Multilevel convergence analysis of multigrid-reduction-in-time

This paper presents a multilevel convergence framework for multigrid-reduction-in-time (MGRIT) as a generalization of previous two-grid estimates. The framework provides a priori upper bounds on the convergence of MGRIT V- and F-cycles, with different relaxation schemes, by deriving the respective residual and error propagation operators. The residual and error operators are functions of the time stepping operator, analyzed directly and bounded in norm, both numerically and analytically. We present various upper bounds of different computational cost and varying sharpness. These upper bounds are complemented by proposing analytic formulae for the approximate convergence factor of V-cycle algorithms that take the number of fine grid time points, the temporal coarsening factors, and the eigenvalues of the time stepping operator as parameters. The paper concludes with supporting numerical investigations of parabolic (anisotropic diffusion) and hyperbolic (wave equation) model problems. We assess the sharpness of the bounds and the quality of the approximate convergence factors. Observations from these numerical investigations demonstrate the value of the proposed multilevel convergence framework for estimating MGRIT convergence a priori and for the design of a convergent algorithm. We further highlight that observations in the literature are captured by the theory, including that two-level Parareal and multilevel MGRIT with F-relaxation do not yield scalable algorithms and the benefit of a stronger relaxation scheme. An important observation is that with increasing numbers of levels MGRIT convergence deteriorates for the hyperbolic model problem, while constant convergence factors can be achieved for the diffusion equation. The theory also indicates that L-stable Runge-Kutta schemes are more amendable to multilevel parallel-in-time integration with MGRIT than A-stable Runge-Kutta schemes.Comment: 26 pages; 17 pages Supplementary Material

Parallel-in-Time Multi-Level Integration of the Shallow-Water Equations on the Rotating Sphere

The modeling of atmospheric processes in the context of weather and climate simulations is an important and computationally expensive challenge. The temporal integration of the underlying PDEs requires a very large number of time steps, even when the terms accounting for the propagation of fast atmospheric waves are treated implicitly. Therefore, the use of parallel-in-time integration schemes to reduce the time-to-solution is of increasing interest, particularly in the numerical weather forecasting field. We present a multi-level parallel-in-time integration method combining the Parallel Full Approximation Scheme in Space and Time (PFASST) with a spatial discretization based on Spherical Harmonics (SH). The iterative algorithm computes multiple time steps concurrently by interweaving parallel high-order fine corrections and serial corrections performed on a coarsened problem. To do that, we design a methodology relying on the spectral basis of the SH to coarsen and interpolate the problem in space. The methods are evaluated on the shallow-water equations on the sphere using a set of tests commonly used in the atmospheric flow community. We assess the convergence of PFASST-SH upon refinement in time. We also investigate the impact of the coarsening strategy on the accuracy of the scheme, and specifically on its ability to capture the high-frequency modes accumulating in the solution. Finally, we study the computational cost of PFASST-SH to demonstrate that our scheme resolves the main features of the solution multiple times faster than the serial schemes

Improved Modeling Of Turbulent Transport: From Noise In Transport Models To The Parareal Algorithm Applied To Full Turbulence Codes

Thesis (Ph.D.) University of Alaska Fairbanks, 2010Turbulence and turbulent transport are ubiquitous in nature and are of fundamental importance in everything from the spread of pollution to confinement in fusion plasmas. In order to study this, turbulence models need to be as realistic as possible and one must also be able to evolve the turbulence and the profiles of the quantities of interest on transport (long) time scales. Improving turbulence simulations by the introduction of new techniques forms the basis of this research. One part of this work involved improving the performance of a 1D transport model by the addition of noise. On a more fundamental level, studying long time dynamics for turbulence simulations is very difficult even with the fastest computers available now or in the near future. To help overcome this difficulty, a new way of simulating turbulence has been presented, namely parallelizing in time. Time parallelization of a fully developed turbulent system is a new application. Parallelizing the space domain to computationally solve partial differential equations has been extensively used and is one of the most common forms of parallelization. In contrast, the Parareal Algorithm parallelizes the time domain and has been found to significantly reduce the computational wall time in many simpler systems. Despite its success in other less complex problems, it has not yet been successfully applied to a turbulent system (to the best of our knowledge). If efficiently applied, this algorithm will allow study of the turbulent transport dynamics on transport time scales - something that has heretofore been very difficult. In this work, the results of applying the Parareal Algorithm to simulations of drift wave turbulence in slab geometry in which the relative dominance of the polarization and E x B nonlinearities are tuned artificially, are presented. These turbulent systems are in many ways similar to neutral fluid turbulence models, so success of the Parareal scheme in them expands the prospect of a broader range of application to many other turbulent problems. This thesis also presents the results of a modification to the algorithm. A model to study and predict the parameters governing the convergence of the scheme is also explored

Numerical wave propagation aided by deep learning

We propose a deep learning approach for wave propagation in media with multiscale wave speed, using a second-order linear wave equation model. We use neural networks to enhance the accuracy of a given inaccurate coarse solver, which under-resolves a class of multiscale wave media and wave fields of interest. Our approach involves generating training data by the given computationally efficient coarse solver and another sufficiently accurate solver, applied to a class of wave media (described by their wave speed profiles) and initial wave fields. We find that the trained neural networks can approximate the nonlinear dependence of the propagation on the wave speed as long as the causality is appropriately sampled in training data. We combine the neural-network-enhanced coarse solver with the parareal algorithm and demonstrate that the coupled approach improves the stability of parareal algorithms for wave propagation and improves the accuracy of the enhanced coarse solvers