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We consider a hyperbolic quasilinear version of the Navier–Stokes equations in $${\mathbb {R}}^2$$ , arising from using a Cattaneo type law instead of a Fourier law. These equations, which depend on a parameter $$\varepsilon $$ , are a way to avoid the infinite speed of propagation which occurs in the classical Navier–Stokes equations. We first prove the existence and uniqueness of solutions to these equations, and then exhibit smallness assumptions on the data, under which the solutions are global in time. In particular, these smallness assumptions disappear when $$\varepsilon $$ vanishes, accordingly to the fact that the solutions of the 2D Navier–Stokes equations are global.

High performance fast multipole method is crucial for the numerical simulation of many physical problems. In a previous study, we have shown that task-based fast multipole method provides the flexibility required to process a wide spectrum of particle distributions efficiently on multicore architectures. In this paper, we now show how such an approach can be extended to fully exploit heterogeneous platforms. For that, we design highly tuned graphics processing unit GPU versions of the two dominant operators P2P and M2L as well as a scheduling strategy that dynamically decides which proportion of subsequent tasks is processed on regular CPU cores and on GPU accelerators. We assess our method with the StarPU runtime system for executing the resulting task flow on an Intel X5650 Nehalem multicore processor possibly enhanced with one, two, or three Nvidia Fermi M2070 or M2090 GPUs Santa Clara, CA, USA. A detailed experimental study on two 30 million particle distributions a cube and an ellipsoid shows that the resulting software consistently achieves high performance across architectures. Copyright © 2015 John Wiley & Sons, Ltd.

We parallelize the TD-BEM on clusters of multicore nodes enhanced with multiple GPU.SpMV is replaced by a more efficient operator by reordering the computation.GPU kernels achieve high Flop-rate using blocking schemes.Idle time is drastically reduced by using a greedy balancing algorithm. The problem of time-domain BEM for the wave equation in acoustics and electromagnetism can be expressed as a sparse linear system composed of multiple interaction/convolution matrices. It can be solved by using sparse matrix-vector products which are inefficient to achieve high Flop-rate neither on CPUs nor GPUs. In this paper we extend the approach proposed in a previous work 1 in which we re-order the computation to get a special matrix structure with one dense vector per row. This new structure is called a slice matrix and is computed with a custom matrix/vector product operator. In this study, we present an optimized implementation of this operator on Nvidia GPUs based on two blocking strategies. We explain how we can obtain multiple block-values from a slice and how these can be computed efficiently on GPUs since we target heterogeneous nodes composed of CPUs and GPUs. In order to deal with different efficiencies of the processing units we use a greedy heuristic that dynamically balances work among the workers. We demonstrate the performance of our system by studying the quality of the balancing heuristic and the sequential Flop-rate of the blocked implementations. Finally, we validate our implementation with an industrial test case on 8 heterogeneous nodes, each composed of 12 CPUs and 3 GPUs.

International audience; With the advent of complex modern architectures, the low-level paradigms long considered sufficient to build High Performance Computing (HPC) numerical codes have met their limits. Achieving efficiency, ensuring portability, while preserving programming tractability on such hardware prompted the HPC community to design new, higher level paradigms while relying on runtime systems to maintain performance. However, the common weakness of these projects is to deeply tie applications to specific expert-only runtime system APIs. The OpenMP specification, which aims at providing common parallel programming means for shared-memory platforms, appears as a good candidate to address this issue thanks to the latest task-based constructs introduced in its revision 4.0. The goal of this paper is to assess the effectiveness and limits of this support for designing a high-performance numerical library, ScalFMM, implementing the fast multipole method (FMM) that we have deeply redesigned with respect to the most advanced features provided by OpenMP 4. We show that OpenMP 4 allows for significant performance improvements over previous OpenMP revisions on recent multicore processors and that extensions to the 4.0 standard allow for strongly improving the performance, bridging the gap with the very high performance that was so far reserved to expert-only runtime system APIs.

