Stefan Langer

Convergence analysis of an inexact iteratively regularized Gauss-Newton method under general source conditions

In this paper we improve existing convergence and convergence rate results for the iteratively regularized Gauss-Newton method in two respects: First we show optimal rates of convergence under general source conditions, and second we assume that the linearized equations are solved only approximately in each Newton step. The latter point is important for large scale problems where the linearized equation can often only be solved iteratively, e.g. by the conjugate gradient method.

The DLR Flow Solver TAU - Status and Recent Algorithmic Developments

The only implicit smoothing method implemented in the DLR Flow Solver TAU is the LU-SGS method. It was chosen several years ago because of its low memory requirements and low operation counts. Since in the past for many examples a severe restriction of the CFL number and loss of robustness was observed, it is the goal of this paper to revisit the LU-SGS implementation and to discuss several alternative implicit smoothing strategies used within an agglomeration multigrid for unstructured meshes. Starting point is a full implicit multistage Runge-Kutta method. Based on this method we develop and suggest several additional features and simplifications such that the implicit method is applicable to high Reynolds number viscous flows, that is the required matrices fit into the fast memory of our cluster hardware and the arising linear systems can be approximately solved efficiently. To this end we focus on simplifications of the Jacobian as well as efficient iterative approximate solution methods. To significantly improve the approximate linear solution methods we take care of grid anisotropy for both approximately solving the linear systems and agglomeration strategy. The procedure creating coarse grid meshes is extended by strategies identifying structured parts of the mesh. This seems to improve the quality of coarse grid meshes in the way that an overall better reliability of multigrid can be observed. Furthermore we exploit grid information within the iterative solution methods for the linear systems. Numerical examples demonstrate the gain with respect to reliability and efficiency.

Acceleration techniques for regularized Newton methods applied to electromagnetic inverse medium scattering problems

We study the construction and updating of spectral preconditioners for regularized Newton methods and their application to electromagnetic inverse medium scattering problems. Moreover, we show how a Lepski-type stopping rule can be implemented efficiently for these methods. In numerical examples, the proposed method compares favorably with other iterative regularization method in terms of work-precision diagrams for exact data. For data perturbed by random noise, the Lepski-type stopping rule performs considerably better than the commonly used discrepancy principle.

Agglomeration multigrid methods with implicit Runge-Kutta smoothers applied to aerodynamic simulations on unstructured grids

For unstructured finite volume methods an agglomeration multigrid with an implicit multistage Runge-Kutta method as a smoother is developed for solving the compressible Reynolds averaged Navier-Stokes (RANS) equations. The implicit Runge-Kutta method is interpreted as a preconditioned explicit Runge-Kutta method. The construction of the preconditioner is based on an approximate derivative. The linear systems are solved approximately with a symmetric Gauss-Seidel method. To significantly improve this solution method grid anisotropy is treated within the Gauss-Seidel iteration in such a way that the strong couplings in the linear system are resolved by tridiagonal systems constructed along these directions of strong coupling. The agglomeration strategy is adapted to this procedure by taking into account exactly these anisotropies in such a way that a directional coarsening is applied along these directions of strong coupling. Turbulence effects are included by a Spalart-Allmaras model, and the additional transport-type equation is approximately solved in a loosely coupled manner with the same method. For two-dimensional and three-dimensional numerical examples and a variety of differently generated meshes we show the wide range of applicability of the solution method. Finally, we exploit the GMRES method to determine approximate spectral information of the linearized RANS equations. This approximate spectral information is used to discuss and compare characteristics of multistage Runge-Kutta methods.

Application of a line implicit method to fully coupled system of equations for turbulent flow problems

For unstructured finite volume methods, we present a line implicit Runge–Kutta method applied as smoother in an agglomerated multigrid algorithm to significantly improve the reliability and convergence rate to approximate steady-state solutions of the Reynolds-averaged Navier–Stokes equations. To describe turbulence, we consider a one-equation Spalart–Allmaras turbulence model. The line implicit Runge–Kutta method extends a basic explicit Runge–Kutta method by a preconditioner given by an approximate derivative of the residual function. The approximate derivative is only constructed along predetermined lines which resolve anisotropies in the given grid. Therefore, the method is a canonical generalisation of point implicit methods. Numerical examples demonstrate the improvements of the line implicit Runge–Kutta when compared with explicit Runge–Kutta methods accelerated with local time stepping.

Application of point implicit Runge-Kutta methods to inviscid and laminar flow problems using AUSM and AUSM+ upwinding

Explicit Runge–Kutta methods preconditioned by a pointwise matrix valued preconditioner can significantly improve the convergence rate to approximate steady state solutions of laminar flows. This has been shown for central discretisation schemes and Roe upwinding. Since the first-order approximation to the inviscid flux assuming constant weighting of the dissipative terms is given by the absolute value of the Roe matrix, the construction of the preconditioner is rather simple compared to other upwind techniques. However, in this article we show that similar improvements in the convergence rates can also be obtained for the AUSM+ scheme. Following the ideas for the central and Roe schemes, the preconditioner is obtained by a first-order approximation to the derivative of the convective flux. Viscous terms are included into the preconditioner considering a thin shear layer approximation. A complete derivation of the derivative terms is shown. In numerical examples, we demonstrate the improved convergence rates when compared with a standard explicit Runge–Kutta method accelerated with local time stepping.

Investigation and Comparison of Implicit Smoothers Applied in Agglomeration Multigrid

A straightforward implicit smoothing method implemented in several codes solving the Reynolds-averaged Navier–Stokes equations is the lower–upper symmetric Gauss–Seidel method. It was proposed several years ago and is attractive to implement because of its low memory requirements and low operation count. Since, for many examples, often a severe restriction of the Courant–Friedrichs–Lewy number and loss of robustness are observed, it is the goal of this paper to revisit the lower–upper symmetric Gauss–Seidel implementation and to discuss several alternative implicit smoothing strategies used within an agglomeration multigrid for unstructured meshes. The starting point is a full implicit multistage Runge–Kutta method. Based on this method, several additional features and simplifications are developed and suggested, such that the implicit method is applicable to high-Reynolds-number viscous flows; that is, the required matrices fit into the fast memory of the cluster hardware and the arising linear systems c...

Point and Line Implicit Methods to Improve the Efficiency and Robustness of the DLR Tau Code

We present a line implicit preconditioned multistage Runge-Kutta method to significantly improve the convergence rate for approximating steady state solutions of high Reynolds number viscous flows. The Runge-Kutta method is used as a smoother in the context of an agglomerated multigrid method for unstructured grids. A simplification of a first order approximation to the Jacobian of the residual function is used as a preconditioner. Predetermined lines identifying mesh regions of high cell stretching are exploited to extract the relevant parts of the Jacobian matrix. The lines are identified using an efficient algorithm based on a weighted graph. This has the advantage that high aspect ratio cells are determined everywhere in the mesh, for example also in the wake of a wing.