Fred Wubs

Matrix Renumbering ILU: An Effective Algebraic Multilevel ILU Preconditioner for Sparse Matrices

In this paper a multilevel-like ILU preconditioner is introduced. The ILU factorization generates its own ordering during the elimination process. Both ordering and dropping depend on the size of the entries. The method can handle structured and unstructured problems. Results are presented for some important classes of matrices and for several well-known test examples. The results illustrate the efficiency of the method and show in several cases near grid independent convergence.

Nested grids ILU-decomposition (NGILU)

A preconditioning technique is described which shows, in many cases, grid-independent convergence. This technique only requires an ordering of the unknowns based on the different levels of multigrid, and an incomplete LU-decomposition based on a drop tolerance. The method is demonstrated on a variety of well-known elliptic test problems including strongly varying coefficients, advective terms and grid refinement.

An explicit-implicit method for a class of time-dependent partial differential equations

For the integration of partial differential equations, we distinguish explicit and implicit methods. In this paper, we consider an explicit-implicit method, which follows from the truncation of the solution process of a fully implicit method. Such a method is of interest because not only better vectorizing properties can be obtained by increasing the explicit part, but the method also fits well in a domain-decomposition approach. In this paper, we focus on the feasibility of such methods by studying their stability and accuracy properties. Nevertheless, we also did some experiments on vectorcomputers to show that for a sufficient degree of explicitness our method is more efficient than fully implicit methods.

Bifurcation analysis of incompressible flow in a driven cavity by the Newton-Picard method

Knowledge of the transition point of steady to periodic flow and the frequency occurring hereafter is becoming increasingly more important in engineering applications. By the Newton-Picard method - a method related to the recursive projection method - periodic solutions can be computed, which makes such knowledge available. In the paper this method is applied to study the bifurcation behavior of the flow in a driven cavity at Reynolds numbers between 7500 and 10000. For the time discretization the θ-method is used and for the space discretization a symmetry-preserving finite-volume method. The implicit relations occurring after linearization are solved by the multilevel ILU solver MRILU. Our results extend findings from earlier work with respect to the transition point.

A tailored solver for bifurcation analysis of ocean-climate models

In this paper, we present a new linear system solver for use in a fully-implicit ocean model. The new solver allows to perform bifurcation analysis of relatively high-resolution primitive-equation ocean-climate models. It is based on a block-ILU approach and takes special advantage of the mathematical structure of the governing equations. In implicit models Jacobian matrices have to be constructed. Analytical construction is hard for complicated but more realistic representations of mixing. This is overcome by evaluating the Jacobian in part numerically. The performance of the new implicit ocean model is demonstrated using (i) a high-resolution model of the wind-forced double-gyre flow problem in a (relatively small) midlatitude spherical basin, and (ii) a medium-resolution model of thermohaline and wind-driven flows in an Atlantic size single-hemispheric basin.

How fast the Laplace equation was solved in 1995

On the occasion of the third centenary of the appointment of Johann Bernoulli at the University of Groningen, a number of linear systems solvers for some Laplace-like equations have been compared during a one-day workshop. CPU times of several advanced solvers measured on the same computer (an HP-755 workstation) are presented, which makes it possible to draw clear conclusions about the performance of these solvers.

Exploiting Multilevel Preconditioning Techniques in Eigenvalue Computations

In the Davidson method, any preconditioner can be exploited for the iterative computation of eigenpairs. However, the convergence of the eigenproblem solver may be poor for a high quality preconditioner. Theoretically, this counter-intuitive phenomenon with the Davidson method is remedied by the Jacobi--Davidson approach, where the preconditioned system is restricted to appropriate subspaces of codimension one. However, it is not clear how the restricted system can be solved accurately and efficiently in the case of a good preconditioner. The obvious approach introduces instabilities that hamper convergence. In this paper, we show how incomplete decomposition based on multilevel approaches can be used in a stable way. We also show how these preconditioners can be efficiently improved when better approximations for the eigenvalue of interest become available. The additional costs for updating the preconditioners are negligible. Furthermore, our approach leads to a good initial guess for the wanted eigenpair and to high quality preconditioners for nearby eigenvalues. We illustrate our ideas for the MRILU preconditioner.

The application of Jacobian-free Newton-Krylov methods to reduce the spin-up time of ocean general circulation models

In present-day forward time stepping ocean-climate models, capturing both the wind-driven and thermohaline components, a substantial amount of CPU time is needed in a so-called spin-up simulation to determine an equilibrium solution. In this paper, we present methodology based on Jacobian-Free Newton-Krylov methods to reduce the computational time for such a spin-up problem. We apply the method to an idealized configuration of a state-of-the-art ocean model, the Modular Ocean Model version 4 (MOM4). It is shown that a typical speed-up of a factor 10-25 with respect to the original MOM4 code can be achieved and that this speed-up increases with increasing horizontal resolution.

Grid-independent convergence based on preconditioning techniques

Today numerical calculations are no longer restricted to a class of simple problems, but cope with complicated simulations and complex geometries. In many situations the accuracy of the numerical solution is determined by the limited amount of computer power and memory. Therefore much attention has been given to the development of numerical methods for solving the large sparse system of equations Ax = b obtained by discretising some partial differential equation. Since direct methods require much computer storage and CPU-time, a large variety of iterative methods has been derived. In this paper we will focus on iterative methods like MICCG and algebraic multigrid. Gustafsson [1] has shown that for several problems the CPU-time using MICCG is O(N 5/4) in 2 dimensions and O(N 7/6) for 3D-problems, where N is the total number of unknowns. Multigrid methods perform even better and for a large class of problems they have an optimal order of convergence: the amount of work and storage is proportial to the number of unknowns N. However, due to the required proper smoothers and the restriction and prolongation operators at each level, the implementation of multigrid for practical problems is much more complicated than that of MICCG. Here we look for a combination of these properties: an incomplete LU-decomposition such that the preconditioned system can be solved with the optimal computational complexity O(N) by a conjugate gradient-like method. The basic idea behind this preconditioning technique is the same as in multigrid methods. In Section 2 a preconditioning technique is described which uses a partition of the unknowns based on the sequence of grids in multigrid.

A Robust Two-Level Incomplete Factorization for (Navier-)Stokes Saddle Point Matrices

We present a new hybrid direct/iterative approach to the solution of a special class of saddle point matrices arising from the discretization of the steady incompressible Navier-Stokes equations on an Arakawa C-grid. The two-level method introduced here has the following properties: (i) it is very robust, even close to the point where the solution becomes unstable; (ii) a single parameter controls fill and convergence, making the method straightforward to use; (iii) the convergence rate is independent of the number of unknowns; (iv) it can be implemented on distributed memory machines in a natural way; (v) the matrix on the second level has the same structure and numerical properties as the original problem, so the method can be applied recursively; (vi) the iteration takes place in the divergence-free space, so the method qualifies as a “constraint preconditioner”; (vii) the approach can also be applied to Poisson problems. This work is also relevant for problems in which similar saddle point matrices occur, for instance, when simulating electrical networks, where one has to satisfy Kirchhoffs conservation law for currents.