A micro/macro parallel-in-time (parareal) algorithm applied to a climate model with discontinuous non-monotone coefficients and oscillatory forcing

We present the application of a micro/macro parareal algorithm for a 1-D energy balance climate model with discontinuous and non-monotone coefficients and forcing terms. The micro/macro parareal method uses a coarse propagator, based on a (macroscopic) 0-D approximation of the underlying (microscopic) 1-D model. We compare the performance of the method using different versions of the macro model, as well as different numerical schemes for the micro propagator, namely an explicit Euler method with constant stepsize and an adaptive library routine. We study convergence of the method and the theoretical gain in computational time in a realization on parallel processors. We show that, in this example and for all settings, the micro/macro parareal method converges in fewer iterations than the number of used parareal subintervals, and that a theoretical gain in performance of up to 10 is possible

Temporal coarsening of transport matrices with Metos3D

Transport matrices represent pre-computed ocean circulation that can be used to efficiently approximate the transport of passive tracers in ocean waters. Metos3D (https://github.com/metos3d) is a software toolkit used for the computation of periodic steady states of biogeochemical and marine ecosystem models. Metos3D makes use of transport matrices. One major disadvantage of transport matrices is the fact that the used time step is hardcoded into each matrix. However, there is a possibility to alter the time step afterwards to some extent. We built in such a conversion technique into the 'metos3d' script using SciPy (https://www.scipy.org/). This technical report explains the details of the 'matrix' subcommand and its usage

Single-precision arithmetic in ECHAM radiation reduces runtime and energy consumption

We converted the radiation part of the atmospheric model ECHAM to a single-precision arithmetic. We analyzed different conversion strategies and finally used a step-by-step change in all modules, subroutines and functions. We found out that a small code portion still requires higher-precision arithmetic. We generated code that can be easily changed from double to single precision and vice versa, basically using a simple switch in one module. We compared the output of the single-precision version in the coarse resolution with observational data and with the original double-precision code. The results of both versions are comparable. We extensively tested different parallelization options with respect to the possible runtime reduction, at both coarse and low resolution. The single-precision radiation itself was accelerated by about 40 %, whereas the runtime reduction for the whole ECHAM model using the converted radiation achieved 18 % in the best configuration. We further measured the energy consumption, which could also be reduced

The Idea and Concept of Metos3D: A Marine Ecosystem Toolkit for Optimization and Simulation in 3-D

The simulation and parameter optimization of coupled ocean circulation and ecosystem models in three space dimensions is one of the most challenging tasks in numerical climate research. Here we present a scientific toolkit that aims at supporting researchers by defining clear coupling interfaces, providing state-of-the-art numerical methods for simulation, parallelization and optimization while using only freely available and (to a great extend) platform-independent software. Besides defining a user-friendly coupling interface (API) for marine ecosystem or biogeochemical models, we heavily rely on the Portable, Extensible Toolkit for Scientific computation [PETSc] developed in Argonne Nat. Lab. [PETSc] for a wide variety of parallel linear and non-linear solvers and optimizers. We specifically focus on the usage of matrix-free Newton-Krylov methods for the fast computation of steady periodic solutions, and make use of the Transport Matrix Method (TMM) introduced by Khatiwala et al. in [KhViCa05]

Aggressive Space Mapping for the Optimization of a Marine Ecosystem Model

In this paper we apply the Aggressive Space Mapping (ASM) algorithm by Bandler et. al. to the parameter optimization of a one-dimensional marine ecosystem model of NPZD type. We show that this approach leads to a very satisfactory solution while yielding a significant reduction in the total optimization cost. The ecosystem model, developed by Oschlies and Garcon, simulates the distribution of nitrogen, phytoplankton, zooplankton and detritus in a water column and is driven by ocean circulation data. A key issue is to optimize model parameters in order to minimize the misfit between the model output and given observational data. In the ASM approach, reducing the overall optimization cost by avoiding expensive function and derivative evaluations is achieved by using a surrogate model that replaces the original one. Furthermore the ASM algorithm solves a nonlinear system of equations which is conditionally equivalent to use this surrogate in the optimization run. We use a coarser time discretization for obtaining a suitable low-fidelity model. This is then corrected to create a physically-based surrogate, where the correction is obtained through a parameter mapping which provides the minimizer of the distance between the fine and the coarse model output. We show that this surrogate provides a good approximation of the fine model. The applicability of the ASM technique to the problem at hand is verified by using synthetic target data. Results are compared to those of the direct fine model optimization. We show that a very reasonable fit of the target data can be obtained with an average reduction in the computational cost of about 65 %

A Fictitious Domain Method for the Numerical Solution of the Stationary Navier-Stokes Equations

An embedding domain method for the stationary incompressible Navier-Stokes equations is presented. The method is useful for solving the equations or complicated-shaped or varying domains. The original domain is embedded in a so-called fictitious one. On the latter an equivalent formulation of the Navier-Stokes equation is derived. Existence and uniqueness results of its solution(s) are presented. The structure of and solution methods for the discrete systems and numerical issues are discussed. An algorithm for finding the trace of the original boundary in the fictitious domain are given. Numerical results for the fictitious domain method are compared with those from computations on the original domain. As test case the 2-D flow around a circular cylinder in a channel is used