Algorithms and Data Structures for Multi-Adaptive Time-Stepping

Multi-adaptive Galerkin methods are extensions of the standard continuous and discontinuous Galerkin methods for the numerical solution of initial value problems for ordinary or partial differential equations. In particular, the multi-adaptive methods allow individual and adaptive time steps to be used for different components or in different regions of space. We present algorithms for efficient multi-adaptive time-stepping, including the recursive construction of time slabs and adaptive time step selection. We also present data structures for efficient storage and interpolation of the multi-adaptive solution. The efficiency of the proposed algorithms and data structures is demonstrated for a series of benchmark problems.Comment: ACM Transactions on Mathematical Software 35(3), 24 pages (2008

Adaptive Quantum Homodyne Tomography

An adaptive optimization technique to improve precision of quantum homodyne tomography is presented. The method is based on the existence of so-called null functions, which have zero average for arbitrary state of radiation. Addition of null functions to the tomographic kernels does not affect their mean values, but changes statistical errors, which can then be reduced by an optimization method that "adapts" kernels to homodyne data. Applications to tomography of the density matrix and other relevant field-observables are studied in detail.Comment: Latex (RevTex class + psfig), 9 Figs, Submitted to PR

A matrix-free high-order discontinuous Galerkin compressible Navier-Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows

Both compressible and incompressible Navier-Stokes solvers can be used and are used to solve incompressible turbulent flow problems. In the compressible case, the Mach number is then considered as a solver parameter that is set to a small value, $\mathrm{M}\approx 0.1$, in order to mimic incompressible flows. This strategy is widely used for high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. The present work raises the question regarding the computational efficiency of compressible DG solvers as compared to a genuinely incompressible formulation. Our contributions to the state-of-the-art are twofold: Firstly, we present a high-performance discontinuous Galerkin solver for the compressible Navier-Stokes equations based on a highly efficient matrix-free implementation that targets modern cache-based multicore architectures. The performance results presented in this work focus on the node-level performance and our results suggest that there is great potential for further performance improvements for current state-of-the-art discontinuous Galerkin implementations of the compressible Navier-Stokes equations. Secondly, this compressible Navier-Stokes solver is put into perspective by comparing it to an incompressible DG solver that uses the same matrix-free implementation. We discuss algorithmic differences between both solution strategies and present an in-depth numerical investigation of the performance. The considered benchmark test cases are the three-dimensional Taylor-Green vortex problem as a representative of transitional flows and the turbulent channel flow problem as a representative of wall-bounded turbulent flows