using preconditioned Krylov subspace methods with recycling

AbstractThe computational bottleneck of topology optimization is the solution of a large number of linear systems arising in the finite element analysis. We propose fast iterative solvers for large three‐dimensional topology optimization problems to address this problem. Since the linear systems in the sequence of optimization steps change slowly from one step to the next, we can significantly reduce the number of iterations and the runtime of the linear solver by recycling selected search spaces from previous linear systems. In addition, we introduce a MINRES (minimum residual method) version with recycling (and a short‐term recurrence) to make recycling more efficient for symmetric problems. Furthermore, we discuss preconditioning to ensure fast convergence. We show that a proper rescaling of the linear systems reduces the huge condition numbers that typically occur in topology optimization to roughly those arising for a problem with constant density. We demonstrate the effectiveness of our solvers by solving a topology optimization problem with more than a million unknowns on a fast PC. Copyright © 2006 John Wiley & Sons, Ltd.

Book Review: Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics

Book Review: Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics

Similarity based interpolation using Catmull–Clark subdivision surfaces

A new method for constructing a Catmull–Clark subdivision surface (CCSS) that interpolates the vertices of a given mesh with arbitrary topology is presented. The new method handles both open and closed meshes. Normals or derivatives specified at any vertices of the mesh (which can actually be anywhere) can also be interpolated. The construction process is based on the assumption that, in addition to interpolating the vertices of the given mesh, the interpolating surface is also similar to the limit surface of the given mesh. Therefore, construction of the interpolating surface can use information from the given mesh as well as its limit surface. This approach, called similarity based interpolation, gives us more control on the smoothness of the interpolating surface and, consequently, avoids the need of shape fairing in the construction of the interpolating surface. The computation of the interpolating surface’s control mesh follows a new approach, which does not require the resulting global linear system to be solvable. An approximate solution provided by any fast iterative linear system solver is sufficient. Nevertheless, interpolation of the given mesh is guaranteed. This is an important improvement over previous methods because with these features, the new method can handle meshes with large number of vertices efficiently. Although the new method is presented for CCSSs, the concept of similarity based interpolation can be used for other subdivision surfaces as well.

A fast iterative solver for scattering by elastic objects in layered media

We developed a fast iterative solver for computing time-harmonic acoustic waves scattered by an elastic object in layered media. The discretization of the problem was performed using a finite element method with linear elements based on a locally body-fitted uniform triangulation. We used a domain decomposition preconditioner in the iterative solution of the resulting system of linear equations. The preconditioner was based on a cyclic reduction type fast direct solver. The solution procedure reduces GMRES iterates onto a sparse subspace which decreases the storage and computational requirements essentially. The numerical results demonstrate the effectiveness of the proposed approach for two-dimensional domains that are hundreds of wavelengths wide and require the solution of linear systems with several millions of unknowns.

A fast and robust iterative solver for nonlinear contact problems using a primal‐dual active set strategy and algebraic multigrid

AbstractFor extending the usability of implicit FE codes for large‐scale forming simulations, the computation time has to be decreased dramatically. In principle this can be achieved by using iterative solvers. In order to facilitate the use of this kind of solvers, one needs a contact algorithm which does not deteriorate the condition number of the system matrix and therefore does not slow down the convergence of iterative solvers like penalty formulations do. Additionally, an algorithm is desirable which does not blow up the size of the system matrix like methods using standard Lagrange multipliers. The work detailed in this paper shows that a contact algorithm based on a primal‐dual active set strategy provides these advantages and therefore is highly efficient with respect to computation time in combination with fast iterative solvers, especially algebraic multigrid methods. Copyright © 2006 John Wiley & Sons, Ltd.

On a nonlinear multigrid algorithm with primal relaxation for the image total variation minimisation

Digital image restoration has drawn much attention in the recent years and a lot of research has been done on effective variational partial differential equation models and their theoretical studies. However there remains an urgent need to develop fast and robust iterative solvers, as the underlying problem sizes are large. This paper proposes a fast multigrid method using primal relaxations. The basic primal relaxation is known to get stuck at a ‘local’ non-stationary minimum of the solution, which is usually believed to be ‘non-smooth’. Our idea is to utilize coarse level corrections, overcoming the deadlock of a basic primal relaxation scheme. A further refinement is to allow non-regular coarse levels to correct the solution, which helps to improve the multilevel method. Numerical experiments on both 1D and 2D images are presented.

Adaptive local multigrid methods for solving time-harmonic eddy-current problems

The efficient computation of large eddy-current problems with finite elements requires adaptive methods and fast optimal iterative solvers such as multigrid methods. This paper provides an overview of the most important implementation aspects of an adaptive multigrid scheme for time-harmonic eddy currents. Numerical experiments show that the standard multigrid scheme can be modified to yield an O(N) complexity even for general adaptive refinement strategies, where the number of unknowns N can grow slowly from one to the next refinement level. Algorithmic details and numerical examples are given.