We designed a fast parallel simulator that solves the acoustic wave equation on a graphics processing unit (GPU) cluster. Solving the acoustic wave equation in an oil exploration industrial context aims at speeding up seismic modeling and reverse time migration (RTM). We considered a finite difference approach on a regular mesh, in both two-dimensional and three-dimensional cases. The acoustic wave equation is solved in a constant density or a variable density domain. All the computations were carried out in single precision (both in the CPU reference implementation and in the GPU implementation), because double precision was not required in our context. We used Compute Unified Device Architecture to take advantage of the GPU computational power. We studied different implementations and their impact on the application performance. The described application handles all the steps of seismic modeling and RTM and is used to solve real-world problems in an industrial production context. We obtained a speedup of 16 for RTM and up to 43 for the modeling application over a sequential code running on general-purpose CPUs. A CPU rack versus a GPU rack comparison was described and showed a 4.3 speedup. Copyright © 2011 John Wiley & Sons, Ltd.

The use of the LCAO (Linear Combination of Atomic Orbitals) method for excited states involves products of orbitals that are known to be linearly dependent. We identify a basis in the space of orbital products that is local for orbitals of finite support and with a residual error that vanishes exponentially with its dimension. As an application of our previously reported technique we compute the Kohn--Sham density response function $\chi_{0}$ for a molecule consisting of $N$ atoms in $N^{2}N_{\omega}$ operations, with $N_{\omega}$ the number of frequency points. We test our construction of $\chi_{0}$ by computing molecular spectra directly from the equations of Petersilka--Gossmann--Gross in $N^{2}N_{\omega}$ operations rather than from Casida's equations which takes $N^{3}$ operations. We consider the good agreement with previously calculated molecular spectra as a validation of our construction of $\chi_{0}$. Ongoing work indicates that our method is well suited for the computation of the GW self-energy $\Sigma=\mathrm{i}GW$ and we expect it to be useful in the analysis of exitonic effects in molecules.

In this work, we simulate the interaction between intense laser radiation and a fully ionized plasma by solving a Vlasov-Maxwell system using the "Particle-In-Cell" (PIC) method. This method provides a very detailed description of the plasma dynamics, but at the expense of large computer resources.Our SPMD 3D PIC code, CALDER, which is fully relativistic, is based on a spatial domain decomposition. Each processor is assigned one subdomain and is in charge of updating its field values and particle coordinates. This paper presents some optimizations to achieve large simulations, such as communication overlapping, cache-friendly data management, and the use of a parallel sparse PCG solves the Poisson's equation.Finally we present the benefits from these optimizations on the IBM SP3 and the physical results for a large case simulation obtained on the CEA/DIF Teraflops parallel computer.

International audience; The Time Splitting Scheme (TSS) has been examined within the context of the one-dimensional (1D) relativistic Vlasov–Maxwell model. In the strongly relativistic regime of the laser–plasma interaction, the TSS cannot be applied to solve the Vlasov equation. We propose a new semi-Lagrangian scheme based on a full 2D advection and study its advantages over the classical Splitting procedure. Details of the underlying integration of the Vlasov equation appear to be important in achieving accurate plasma simulations. Examples are given which are related to the relativistic modulational instability and the self-induced transparency of an ultra-intense electromagnetic pulse in the relativistic regime.

Resume Nous proposons un schema de discretisation en temps pour le couplage entre une methode de continuum pour calculer les effets electrostatiques et la dynamique moleculaire. Nous montrons que l'algorithme conserve l'energie totale du systeme et que le schema est symplectique. De plus, nous donnons une variante qui conserve le volume de simulation pour les dynamiques NVE.

A new and efficient algorithm is presented for the coordinate transformation problem of massively three-dimensional networks formed, e.g., by the atoms of crystal fragments or molecular clusters. The new algorithm is based on a divide-and-conquer technique to perform iterative coordinate transformation, applicable even for three-dimensional networks, with linear scaling memory and near linear scaling CPU time requirements. The new algorithm proved to be very fast in the coordinate transformation problems and geometry optimization of diamond fragments, water clusters, globular proteins, and proteins in solvent.

