2023Activity reportProject-TeamCONCACE

3.1 Numerical algebra methodologies in model and data-driven scientific computing

At the core of many simulations, one has to solve a linear algebra problem that is defined in a vector space and that involves linear operators, vectors and scalars, the unknowns being usually vectors or scalars, e.g. for the solution of a linear system or an eigenvalue problem. For many years, in particular in model-driven simulations, the problems have been reformulated in classical matrix formalism possibly unfolding the spaces where the vectors naturally live (typically 3D PDEs) to end up with classical vectors in $R^n$ or $C^n$. For some problems, defined in higher dimension (e.g., time dependent 3D PDE), the other dimensions are dealt in a problem specific fashion as unfolding those dimensions would lead to too large matrices/vectors. The concace research program on numerical methodology intends to address the study of novel numerical algorithms to continue addressing the mainstream approaches relying on classical matrix formalism but also to investigate alternatives where the structure of the underlying problem is kept preserved and all dimensions are dealt with equally. This latter research activity mostly concerns linear algebra in tensor spaces. In terms of algorithmic principles, we will lay an emphasis on hierarchy as a unifying principle for the numerical algorithms, the data representation and processing (including the current hierarchy of arithmetic) and the parallel implementation towards scalability.

3.2 Composition of parallel numerical algorithms from a sequential expression

A major breakthrough for exploiting multicore machine  42 is based on a data format and computational technique originally used in an out-of-core context  54. This is itself a refinement of a broader class of numerical algorithms – namely, “updating techniques” – that were not originally developed with specific hardware considerations in mind. This historical anecdote perfectly illustrates the need to separate data representation, algorithmic and architectural concerns when developing numerical methodologies. In the recent past, we have contributed to the study of the sequential task flow (STF) programming paradigm, that enabled us to abstract the complexity of the underlying computer architecture  32, 33, 31. In the concace project, we intend to go further by abstracting the numerical algorithms and their dedicated data structures. We strongly believe that combining these two abstractions will allow us to easily compose toolbox algorithms and data representations in order to study combinatorial alternatives towards numerical and parallel computational efficiency. We have demonstrated this potential on domain decomposition methods for solving sparse linear systems arising from the discretisation of PEDs, that has been implemented in the maphys++ parallel package.

Regarding the abstraction of the target architecture in the design of numerical algorithms, the STF paradigm has been shown to significantly reduce the difficulty of programming these complex machines while ensuring high computational efficiency. However, some challenges remain. The first major difficulty is related to the scalability of the model at large scale where handling the full task graph associated with the STF model becomes a severe bottleneck. Another major difficulty is the inability (at a reasonable runtime cost) to efficiently handle fine-grained dynamic parallelism, such as numerical pivoting in the Gaussian elimination where the decision to be made depends on the outcome of the current calculation and cannot be known in advance or described in a task graph. These two challenges are the ones we intend to study first.

With respect to the second ingredient, namely the abstraction of the algorithms and data representation, we will also explore whether we can provide additional separation of concerns beyond that offered by a task-based design. As a seemingly simple example, we will investigate the possibility of abstracting the matrix-vector product, basic kernel at the core of many numerical linear algebra methods, to cover the case of the fast multipole method (FMM, at the core of the ScalFMM library). FMM is mathematically a block matrix-vector product where some of the operations involving the extra-diagonal blocks with hierachical structure would be compressed analytically. Such a methodological step forward will consequently allow the factorisation of a significant part of codes (so far completely independent because no bridge has been made upstream) including in particular the ones dealing with $\mathscr{H}\text{-matrices}$. The easy composition of these different algorithms will make it possible to explore the combinatorial nature of the possible options in order to best adapt them to the size of the problem to be treated and the characteristics of the target computer. *Offering such a continuum of numerical methods rather than a discrete set of tools is part of the team's objectives* It is a very demanding effort in terms of HPC software engineering expertise to coordinate the overall technical effort.

We intend to strengthen our engagement in reproducible and open science. Consequently, we will continue our joint effort to ensure consistent deployment of our parallel software; this will contribute to improve its impact on academic and industrial users. The software engineering challenge is related to the increasing number of software dependencies induced by the desired capability of combining the functionality of different numerical building boxes, e.g., a domain decomposition solver (such as maphys++) that requires advanced iterative schemes (such as those provided by fabulous) as well as state-of-the-art direct methods (such as pastix, mumps, or qr_mumps), deploying the resulting software stack can become tedious  36.