A multidomain boundary element method for two equation turbulence models

The paper deals with the multidomain Boundary Element Method (BEM) for modelling 2D complex turbulent flow using low Reynolds two equation turbulence models. While the BEM is widely accepted for laminar flow this is the first case, where this method is applied for a complex flow problems using k– ε turbulence model. The integral boundary domain equations are discretised using mixed boundary elements and a multidomain method also known as subdomain technique. The resulting system matrix is overdetermined, sparse, block banded and solved using fast iterative linear least squares solver. The simulation of turbulent flow over a backward step is in excellent agreement with the finite volume method using the same turbulent model.

A fast iterative solver for the variable coefficient diffusion equation on a disk

We present an efficient iterative method for solving the variable coefficient diffusion equation on a unit disk. The equation is written in polar coordinates and is discretized by the standard centered difference approximation under the grid arrangement by shifting half radial mesh away from the origin so that the coordinate singularity can be handled naturally without pole conditions. The resultant matrix is symmetric positive definite so the preconditioned conjugate gradient (PCG) method can be applied. Different preconditioners have been tested for comparison, in particular, a fast direct solver derived from the equation and the semi-coarsening multigrid are shown to be almost scalable with the problem size and outperform other preconditioners significantly. The present elliptic solver has been applied to study the vortex dynamics of the Ginzburg–Landau equation with a variable diffusion coefficient.

Fast Iterative Solvers for Discrete Stokes Equations

We consider saddle point problems that result from the finite element discretization of stationary and nonstationary Stokes equations. Three efficient iterative solvers for these problems are treated, namely, the preconditioned conjugate gradient method introduced by Bramble and Pasciak, the preconditioned MINRES method, and a method due to Bank et al. We give a detailed overview of algorithmic aspects and theoretical convergence results. For the method of Bank et al. a new convergence analysis is presented. A comparative study of the three methods for a three-dimensional Stokes problem discretized by the Hood--Taylor P2-P1 finite element pair is given.

Mini-Workshop: Numerical Methods for Instationary Control Problems

The topic for the current Mini-Workshop organized by Karl Kunisch (Univ. Graz), Angela Kunoth (Univ. Bonn) and Rolf Rannacher (Univ. Heidelberg) emerged from the Oberwolfach Workshop “Numerical Techniques for Optimization Problems with PDE Constraints” which was held February 16–22, 2003. It was realized that numerically solving control problems which are constrained by time-dependent nonlinear PDEs (Partial Differential Equations) are particularly challenging with respect to the complexity of the problem. Mathematically, one has to minimize a functional under PDE constraints and possibly additional constraints on the state and the control. Standard discretizations on uniform grids in space and time will only yield solutions where the inherent structures of the problem (nonlinearity, constraints) are not sufficiently captured. Certain optimization problems for large coupled systems of partial differential equations are currently not yet numerically treatable or do not satisfy the time constraints required in practice. Overcoming this barrier can only be achieved by designing new mathematically founded algorithmic approaches. The road towards this goal leads to many interesting problems in optimization, linear algebra, numerics, analysis, and approximation theory. The conference had 21 participants which represented continuous optimization, numerical analysis and scientific computing. 18 talks were given. The 10 longer, overview-style talks were on optimization with PDEs, focussing on These talks were intended to bridge the gap between the different research fields Optimization and Numerics. They were complemented by 8 shorter talks on more specialized research topics, ranging from efficiency indices for optimization over iterative methods for the coupled systems and multigrid acceleration to modelling issues in optimization, and data compression by means of proper orthogonal decompositions and central Voronoi tesselations. Different modern approaches to overcome the complexity issues in numerical simulations for PDE-constrained optimization have been presented and discussed. One of the approaches is to employ fast iterative solvers like multigrid on uniform grids. The methodology which conceptually provides the largest potential is to introduce adaptivity. This drastically reduces complexity but depending on the context may require solving an additional problem. Wavelet approaches particularly allow to resolve each of the variables separately and in addition provide a built-in preconditioning. Yet another approach uses compressed information in order to efficiently solve the primal-dual system. All these issues are currently very active research areas. The extensive discussions held during this workshop have produced a number of new ideas and connections. It was agreed upon that a mere black-box-style matching of efficient PDE codes with optimization tool boxes would on one hand remain much below its potential and on the other hand not help overcoming complexity barriers. Some concepts from automatic differentiation seem to carry over to adaptive methods. Even combining adaptivity with proper orthogonal decompositions may be a very promising direction. The many new ideas discussed during this workshop will have to be further elaborated in future.

Efficient numerical methods for radiation in gas turbines

Numerical methods for radiative heat transfer equations coupled to a temperature equation are considered. Efficient solution methods and approximate equations for this system are investigated and a comparative numerical study of the different approximations is given. The approximate equations considered in this paper include moment methods and diffusive approximations. Fast iterative solvers for the problem like multilevel methods with suitable preconditioning are considered in detail. Numerical experiments and comparisons in different space dimensions and for various physical situations are presented.