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.     

Accelerating Galerkin BEM for linear elasticity using adaptive cross approximation

The adaptive cross approximation (ACA) algorithm (Numer. Math. 2000; 86 :565–589; Computing 2003; 70 (1):1–24) provides a means to compute data‐sparse approximants of discrete integral formulations of elliptic boundary value problems with almost linear complexity. ACA uses only few of the original entries for the approximation of the whole matrix and is therefore well‐suited to speed up existing computer codes. In this article we extend the convergence proof of ACA to Galerkin discretizations. Additionally, we prove that ACA can be applied to integral formulations of systems of second‐order elliptic operators without adaptation to the respective problem. The results of applying ACA to boundary integral formulations of linear elasticity are reported. Furthermore, we comment on recent implementation issues of ACA for non‐smooth boundaries. Copyright © 2006 John Wiley & Sons, Ltd.     

Discontinuous Galerkin methods on graphics processing units for nonlinear hyperbolic conservation laws

We present a novel implementation of the modal DG method for hyperbolic conservation laws in two dimensions on graphics processing units (GPUs) using NVIDIA's Compute Unified Device Architecture. Both flexible and highly accurate, DG methods accommodate parallel architectures well as their discontinuous nature produces element‐local approximations. High‐performance scientific computing suits GPUs well, as these powerful, massively parallel, cost‐effective devices have recently included support for double‐precision floating‐point numbers. Computed examples for Euler equations over unstructured triangle meshes demonstrate the effectiveness of our implementation on an NVIDIA GTX 580 device. Profiling of our method reveals performance comparable with an existing nodal DG‐GPU implementation for linear problems. Copyright © 2014 John Wiley & Sons, Ltd.     

Stability of algebraic multigrid for Stokes problems

An investigation is made of the performance of algebraic multigrid (AMG) solvers for the discrete Stokes problem. The saddle‐point formulations are based on the direct enforcement of the fundamental conservation laws in discrete spaces and subsequently stabilised with the aid of a regular splitting of the diffusion operator. AMG solvers based on an independent coarsening of the fields (the unknown approach) and also on a common coarsening (the point approach) are investigated. Both mixed‐order and equal‐order interpolations are considered. The dependence of convergence on the ‘degree of coarsening’ is investigated by studying the ‘convergence versus coarsening’ characteristics and their variation with mesh resolution. They show a consistency in shape, which reveals two distinct performance zones, one convergent the other divergent. The transition from the convergent to the divergent zones is discontinuous and occurs at a critical coarsening factor that is largely mesh independent. It signals a breakdown in the stability of the smoothing at the coarser levels of coarse grid approximation. It is shown that the previously observed, mesh‐dependent, scaling of convergence factors, which had suggested inconsistencies in the coarse grid approximation, is not a reliable marker of inconsistency. It is an indirect consequence of the breakdown in the stability of smoothing. For stable smoothing, reduction factors are shown to be largely mesh independent. The ability of mixed‐order interpolation to permit stable smoothing and therefore to deliver mesh‐independent convergence is explained. Two expedient options are suggested for obtaining mesh‐independent convergence for those AMG codes that are based on an equal‐order interpolation. Copyright © 2012 John Wiley & Sons, Ltd.     

The discontinuous profile method for simulating two‐phase flow in pipes using the single component approximation

Godunov‐type algorithms are very attractive for the numerical solution of discontinuous flows. The reconstruction of the profile inside the cells is crucial to scheme performance. The non‐linear generalization of the discontinuous profile method (DPM) presented here for the modelling of two‐phase flow in pipes uses a discontinuous reconstruction in order to capture shocks more efficiently than schemes using continuous functions. The reconstructed profile is used to define the Riemann problem at cell interfaces by averaging of the components of the variable in the base of eigenvectors over their domain of dependence. Intercell fluxes are computed by solving the Riemann problem with an approximate‐state solver. The adapted treatment of boundary conditions is essential to ensure the quality of the computational results and a specific procedure using virtual cells at both extremities of the computational domain is required. Internal boundary conditions can be treated in the same way as external ones. Application of the DPM to test cases is shown to improve the quality of computational results significantly. Copyright © 2001 John Wiley & Sons, Ltd.     