In that context, we will consider the benefit of using hybridization between simulation and learning in order to reduce the complexity of classical approaches by diminishing the problem size or improving preconditioning techniques. In a longer term perspective, we will also conduct an active technological watch activity with respect to quantum computing to better understand how such a advanced computing technology can be synergized with classical scientific computing.

4.1 Aeroacoustics Simulation

Participants: Emmanuel Agullo, Carola Kruse, Paul Mycek, Pierre Benjamin, Marek Felsoci, Luc Giraud, Gilles Marait, Guillaume Sylvand.

This domains is in the context of a long term collaboration with Airbus Research Centers. Wave propagation phenomena intervene in many different aspects of systems design at Airbus. They drive the level of acoustic vibrations that mechanical components have to sustain, a level that one may want to diminish for comfort reason (in the case of aircraft passengers, for instance) or for safety reason (to avoid damage in the case of a payload in a rocket fairing at take-off). Numerical simulations of these phenomena plays a central part in the upstream design phase of any such project  44. Airbus Central R & T has developed over the last decades an in-depth knowledge in the field of Boundary Element Method (BEM) for the simulation of wave propagation in homogeneous media and in frequency domain. To tackle heterogeneous media (such as the jet engine flows, in the case of acoustic simulation), these BEM approaches are coupled with volumic finite elements (FEM). We end up with the need to solve large (several millions unknowns) linear systems of equations composed of a dense part (coming for the BEM domain) and a sparse part (coming from the FEM domain). Various parallel solution techniques are available today, mixing tools created by the academic world (such as the Mumps and Pastix sparse solvers) as well as parallel software tools developed in-house at Airbus (dense solver SPIDO, multipole solver, $\mathscr{H}$-matrix solver with an open sequential version available online). In the current state of knowledge and technologies, these methods do not permit to tackle the simulation of aeroacoustics problems at the highest acoustic frequencies (between 5 and 20 kHz, upper limits of human audition) while considering the whole complexity of geometries and phenomena involved (higher acoustic frequency implies smaller mesh sizes that lead to larger unknowns number, a number that grows like $f^2$ for BEM and $f^3$ for FEM, where f is the studied frequency). The purpose of the study in this domain is to develop advanced solvers able to tackle this kind of mixed dense/sparse linear systems efficiently on parallel architectures.

5.1 New software

6.1 Towards a direct task-based solver for sparse/dense FEM/BEM linear systems

We are interested in the direct solution of very large linear systems composed of both sparse and dense parts. Coupled hollow/dense systems appear in various physical problems, such as the simulation of acoustic wave propagation around aircraft. To produce a physically realistic result, the number of unknowns in the system can be extremely high, making its handling a real challenge. Thanks to the building blocks provided by state-of-the-art hollow and dense solvers, we can compose a coupled hollow/dense solver. To reduce the computation time and memory consumption of direct methods, some solvers implement advanced features such as numerical compression, out-of-core computation and distributed memory parallelism. These functionalities can be easily applied within the individual building blocks, but this is not trivial at the articulation between the hollow solver bricks and the dense solver bricks. Their programming interface (API) has not been designed for this purpose. We have previously proposed solver coupling schemes that still allow the use of these well-optimized solvers with advanced functionalities. The idea is to apply the existing API to carefully selected sub-arrays of coupled systems so as to take full advantage of digital compression and out-of-core computation in both shared and distributed memory. Although capable of handling considerably larger coupled systems compared to the state of the art, these schemes remain sub-optimal due to intrinsic design limitations. We therefore explore an alternative coupling scheme based on direct task-based solvers that use the same execution engine. The aim is to improve composability and facilitate data passing between hollow and dense solvers for more efficient computation. Before considering the integration of this approach into the complex code of a full community solver, we implemented a proof-of-concept without some advanced features. A preliminary experimental study enabled us to validate our prototype and demonstrate its competitiveness against other approaches.

For more details on this work we refer to  17.

