Characteristic Evolution and Matching

I review the development of numerical evolution codes for general relativity based upon the characteristic initial value problem. Progress in characteristic evolution is traced from the early stage of 1D feasibility studies to 2D axisymmetric codes that accurately simulate the oscillations and gravitational collapse of relativistic stars and to current 3D codes that provide pieces of a binary black hole spacetime. Cauchy codes have now been successful at simulating all aspects of the binary black hole problem inside an artificially constructed outer boundary. A prime application of characteristic evolution is to extend such simulations to null infinity where the waveform from the binary inspiral and merger can be unambiguously computed. This has now been accomplished by Cauchy-characteristic extraction, where data for the characteristic evolution is supplied by Cauchy data on an extraction worldtube inside the artificial outer boundary. The ultimate application of characteristic evolution is to eliminate the role of this outer boundary by constructing a global solution via Cauchy-characteristic matching. Progress in this direction is discussed.Comment: New version to appear in Living Reviews 2012. arXiv admin note: updated version of arXiv:gr-qc/050809

Distributed stochastic optimization via matrix exponential learning

In this paper, we investigate a distributed learning scheme for a broad class of stochastic optimization problems and games that arise in signal processing and wireless communications. The proposed algorithm relies on the method of matrix exponential learning (MXL) and only requires locally computable gradient observations that are possibly imperfect and/or obsolete. To analyze it, we introduce the notion of a stable Nash equilibrium and we show that the algorithm is globally convergent to such equilibria - or locally convergent when an equilibrium is only locally stable. We also derive an explicit linear bound for the algorithm's convergence speed, which remains valid under measurement errors and uncertainty of arbitrarily high variance. To validate our theoretical analysis, we test the algorithm in realistic multi-carrier/multiple-antenna wireless scenarios where several users seek to maximize their energy efficiency. Our results show that learning allows users to attain a net increase between 100% and 500% in energy efficiency, even under very high uncertainty.Comment: 31 pages, 3 figure

A Hybrid Scheme for Gas-Dust Systems Stiffly Coupled via Viscous Drag

We present a stable and convergent method for studying a system of gas and dust, coupled through viscous drag in both non-stiff and stiff regimes. To account for the effects of dust drag in the update of the fluid quantities, we employ a fluid description of the dust component and study the modified gas-dust hyperbolic system following the approach in Miniati & Colella (2007). In addition to two entropy waves for the gas and dust components, respectively, the extended system includes three waves driven partially by gas pressure and partially by dust drift, which, in the limit of vanishing coupling, tend to the two original acoustic waves and the unhindered dust streaming. Based on this analysis we formulate a predictor step providing first order accurate reconstruction of the time-averaged state variables at cell interfaces, whence a second order accurate estimate of the conservative fluxes can be obtained through a suitable linearized Riemann solver. The final source term update is carried out using a one-step, second order accurate, L-stable, predictor corrector asymptotic method (the alpha-QSS method suggested by Mott et. al. 2000). This procedure completely defines a two-fluid method for gas-dust system. Using the updated fluid solution allows us to then advance the individual particle solutions, including self-consistently the time evolution of the gas velocity in the estimate of the drag force. This is done with a suitable particle scheme also based on the alpha-QSS method. A set of benchmark problems shows that our method is stable and convergent. When dust is modeled as a fluid (two-fluid) second order accuracy is achieved in both stiff and non-stiff regimes, whereas when dust is modeled with particles (hybrid) second order is achieved in the non-stiff regime and first order otherwise.Comment: 41 pages, 3 figures, 14 tables, accepted to J. Comp. Phys

A Moving Frame Algorithm for High Mach Number Hydrodynamics

We present a new approach to Eulerian computational fluid dynamics that is designed to work at high Mach numbers encountered in astrophysical hydrodynamic simulations. The Eulerian fluid conservation equations are solved in an adaptive frame moving with the fluid where Mach numbers are minimized. The moving frame approach uses a velocity decomposition technique to define local kinetic variables while storing the bulk kinetic components in a smoothed background velocity field that is associated with the grid velocity. Gravitationally induced accelerations are added to the grid, thereby minimizing the spurious heating problem encountered in cold gas flows. Separately tracking local and bulk flow components allows thermodynamic variables to be accurately calculated in both subsonic and supersonic regions. A main feature of the algorithm, that is not possible in previous Eulerian implementations, is the ability to resolve shocks and prevent spurious heating where both the preshock and postshock Mach numbers are high. The hybrid algorithm combines the high resolution shock capturing ability of the second-order accurate Eulerian TVD scheme with a low-diffusion Lagrangian advection scheme. We have implemented a cosmological code where the hydrodynamic evolution of the baryons is captured using the moving frame algorithm while the gravitational evolution of the collisionless dark matter is tracked using a particle-mesh N-body algorithm. The MACH code is highly suited for simulating the evolution of the IGM where accurate thermodynamic evolution is needed for studies of the Lyman alpha forest, the Sunyaev-Zeldovich effect, and the X-ray background. Hydrodynamic and cosmological tests are described and results presented. The current code is fast, memory-friendly, and parallelized for shared-memory machines.Comment: 19 pages, 5 figure