This article presents a new algorithm to solve the coordinate transformation problem of molecular geometry optimization. The algorithm is very fast and its CPU time consumption scales linearly with the system size. It makes use of the locality of internal coordinates by efficient sparse matrix techniques. The new algorithm drastically reduces the time needed for coordinate transformations as demonstrated by test calculations on polyalanine and carbone nanotube systems: for a 2000 atom system it requires just seven seconds, instead of the hours consumed by traditional schemes.

Interaction of relativistically strong laser pulses with underdense and overdense plasmas is investigated by a semi-Lagrangian Vlasov code. These Vlasov simulations revealed a rich variety of phenomena associated with the fast particle dynamics induced by the electromagnetic wave as electron trapping, particle acceleration, and electron plasma wavebreaking. To describe the distribution of accelerated particle momenta and energy will require a very detailed analysis of the kinetic and time history of the plasma wave evolution. The semi-Lagrangian Vlasov code allows us to handle the interaction of ultrashort electromagnetic pulse with plasma at strongly relativistic intensities with a great deal of resolution in phase space.

A block version of the Shake method for heavy atom simulation in biological systems is presented in this paper. The method solves successively, independent blocks of constraints of small size by a Newton method. This algorithm is implemented in TAKAKAW, an efficient parallel molecular dynamics code. This method has been tested on a small system and on an ionic canal of 67671 atoms.

We describe the parallel implementation of semi-Lagrangian Vlasov solvers, which are an alternative to particle-in-cell (PIC) simulations for the numerical investigation of the behaviour of charged particles in their self-consistent electromagnetic fields. The semi-Lagrangian method, which couples the Lagrangian and Eulerian points of view, is particularly interesting on parallel computers, since the solution is computed on grid points, the number of which remains constant in time on each processor, unlike the number of particles in PIC simulations, and thus greatly simplifies the parallelization process.

This paper describes a high level C++ interface for message passing applications. Our interface is built on top of PVM and MPI. The two main contributions are to allow a quicker design of parallel applications without any important drop of performances. We introduce two levels of tasks and use C++ streams for communications. We also present a performance study over both PVM and MPI to show the overhead of our implementation. Finally, we detail two applications based on the heat equation to explain how Para++ can be used for SPMD and MPMD applications .

Dans ce papier nous decrivons le comportement en temps et en espace du champ d'induction magnetique cree par un courant alternatif a haute frequence impose autour d'un conducteur solide. Pour cela, nous introduisons deux echelles de temps et nous decomposons le champ d'induction en un champ moyen et un champ oscillant. A l'aide de la theorie des perturbations singulieres et des methodes d'echelles multiples nous obtenons deux modeles decouples, un pour les grandes echelles de temps et l'autre pour les hautres frequences. Dans le cas d'une coupe d'une colonne de metal, nous construisons les deux premiers termes du developpement asymptotique du champ d'induction. De plus, pour un courant harmonique mono-frequence nous justifions l'approximation harmonique pour les temps grands. Finalement, nous appliquons les resultats obtenus dans le cas d'une colonne cylindrique, en calculant numeriquement le champ d'induction pour differentes frequences.

We study the long time behaviour of the solutions of the third grade fluids׳ equations in dimension 2. Introducing scaled variables and performing several energy estimates in weighted Sobolev spaces, we describe the first order of an asymptotic expansion of these solutions. It shows in particular that, under smallness assumptions on the data, the solutions of the third grade fluids׳ equations converge to self-similar solutions of the heat equations, which can be computed explicitly from the data.

International audience; A new variational algorithm called adaptive vibrational configuration interaction (A-VCI) intended for the resolution of the vibrational Schrödinger equation was developed. The main advantage of this approach is to efficiently reduce the dimension of the active space generated into the configuration interaction (CI) process. Here, we assume that the Hamiltonian writes as a sum of products of operators. This adaptive algorithm was developed with the use of three correlated conditions i.e. a suitable starting space ; a criterion for convergence, and a procedure to expand the approximate space. The velocity of the algorithm was increased with the use of a posteriori error estimator (residue) to select the most relevant direction to increase the space. Two examples have been selected for benchmark. In the case of H 2 CO, we mainly study the performance of A-VCI algorithm: comparison with the variation-perturbation method, choice of the initial space, residual contributions. For CH 3 CN, we compare the A-VCI results with a computed reference spectrum using the same potential energy surface and for an active space reduced by about 90 %.