6.2 On the Arithmetic Intensity of Distributed-Memory Dense Matrix Multiplication Involving a Symmetric Input Matrix

Dense matrix multiplication involving a symmetric input matrix (SYMM) is implemented in reference distributed-memory codes with the same data distribution as its general analogue (GEMM). We show that, when the symmetric matrix is dominant, such a 2D block-cyclic (2D BC) scheme leads to a lower arithmetic intensity (AI) of SYMM than that of GEMM by a factor of 2. We propose alternative data distributions preserving the memory benefit of SYMM of storing only half of the matrix while achieving up to the same AI as GEMM. We also show that, in the case we can afford the same memory footprint as GEMM, SYMM can achieve a higher AI. We propose a task-based design of SYMM independent of the data distribution. This design allows for scalable A-stationary SYMM with which all discussed data distributions, may they be very irregular, can be easily assessed. We have integrated the resulting code in a reduction dimension algorithm involving a randomized singular value decomposition dominated by SYMM. An experimental study shows a compelling impact on performance.

For more details on this work we refer to  15.

6.3 Task-based parallel programming for scalable matrix product algorithms

Task-based programming models have succeeded in gaining the interest of the high-performance mathematical software community because they relieve part of the burden of developing and implementing distributed-memory parallel algorithms in an efficient and portable way.In increasingly larger, more heterogeneous clusters of computers, these models appear as a way to maintain and enhance more complex algorithms. However, task-based programming models lack the flexibility and the features that are necessary to express in an elegant and compact way scalable algorithms that rely on advanced communication patterns. We show that the Sequential Task Flow paradigm can be extended to write compact yet efficient and scalable routines for linear algebra computations. Although, this work focuses on dense General Matrix Multiplication, the proposed features enable the implementation of more complex algorithms. We describe the implementation of these features and of the resulting GEMM operation. Finally, we present an experimental analysis on two homogeneous supercomputers showing that our approach is competitive up to 32,768 CPU cores with state-of-the-art libraries and may outperform them for some problem dimensions. Although our code can use GPUs straightforwardly, we do not deal with this case because it implies other issues which are out of the scope of this work.

For more details on this work we refer to  13.

6.4 Combining reduction with synchronization barrier on multi-core processors

With the rise of multi-core processors with a large number of cores the need of shared memory reduction that perform efficiently on a large number of core is more pressing. Efficient shared memory reduction on these multi-core processors will help share memory programs being more efficient on these one. In this paper, we propose a reduction combined with barrier method that uses SIMD instructions to combine barriers signaling and reduction value read/write to minimize memory/cache traffic between cores thus, reducing barrier latency. We compare different barriers and reduction methods on three multi-core processors and show that proposed combining barrier/reduction method are 4 and 3.5 times faster than respectively GCC 11.1 and Intel 21.2 OpenMP 4.5 reduction.

For more details on this work we refer to  14.

6.5 A note on GMRES algorithm in Tensor Train format for the solution of parametric linear systems

We consider the solution of linear systems with tensor product structure using a GMRES algorithm. To cope with the computational complexity in large dimension both in terms of floating point operations and memory requirement, our algorithm is based on low-rank tensor representation, namely the Tensor Train format. In a backward error analysis framework, we show how the tensor approximation affects the accuracy of the computed solution. With the backward perspective, we investigate the situations where the $(d+1)$-dimensional problem to be solved results from the concatenation of a sequence of $d$-dimensional problems (like parametric linear operator or parametric right-hand side problems), we provide backward error bounds to relate the accuracy of the $(d+1)$-dimensional computed solution with the numerical quality of the sequence of $d$-dimensional solutions that can be extracted form it. This enables to prescribe convergence threshold when solving the $(d+1)$-dimensional problem that ensures the numerical quality of the $d$-dimensional solutions that will be extracted from the $(d+1)$-dimensional computed solution once the solver has converged. The above mentioned features are illustrated on a set of academic examples of varying dimensions and sizes.

6.6 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 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.

This work was presented in two international conferences 28, 29 from different scientific communities.

For more details on this work we refer to the revised version of the scientific report  48 to be published.

6.7 Neural network preconditioned subspace methods for the solution of the Helmholtz equation

In recent years, scientific machine learning, utilizing deep learning methodologies, has found widespread application in the fields of scientific computing and computational engineering. Nevertheless, while these data-driven deep learning solvers can be highly effective once appropriately trained, they often yield solutions of limited accuracy. Additionally, the computational expenses incurred during the training phase can be prohibitively high. In this talk, we first presents the details of training various learning solvers, incorporating with different neural network architectures, for solving the heterogeneous Helmholtz equation. Some mathematical ingredients from classical iterative solver are considered into the training phase to enhance robustness and speed. Moreover, once the neural network solvers are adequately trained, their inferences can be applied as a nonlinear preconditioner in the classical subspace methods, like the flexible GMRES or flexible FOM method. This presentation demonstrates the efficiency of employing neural networks as preconditioner and showcases the evident advantages of these neural network preconditioned approaches. They outperform both the newly emerging deep neural network methods and the classical subspace methods in both computational efficiency and solution accuracy.

