A Parallel Iterative Method for Computing Molecular Absorption Spectra

We describe a fast parallel iterative method for computing molecular absorption spectra within TDDFT linear response and using the LCAO method. We use a local basis of "dominant products" to parametrize the space of orbital products that occur in the LCAO approach. In this basis, the dynamical polarizability is computed iteratively within an appropriate Krylov subspace. The iterative procedure uses a a matrix-free GMRES method to determine the (interacting) density response. The resulting code is about one order of magnitude faster than our previous full-matrix method. This acceleration makes the speed of our TDDFT code comparable with codes based on Casida's equation. The implementation of our method uses hybrid MPI and OpenMP parallelization in which load balancing and memory access are optimized. To validate our approach and to establish benchmarks, we compute spectra of large molecules on various types of parallel machines. The methods developed here are fairly general and we believe they will find useful applications in molecular physics/chemistry, even for problems that are beyond TDDFT, such as organic semiconductors, particularly in photovoltaics.Comment: 20 pages, 17 figures, 3 table

Modélisation hautes fréquences en magnétohydrodynamique : aspect magnétique

Dans ce papier nous rappelons la modélisation de l'induction magnétique générée par des courants alternatifs de haute fréquence pour le traitement des métaux liquides. A l'aide des méthodes asymptotiques nous construisons des modèles limites lorsque la fréquence du courant imposée est grande

Numerical approximation of free boundary problem arising in electromagnetic shaping

We approximate numerically the section of a column of liquid metal submitted to an external electromagnetic field. The bidimensional model used here leads us to solve by a spectral method a nonlinear boundary problem posed on the exterior of the unit disk to find a convenient conformal mapping. We give accurates bound of the errors obtained during our approximation process. We present some examples of shapes for both exterior and interior shaping

Pipelining the Fast Multipole Method over a Runtime System

Fast Multipole Methods (FMM) are a fundamental operation for the simulation of many physical problems. The high performance design of such methods usually requires to carefully tune the algorithm for both the targeted physics and the hardware. In this paper, we propose a new approach that achieves high performance across architectures. Our method consists of expressing the FMM algorithm as a task flow and employing a state-of-the-art runtime system, StarPU, in order to process the tasks on the different processing units. We carefully design the task flow, the mathematical operators, their Central Processing Unit (CPU) and Graphics Processing Unit (GPU) implementations, as well as scheduling schemes. We compute potentials and forces of 200 million particles in 48.7 seconds on a homogeneous 160 cores SGI Altix UV 100 and of 38 million particles in 13.34 seconds on a heterogeneous 12 cores Intel Nehalem processor enhanced with 3 Nvidia M2090 Fermi GPUs.Comment: No. RR-7981 (2012

Asymptotic Analysis of Magnetic Induction with High Frequency for Solid Conducteurs

In this paper we describe the behaviour both in time and in space of an induction field created by an imposed high frequency alternating current around a solid conductor. To do this, we introduce two time scales and we decompose the induction field in a mean field and an oscillating field. With the help of singular perturbations theory and multiple scales method we obtain two uncoupled models; one for the large time scale and the other for the high frequencies. For a cross section of a solid column of metal, we build the two first terms of the asymptotic expansion of the induction field. Moreover, we justify the classical harmonic approximation used in such configuration. Finally in the case of a cylinder column, we apply the previous results by computing numerically the induction for different values of the frequency

On some orthogonalization schemes in Tensor Train format

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 experimentally study 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 overcome the curse of dimensionality, we represent tensors with a low-rank approximation using the Tensor Train (TT) formalism. In addition, we introduce recompression steps in the standard algorithm outline through the TT-rounding method at a prescribed accuracy. After describing the structure and properties of the algorithms, we illustrate their loss of orthogonality with numerical experiments. The theoretical bounds from the classical matrix computation round-off analysis, obtained over several decades, seem to be maintained, with the unit round-off replaced by the TT-rounding accuracy. The computational analysis for each orthogonalization kernel in terms of the memory requirements and the computational complexity measured as a function of the number of TT-rounding, which happens to be the most computationally expensive operation, completes the study

Contributions algorithmiques pour les simulations complexes en physique des matériaux

Because computer performance is always increasing, the numerical simulations of physical phenomena become much harder. Where does this complexity come from ? On the one hand, for a higher fidelity of the solution problem, more advanced physical models are introduced ; on the other hand, the size of the discretization is shrinking which leads to very large-scale numerical problems. In addition the computers themselves become more complex, for instance with hierar- chical memories and hierarchical process units. Hence the need to redesign algorithms, numerical schemes and software to make them more efficient and effective on emerging architectures.The work layout in this document is divided in two mainlines. First, the development and the parallel design of the fast multipole method to compute pair-wise interactions. Then the coupling of models/methods and codes in material physics at atomistic scale, and the computational steering of these simulations as well.Avec l’accroissement de la puissance des ordinateurs, les simulations numériques de phénomènes physiques deviennent de plus en plus complexes. Cette complexité provient d’une part, de l’ajout de modèles physiques plus compliqués pour mieux représenter la physique et d’autre part, par une discrétisation plus fine du problème conduisant à des problèmes de grande taille. Une difficulté supplémentaire s’ajoute avec la complexification des ordinateurs notamment par des mémoires et des unités de calculs hiérarchiques. Il est donc nécessaire de redessiner les algorithmiques, les schémas numériques et les logiciels pour les rendre plus efficaces sur ces architectures émergentes. Le travail présenté dans ce document correspond à deux axes de recherche. Le premier porte sur le développement et la parallélisation de la méthode des multipôles rapides pour calculer rapidement des interactions de paires. Le deuxième axe concerne le couplage de méthodes et de codes en physique des matériaux à échelle atomique ainsi que le pilotage de ces simulations