The goal of Matrices Over Runtime Systems at Exascale (MORSE) project is to design dense and sparse linear algebra methods that achieve the fastest possible time to an accurate solution on large-scale multicore systems with GPU accelerators, using all the processing power that future high end systems can make available. In this poster, we propose a framework for describing linear algebra algorithms at a high level of abstraction and delegating the actual execution to a runtime system in order to design software whose performance is portable accross architectures. We illustrate our methodology on three classes of problems: dense linear algebra, sparse direct methods and fast multipole methods. The resulting codes have been incorporated into Magma, Pastix and ScalFMM solvers, respectively.

The goal of Matrices Over Runtime Systems at Exascale (MORSE) project is to design dense and sparse linear algebra methods that achieve the fastest possible time to an accurate solution on large-scale multicore systems with GPU accelerators, using all the processing power that future high end systems can make available. In this poster, we propose a framework for describing linear algebra algorithms at a high level of abstraction and delegating the actual execution to a runtime system in order to design software whose performance is portable accross architectures. We illustrate our methodology on three classes of problems: dense linear algebra, sparse direct methods and fast multipole methods. The resulting codes have been incorporated into Magma, Pastix and ScalFMM solvers, respectively.

This paper deals with the numerical approximation of the section of a column of liquid metal shaped by an external electromagnetic field. The bidimensional model used here leads up to solve, by a spectral method, a nonlinear boundary value problem posed on the exterior of the unit disk to find a convenient conformal mapping. We give accurate bounds on the errors in the approximation process. Some examples of shapes for both exterior and interior shaping are presented.

We study a nonlinear boundary value problem posed in the interior or the exterior of the unit disk in R2. Using capacity operator, we transform it into a pseudo-differential problem on the unit circle. The Galerkin method together with Fourier expansion, is used to approximate our problem. We show the convergence of the fixed-point scheme and we give an accurate bound of the L2-norm of the error. Numerical results coming from a problem arising in electromagnetic casting are also presented.

We present in this paper multi-thread and multi-process parallelizations of the Fast Multipole Method (FMM) for Laplace equation, for uniform and non uniform distributions. These parallelizations apply to the original FMM formulation and to our new matrix formulation with BLAS (Basic Linear Algebra Subprograms) routines. Differences between the multi-thread and the multi-process versions are detailed, and a hybrid MPI-thread approach enables to gain parallel efficiency and memory scalability over the pure MPI one on clusters of SMP nodes. On 128 processors, we obtain 85% (respectively 75%) parallel efficiency for uniform (respectively non uniform) distributions with up to 100 million particles.

International audience; In the context of scientific computing, the computational steering consists in the coupling of numerical simulations with 3D visualization systems through the network. This allows scientists to monitor online the intermediate results of their computations in a more interactive way than the batch mode, and allows them to modify the simulation parameters on-the-fly. While most of existing computational steering environments support parallel simulations, they are often limited to sequential visualization systems. This may lead to an important bottleneck and increased rendering time. To achieve the required performance for online visualization, we have designed the EPSN framework, a computational steering environment that enables to interconnect legacy parallel simulations with parallel visualization systems. For this, we have introduced a redistribution algorithm for unstructured data, that is well adapted to the context of M × N computational steering. Then, we focus on the design of our parallel viewer and present some experimental results obtained with a particle-based simulation in astrophysics.

We present a parallel implementation of a multi-scale coupling method for material simulations. We describe the models used for our application and we focus on the design of our numerical simulation, which is based on a coupling involving a parallel code for molecular dynamics and a parallel finite element code for continuum mechanics. Data redistribution and atom migration issues are discussed, and an experimental validation on a real crack propagation problem is provided.