For more details on this work we refer to the revised version of the scientific report 27, 30

6.8 Machine learning techniques to predict the rank H-matrices

The discretization of spatial operators using boundary element techniques leads to dense linear systems. The representation of full matrices in $\mathscr{H}$-matrix (Hierarchical Matrices) format is based on a bisection of space, leading to a binary tree defined on the operator definition space. A block of the matrix representing the interaction between two subsets of unknowns can be interpreted as the interaction between two nodes of the binary tree. An admissibility condition is used to determine whether this matrix block admits a low-rank representation or whether this approximation should be considered at the level of the sons of these nodes. These admissibility conditions have been studied theoretically in certain configurations. The goal of this work is to propose an admissibility condition learned by a neural network from computationally inexpensive information. The training data will be extracted from a set of simulations performed by Airbus CR&T.

6.9 Multivariate extensions of the Multilevel Best Linear Unbiased Estimator for ensemble-variational data assimilation

Multilevel estimators aim at reducing the variance of Monte Carlo statistical estimators, by combining samples generated with simulators of different costs and accuracies. In particular, the recent work of Schaden and Ullmann (2020) on the multilevel best linear unbiased estimator (MLBLUE) introduces a framework unifying several multilevel and multifidelity techniques. The MLBLUE is reintroduced here using a variance minimization approach rather than the regression approach of Schaden and Ullmann. We then discuss possible extensions of the scalar MLBLUE to a multidimensional setting, i.e. from the expectation of scalar random variables to the expectation of random vectors. Several estimators of increasing complexity are proposed: a) multilevel estimators with scalar weights, b) with element-wise weights, c) with spectral weights and d) with general matrix weights. The computational cost of each method is discussed. We finally extend the MLBLUE to the estimation of second-order moments in the multidimensional case, i.e. to the estimation of covariance matrices. The multilevel estimators proposed are d) a multilevel estimator with scalar weights and e) with element-wise weights. In large-dimension applications such as data assimilation for geosciences, the latter estimator is computationnally unaffordable. As a remedy, we also propose f) a multilevel covariance matrix estimator with optimal multilevel localization, inspired by the optimal localization theory of Ménétrier and Auligné (2015).

For more details on this work we refer to the revised version of the scientific report 24

6.10 A filtered multilevel Monte Carlo method for estimating the expectation of discretized random fields

We investigate the use of multilevel Monte Carlo (MLMC) methods for estimating the expectation of discretized random fields. Specifically, we consider a setting in which the input and output vectors of the numerical simulators have inconsistent dimensions across the multilevel hierarchy. This requires the introduction of grid transfer operators borrowed from multigrid methods. Starting from a simple 1D illustration, we demonstrate numerically that the resulting MLMC estimator deteriorates the estimation of high-frequency components of the discretized expectation field compared to a Monte Carlo (MC) estimator. By adapting mathematical tools initially developed for multigrid methods, we perform a theoretical spectral analysis of the MLMC estimator of the expectation of discretized random fields, in the specific case of linear, symmetric and circulant simulators. This analysis provides a spectral decomposition of the variance into contributions associated with each scale component of the discretized field. We then propose improved MLMC estimators using a filtering mechanism similar to the smoothing process of multigrid methods. The filtering operators improve the estimation of both the small- and large-scale components of the variance, resulting in a reduction of the total variance of the estimator. These improvements are quantified for the specific class of simulators considered in our spectral analysis. The resulting filtered MLMC (F-MLMC) estimator is applied to the problem of estimating the discretized variance field of a diffusion-based covariance operator, which amounts to estimating the expectation of a discretized random field. The numerical experiments support the conclusions of the theoretical analysis even with non-linear simulators, and demonstrate the improvements brought by the proposed F-MLMC estimator compared to both a crude MC and an unfiltered MLMC estimator.