Preconditioning techniques for iterative solvers in the Discrete Sources Method

Different preconditioning techniques for the iterative method MinRes as solver for the Discrete Sources Method (DSM) are presented. This semi-analytical method is used for light scattering computations by particles in the Mie scattering regime. Its numerical schema includes a linear least-squares problem commonly solved using the QR decomposition method. This could be the subject of numerical difficulties and instabilities for very large particles or particles with extreme geometry. In these cases, we showed that iterative methods with preconditioning techniques can provide a satisfying solution.In our previous paper, we studied four different iterative solvers (RGMRES, BiCGStab, BiCGStab(l), and MinRes) considering the performance and the accuracy of a solution. Here, we study several preconditioning techniques for the MinRes method for a variety of oblate and prolate spheroidal particles of different size and geometrical aspect ratio. Using preconditioning techniques we highly accelerated the iterative process especially for particles with a higher aspect ratio.     

Implementation and investigation of iterative solvers in the Discrete Sources Method

The implementation of iterative methods as solvers for the Discrete Sources Method (DSM) is presented. In this method, light scattering computation linear systems with dense and relative small matrices are generated. The linear systems are traditionally solved using the QR-decomposition method. For large particles or particles with extreme geometries even this commonly stable solver can fail. In these cases, we expect that iterative methods can provide a satisfying solution nevertheless.We will present our investigation in two consecutive papers. Here, we study four different iterative solvers (RGMRES, BiCGStab, BiCGStab(l), and MinRes) considering the performance and the accuracy for typical light scattering problems. Using these iterative methods we increased the quality of a solution, especially for oblate spheroids with a higher aspect ratio. Preconditioning technique is considered in the following paper.     

A hybrid FVM–LBM method for single and multi‐fluid compressible flow problems

The lattice Boltzmann method (LBM) has established itself as an alternative approach to solve the fluid flow equations. In this work we combine LBM with the conventional finite volume method (FVM), and propose a non‐iterative hybrid method for the simulation of compressible flows. LBM is used to calculate the inter‐cell face fluxes and FVM is used to calculate the node parameters. The hybrid method is benchmarked for several one‐dimensional and two‐dimensional test cases. The results obtained by the hybrid method show a steeper and more accurate shock profile as compared with the results obtained by the widely used Godunov scheme or by a representative flux vector splitting scheme. Additional features of the proposed scheme are that it can be implemented on a non‐uniform grid, study of multi‐fluid problems is possible, and it is easily extendable to multi‐dimensions. These features have been demonstrated in this work. The proposed method is therefore robust and can possibly be applied to a variety of compressible flow situations. Copyright © 2009 John Wiley & Sons, Ltd.     

L2Roe: a low dissipation version of Roe's approximate Riemann solver for low Mach numbers

A modification of the Roe scheme called L²Roe for low dissipation low Mach Roe is presented. It reduces the dissipation of kinetic energy at the highest resolved wave numbers in a low Mach number test case of decaying isotropic turbulence. This is achieved by scaling the jumps in all discrete velocity components within the numerical flux function. An asymptotic analysis is used to show the correct pressure scaling at low Mach numbers and to identify the reduced numerical dissipation in that regime. Furthermore, the analysis allows a comparison with two other schemes that employ different scaling of discrete velocity jumps, namely, LMRoe and a method of Thornber et al. To this end, we present for the first time an asymptotic analysis of the last method. Numerical tests on cases ranging from low Mach number (M_∞=0.001) to hypersonic (M_∞=5) viscous flows are used to illustrate the differences between the methods and to show the correct behavior of L²Roe. No conflict is observed between the reduced numerical dissipation and the accuracy or stability of the scheme in any of the investigated test cases. Copyright © 2015 John Wiley & Sons, Ltd.     

BiCGstab(l) and other hybrid Bi-CG methods

It is well-known that Bi-CG can be adapted so that the operations withA ^T can be avoided, and hybrid methods can be constructed in which it is attempted to further improve the convergence behaviour. Examples of this are CGS, Bi-CGSTAB, and the more general BiCGstab(l) method. In this paper it is shown that BiCGstab(l) can be implemented in different ways. Each of the suggested approaches has its own advantages and disadvantages. Our implementations allow for combinations of Bi-CG with arbitrary polynomial methods. The choice for a specific implementation can also be made for reasons of numerical stability. This aspect receives much attention. Various effects have been illustrated by numerical examples.