Publisher Summary This chapter discusses the first step toward a computational steering environment based on CORBA (common object request broker architecture). This environment, called EPSN, allows the control, the data exploration and the data modification for numerical simulations, involving an iterative process. In order to be as generic as possible, an abstract model of steerable simulations is introduced. This abstraction allows building steering clients independently of a given simulation. This model is described with XML syntax and is used in the simulation by some source code annotations. EPSN takes advantage of the CORBA technology to design a communication infrastructure with portability, interoperability, and network transparency. In addition, the in-progress parallel CORBA objects give a framework for extending the steering to parallel and distributed simulations. The prototype Epsilon reveals itself as a really light and easy to use steering platform, providing great capabilities of interaction. Epsilon validates the model of integration based on both a source code annotation and a XML description. The developments of EPSN are now oriented on the integration of parallel and distributed simulations with irregular data distributions.

This paper deals with the introduction of a nonlinear term into Darcy's equation to describe inertial effects in a porous medium. The method chosen is the numerical resolution of flow equations at a pore scale. The medium is modelled by cylinders of either equal or unequal diameters arranged in a regular pattern with a square or triangular base. For a given flow through this medium the pressure drop is evaluated numerically.The Navier-Stokes equations are discretized by the mixed finite-element method. The numerical solution is based on operator-splitting methods whose purpose is to separate the difficulties due to the nonlinear operator in the equation of motion and the necessity of taking into account the continuity equation. The associated Stokes problems are solved by a mixed formulation proposed by Glowinski & Pironneau.For Reynolds numbers lower than 1, the relationship between the global pressure gradient and the filtration velocity is linear as predicted by Darcy's law. For higher values of the Reynolds number the pressure drop is influenced by inertial effects which can be interpreted by the addition of a quadratic term in Darcy's law.On the one hand this study confirms the presence of a nonlinear term in the motion equation as experimentally predicted by several authors, and on the other hand analyses the fluid behaviour in simple media. In addition to the detailed numerical solutions, an estimation of the hydrodynamical constants in the Forchheimer equation is given in terms of porosity and the geometrical characteristics of the models studied.

In the framework of tensor spaces, we consider orthogonalization kernels to generate an orthogonal basis of a tensor subspace from a set of linearly independent tensors. In particular, we investigate numerically the loss of orthogonality of six orthogonalization methods, namely Classical and Modified Gram-Schmidt with (CGS2, MGS2) and without (CGS, MGS) re-orthogonalization, the Gram approach, and the Householder transformation. To tackle the curse of dimensionality, we represent tensor with low rank approximation using the Tensor Train (TT) formalism, and we introduce recompression steps in the standard algorithm outline through the TT-rounding method at a prescribed accuracy. After describing the algorithm structure and properties, we illustrate numerically that the theoretical bounds for the loss of orthogonality in the classical matrix computation round-off analysis results are maintained, with the unit round-off replaced by the TT-rounding accuracy. The computational analysis for each orthogonalization kernel in terms of the memory requirement and the computational complexity measured as a function of the number of TT-rounding, which happens to be the computational most expensive operation, completes the study.; Dans le contexte de l'espace tensoriel, nous considérons les noyaux d'orthogonalisation pour générer une base orthogonale d'un sous-espace tensoriel à partir d'un ensemble de tenseurs linéairement indépendants. En particulier, nous étudions numériquement la perte d'orthogonalité de six méthodes d'orthogonalisation, à savoir les méthodes de Gram-Schmidt classique et modifiée avec (CGS2, MGS2) et sans (CGS, MGS) réorthogonalisation, l'approche de Gram et la transformation de Householder. Pour lutter contre la malédiction de la dimensionnalité, nous représentons les tenseurs avec le formalisme Train de tenseurs (TT), et nous introduisons des étapes de recompression dans le schéma de l'algorithme standard par la méthode TT-rounding à une précision prescrite. Après avoir décrit la structure et les propriétés de l'algorithme, nous illustrons numériquement que les limites théoriques de la perte d'orthogonalité dans le calcul matriciel classique sont maintenues, l'arrondi unitaire étant remplacé par la précision de l'arrondi TT. L'analyse pour chaque noyau d'orthogonalisation de l'exigence de mémoire et de la complexité de calcul en termes d'arrondi TT, qui se trouve être l'opération la plus coûteuse en termes de calcul, complète l'étude.

