Jan Mohring

Hardware-Based Efficiency Advances in the EXA-DUNE Project

We present advances concerning efficient finite element assembly and linear solvers on current and upcoming HPC architectures obtained in the frame of the Exa-Dune project, part of the DFG priority program 1648 Software for Exascale Computing (SPPEXA). In this project, we aim at the development of both flexible and efficient hardware-aware software components for the solution of PDEs based on the DUNE platform and the FEAST library. In this contribution, we focus on node-level performance and accelerator integration, which will complement the proven MPI-level scalability of the framework. The higher-level aspects of the Exa-Dune project, in particular multiscale methods and uncertainty quantification, are detailed in the companion paper (Bastian et al., Advances concerning multiscale methods and uncertainty quantification in Exa-Dune. In: Proceedings of the SPPEXA Symposium, 2016).

Uncertainty Quantification for Porous Media Flow Using Multilevel Monte Carlo

Uncertainty quantification (UQ) for porous media flow is of great importance for many societal, environmental and industrial problems. An obstacle for progress in this area is the extreme computational effort needed for solving realistic problems. It is expected that exa-scale computers will open the door for a significant progress in this area. We demonstrate how new features of the Distributed and Unified Numerics Environment DUNE [1] address these challenges. In the frame of the DFG funded project EXA-DUNE the software has been extended by multiscale finite element methods (MsFEM) and by a parallel framework for the multilevel Monte Carlo (MLMC) approach. This is a general concept for computing expected values of simulation results depending on random fields, e.g. the permeability of porous media. It belongs to the class of variance reduction methods and overcomes the slow convergence of classical Monte Carlo by combining cheap/inexact and expensive/accurate solutions in an optimal ratio.

Interpolation Strategy for BT-Based Parametric MOR of Gas Pipeline-Networks

Proceeding from balanced truncation-based parametric reduced order models (BT-pROM) a matrix interpolation strategy is presented that allows the cheap evaluation of reduced order models at new parameter sets. The method extends the framework of model order reduction (MOR) for high-order parameter-dependent linear time invariant systems in descriptor form by Geuss (2013) by treating not only permutations and rotations but also distortions of reduced order basis vectors. The applicability of the interpolation strategy and different variants is shown on BT-pROMs for gas transport in pipeline-networks.

Advances Concerning Multiscale Methods and Uncertainty Quantification in EXA-DUNE

In this contribution we present advances concerning efficient parallel multiscale methods and uncertainty quantification that have been obtained in the frame of the DFG priority program 1648 Software for Exascale Computing (SPPEXA) within the funded project Exa-Dune. This project aims at the development of flexible but nevertheless hardware-specific software components and scalable high-level algorithms for the solution of partial differential equations based on the DUNE platform. While the development of hardware-based concepts and software components is detailed in the companion paper (Bastian et al., Hardware-based efficiency advances in the Exa-Dune project. In: Proceedings of the SPPEXA Symposium 2016, Munich, 25–27 Jan 2016), we focus here on the development of scalable multiscale methods in the context of uncertainty quantification. Such problems add additional layers of coarse grained parallelism, as the underlying problems require the solution of many local or global partial differential equations in parallel that are only weakly coupled.

Renormalization Based MLMC Method for Scalar Elliptic SPDE

Previously the authors have presented MLMC algorithms exploiting Multiscale Finite Elements and Reduced Bases as a basis for the coarser levels in the MLMC algorithm. In this paper a Renormalization based Multilevel Monte Carlo algorithm is discussed. The advantage of the renormalization as a basis for the coarse levels in MLMC is that it allows in a cheap way to create a reduced dimensional space with a variation which is very close to the variation at the finest level. This leads to especially efficient MLMC algorithms. Parallelization of the proposed algorithm is also considered and results from numerical experiments are presented.

Stability-Preserving Interpolation Strategy for Parametric MOR of Gas Pipeline-Networks

Optimization and control of large transient gas networks require the fast simulation of the underlying parametric partial differential algebraic systems. Surrogate modeling techniques based on linearization around specific stationary states, spatial semi-discretization and model order reduction allow for the set-up of parametric reduced order models that can act as basis sample to cover a wide parameter range by means of matrix interpolations. However, the interpolated models are often not stable. In this paper, we develop a stability-preserving interpolation method.

MOR via quadratic-linear representation of nonlinear-parametric PDEs

This work deals with the model order reduction (MOR) of a nonlinear-parametric system of partial differential equations (PDEs). Applying a semidiscretization in space and replacing the nonlinearities by introducing new state variables, we set up quadratic-linear differential algebraic systems (QLDAE) and use a Krylov-subspace MOR. The approach is investigated for gas pipeline modeling.