Vincent Heuveline

A posteriori error control for finite element approximations of elliptic eigenvalue problems

We develop a new approach to a posteriori error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for nonlinear variational equations. In this context residual-based a posteriori error representations are available with explicitly given remainder terms. The careful evaluation of these error representations for the concrete situation of an eigenvalue problem results in a posteriori error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local error indicators that are used in the mesh refinement process.

Duality-based adaptivity in the hp -finite element method

In this paper a duality-based a posteriori error analysis is developed for the conforming hp Galerkin finite element approximation of second-order elliptic problems. Duality arguments combined with Galerkin orthogonality yield representations of the error in arbitrary quantities of interest. From these error estimates, criteria are derived for the simultaneous adaptation of the mesh size h and the polynomial degree p. The effectivity of this procedure is confirmed by numerical tests.

Framework for Modular, Flexible and Efficient Solving the Cardiac Bidomain Equations Using PETSc

In this work, a new framework is presented that is suitable to solve the cardiac bidomain equation efficiently using the scientific computing library PETSc. Furthermore, the framework is able to modularly combine different ionic channels and is flexible enough to include arbitrary heterogeneities in ionic or coupling channel density. The ability of this framework is demonstrated in an example simulation in which the three-dimensional electrophysiological heterogeneity was adjusted in order to get a positive T-wave in the body electrocardiogram (ECG).

Towards a hybrid parallelization of lattice Boltzmann methods

Ongoing research towards the development of a hybrid parallelization concept for lattice Boltzmann methods is presented. It allows coping with platforms sharing both the properties of shared and distributed architectures. The proposed approach relies on spatial domain decomposition where each domain represents a basic block entity which is solved on a symmetric multi-processing (SMP) system. Emphasis is placed on the software design and the reworking needed to achieve good performance using OpenMP in that context. Those ideas are implemented in the C++ project OpenLB, which is also sketched in this article. The efficiency of the proposed approaches is tested on a 3D benchmark problem and compared with a purely MPI based approach.

Adjoint-based fluid flow control and optimisation with lattice Boltzmann methods

A lattice Boltzmann (LB) framework to solve fluid flow control and optimisation problems numerically is presented. Problems are formulated on a mesoscopic basis. In a side condition, the dynamics of a Newtonian fluid is described by a family of simplified Boltzmann-like equations, namely BGK-Boltzmann equations, which are linked to an incompressible Navier-Stokes equation. It is proposed to solve the non-linear optimisation problem by a line search algorithm. The needed derivatives are obtained by deriving the adjoint equations, referred to as adjoint BGK-Boltzmann equations. The primal equations are discretised by standard lattice Boltzmann methods (LBM) while for the adjoint equations a novel discretisation strategy is introduced. The approach follows the main ideas behind LBM and is therefore referred to as adjoint lattice Boltzmann methods (ALBM). The corresponding algorithm retains most of the basic features of LB algorithms. In particular, it enables a highly-efficient parallel implementation and thus solving large-scale fluid flow control and optimisation problems. The overall solution strategy, the derivation of a prototype adjoint BGK-Boltzmann equation, the novel ALBM and its parallel realisation as well as its validation are discussed in detail in this article. Numerical and performance results are presented for a series of steady-state distributed control problems with up to approximately 1.6 million unknown control parameters obtained on a high performance computer with up to 256 processing units.

Optimized Stencil Computation Using In-Place Calculation on Modern Multicore Systems

Numerical algorithms on parallel systems built upon modern multicore processors are facing two challenging obstacles that keep realistic applications from reaching the theoretically available compute performance. First, the parallelization on several system levels has to be exploited to the full extent. Second, provision of data to the compute cores needs to be adapted to the constraints of a hardware-controlled nested cache hierarchy with shared resources. In this paper we analyze dedicated optimization techniques on modern multicore systems for stencil kernels on regular three-dimensional grids. We combine various methods like a compressed grid algorithm with finite shifts in each time step and loop skewing into an optimized parallel in-place stencil implementation of the three-dimensional Laplacian operator. In that context, memory requirements are reduced by a factor of approximately two while considerable performance gains are observed on modern Intel and AMD based multicore systems.

Adjoint-based estimation and optimization for column liquid chromatography models

Abstract Simulation and optimization of chromatographic processes are continuously gaining practical importance, as they allow for faster and cheaper process development. Although a lot of effort has been put into developing numerical schemes for simulation, fast optimization and estimation algorithms also are of importance. To determine parameters for an a priori defined model, a suited approach is the inverse method that fits the measurement data to the model response. This paper presents an adjoint method to compute model parameter derivatives for a wide range of differentiable liquid chromatography models and provides practical information for the implementation in a generic simulation framework by the example of ion-exchange chromatography. The example shows that the approach is effective for parameter estimation of model proteins and superior to forward sensitivities in terms of computational effort. An optimization of peak separation in salt step elution demonstrates that the method is not restricted to inverse parameter estimation.

A survey on hardware-aware and heterogeneous computing on multicore processors and accelerators

In the last few years, the landscape of parallel computing has been subject to profound and highly dynamic changes. The paradigm shift towards multicore and manycore technologies coupled with accelerators in a heterogeneous environment is offering a great potential of computing power for scientific and industrial applications. However, for one to take full advantage of these new technologies, holistic approaches coupling the expertise ranging from hardware architecture and software design to numerical algorithms are a pressing necessity. Parallel computing is no longer limited to supercomputers and is now much more diversified – with a multitude of technologies, architectures, and programming approaches leading to increased complexity for developers and engineers.

Analysis and optimization of power consumption in the iterative solution of sparse linear systems on multi-core and many-core platforms

Energy efficiency is a major concern in modern high-performance-computing. Still, few studies provide a deep insight into the power consumption of scientific applications. Especially for algorithms running on hybrid platforms equipped with hardware accelerators, like graphics processors, a detailed energy analysis is essential to identify the most costly parts, and to evaluate possible improvement strategies. In this paper we analyze the computational and power performance of iterative linear solvers applied to sparse systems arising in several scientific applications. We also study the gains yield by dynamic voltage/frequency scaling (DVFS), and illustrate that this technique alone cannot to reduce the energy cost to a considerable amount for iterative linear solvers. We then apply techniques that set the (multi-core processor in the) host system to a low-consuming state for the time that the GPU is executing. Our experiments conclusively reveal how the combination of these two techniques deliver a notable reduction of energy consumption without a noticeable impact on computational performance.

Energy efficiency of mixed precision iterative refinement methods using hybrid hardware platforms

In this paper we evaluate the possibility of using mixed precision algorithms on different hardware platforms to obtain energy-efficient solvers for linear systems of equations. Our test-cases arise in the context of computational fluid dynamics.Therefore, we analyze the energy efficiency of common cluster nodes and a hybrid, GPU-accelerated cluster node, when applying a linear solver, that can benefit from the use of different precision formats.We show the high potential of hardware-aware computing in terms of performance and energy efficiency.