Preliminary version of the book chapter entitled "Deflation and augmentation techniques in Krylov linear solvers" published in "Developments in Parallel, Distributed, Grid and Cloud Computing for Engineering", ed. Topping, B.H.V and Ivanyi, P., Saxe-Coburg Publications, Kippen, Stirlingshire, United Kingdom, ISBN 978-1-874672-62-3, p. 249-275, 2013; In this paper we present deflation and augmentation techniques that have been designed to accelerate the convergence of Krylov subspace methods for the solution of linear systems of equations. We review numerical approaches both for linear systems with a non-Hermitian coefficient matrix, mainly within the Arnoldi framework, and for Hermitian positive definite problems with the conjugate gradient method.; Dans ce rapport nous présentons des techniques de déflation et d'augmentation qui ont été développées pour accélérer la convergence des méthodes de Krylov pour la solution de systémes d'équations linéaires. Nous passons en revue des approches pour des systémes linéaires dont les matrices sont non-hermitiennes, principalement dans le contexte de la méthode d'Arnoldi, et pour des matrices hermitiennes définies positives avec la méthode du gradient conjugué.

International audience; We have designed a fast parallel simulator that solves the acoustic wave equation on a GPU cluster. Solving the acoustic wave equation in an oil exploration industrial context aims at speeding up seismic modeling and Reverse Time Migration. We use a finite difference approach on a regular mesh, in both 2D and 3D cases. The acoustic wave equation is solved in either a constant density or a variable density domain. We use CUDA to take advantage of the GPUs computational power. We study different implementations and their impact on the application performance. We obtain a speedup of 11 for Reverse Time Migration and up to 30 for the modeling application over a sequential code running on general purpose CPU.

International audience

International audience; Learn about the fast multipole method (FMM) and its optimization on NVIDIA GPUs. The FMM is a well-known algorithm with a variety of applications in areas like galaxy simulation, electrostatic potential calculations, boundary element methods, integral equations, dislocations dynamics, etc. The FMM offers several difficulties when running on parallel heterogeneous platforms such as multicore processors with GPUs. Some parts of the calculation suffer from limited concurrency, and load-balancing can be very uneven for certain distributions of particles. We will present a new API and runtime system, called StarPU, that allows expressing a calculation as a graph of tasks, with dependencies, and contains a runtime system that can optimally schedule those tasks on a parallel machine. StarPU supports conventional multicore processors as well as NVIDIA

This paper presents an extension of Correspondence Analysis (CA) to tensors through High Order Singular Value Decomposition (HOSVD) from a geometric viewpoint. Correspondence analysis is a well-known tool, developed from principal component analysis, for studying contingency tables. Different algebraic extensions of CA to multi-way tables have been proposed over the years, nevertheless neglecting its geometric meaning. Relying on the Tucker model and the HOSVD, we propose a direct way to associate with each tensor mode a point cloud. We prove that the point clouds are related to each other. Specifically using the CA metrics we show that the barycentric relation is still true in the tensor framework. Finally two data sets are used to underline the advantages and the drawbacks of our strategy with respect to the classical matrix approaches.; Ce document présente une extension de l'analyse des correspondances aux tenseurs par la décomposition en valeurs singulières d'ordre élevé (HOSVD) d'un point de vue géométrique. L'analyse des correspondances est un outil bien connu, développé à partir de l'analyse en composantes principales, pour étudier les tables de contingence. Différentes extensions algébriques de l'analyse des correspondances aux tables à voies multiples ont été proposées au fil des ans. En nous appuyant sur le modèle de Tucker et la HOSVD, nous proposons d'associer à chaque mode d'un tenseur un nuage de points. Nous établissons un lien entre les cordonnées de ces différents nuages. Une telle relation est classique en Analyse Factorielle des Correspondances (AFC) pour justifier la projection simultanée des profils lignes et profils colonnes d'une table de contingence (d'où le nom de correspondance). Nous étendons une telle relation barycentrique aux liens entre les nuages de points associés aux différents modes de l'Analyse Factorielle des Correspondances Multiple d'un tenseur, construite via la HOSVD avec les métriques de l'AFC.

International audience; We propose here an efficient method to define a representative approximation subspace to compute the first eigenvalues of the vibrational Hamiltonian which are those of interest in the experimental results.