For more details on this work we refer to the revised version of the scientific report 23

6.11 Multilevel Surrogate-based Control Variates

Monte Carlo (MC) sampling is a popular method for estimating the statistics (e.g. expectation and variance) of a random variable. Its slow convergence has led to the emergence of advanced techniques to reduce the variance of the MC estimator for the outputs of computationally expensive solvers. The control variates (CV) method corrects the MC estimator with a term derived from auxiliary random variables that are highly correlated with the original random variable. These auxiliary variables may come from surrogate models. Such a surrogate-based CV strategy is extended here to the multilevel Monte Carlo (MLMC) framework, which relies on a sequence of levels corresponding to numerical simulators with increasing accuracy and computational cost. MLMC combines output samples obtained across levels, into a telescopic sum of differences between MC estimators for successive fidelities. In this paper, we introduce three multilevel variance reduction strategies that rely on surrogate-based CV and MLMC. MLCV is presented as an extension of CV where the correction terms devised from surrogate models for simulators of different levels add up. MLMC-CV improves the MLMC estimator by using a CV based on a surrogate of the correction term at each level. Further variance reduction is achieved by using the surrogate-based CVs of all the levels in the MLMC-MLCV strategy. Alternative solutions that reduce the subset of surrogates used for the multilevel estimation are also introduced. The proposed methods are tested on a test case from the literature consisting of a spectral discretization of an uncertain 1D heat equation, where the statistic of interest is the expected value of the integrated temperature along the domain at a given time. The results are assessed in terms of the accuracy and computational cost of the multilevel estimators, depending on whether the construction of the surrogates, and the associated computational cost, precede the evaluation of the estimator. It was shown that when the lower fidelity outputs are strongly correlated with the high-fidelity outputs, a significant variance reduction is obtained when using surrogate models for the coarser levels only. It was also shown that taking advantage of pre-existing surrogate models proves to be an even more efficient strategy.

For more details on this work we refer to the revised version of the scientific report  25

6.12 Inferences in Hybrid Bayesian Networks using Quadrature Rules

Probabilistic inference in high-dimensional continuous (or hybrid) domains is a challenging problem typically addressed through discretization, sampling, or reliance on often naive parametric assumptions. The drawbacks of these methods are well-known: slow computational speeds and/or highly inaccurate results.

This paper introduces a novel deterministic and general inference algorithm designed for hybrid Bayesian networks featuring both discrete and continuous variables. The algorithm avoids the discretization of continuous densities into histograms by employing quadrature rules to compute continuous integrals, thus transforming the process of marginalizing continuous random variables into summations. These summations are subsequently computed using classical sum-product algorithms within an auxiliary discrete Bayesian network, appropriately constructed for this purpose.

Numerous experiments are conducted using either the conditional linear Gaussian model for reference, or non-Gaussian models for the sake of generality. The algorithm shows remarkable performances both in speed and accuracy when compared with discretization, kernel smoothing or Gaussian assumption. This establishes the algorithm’s efficacy across a spectrum of scenarios, and proving its potential as a robust tool for hybrid Bayesian network inferences.

7.1 Bilateral Grants with Industry

Some on the ongoing PhD thesis are developed within bilareal contract with industry for PhD advisory such as

• Airbus CR&T for the PhD thesis of Marek Felsoci.

In addition two post-docs, namely Maksym Shpakovych and Marvin Lasserre, are funded by the "plan de relance".

8.1 International initiatives

8.2 European initiatives

8.3 National initiatives

8.4 Regional initiatives

• Title:
  HPC-Ecosystem
• Duration:
  2018 – 2023
• Coordinator:
  Emmanuel Agullo
• Concace contact:
  Emmanuel Agullo
• Partners:
  • STORM, TADAAM, TOPAL from Inria Bordeaux Sud-Ouest
  • Airbus Central R&T
  • CEA Commissariat à l’énergie atomique et aux énergies alternatives
• Summary:
  Numerical simulation is today integrated in all cycles of scientific design and studies, whether academic or industrial, to predict or understand the behavior of complex phenomena often coupled or multi-physical. The quality of the prediction requires having precise and adapted models, but also to have computation algorithms efficiently implemented on computers with architectures in permanent evolution. Given the ever increasing size and sophistication of simulations implemented, the use of parallel computing on computers with up to several hundred thousand computing cores and consuming / generating massive volumes of data becomes unavoidable; this domain corresponds to what is now called High Performance Computing (HPC). On the other hand, the digitization of many processes and the proliferation of connected objects of all kinds generate ever-increasing volumes of data that contain multiple valuable information; these can only be highlighted through sophisticated treatments; we are talking about Big Data. The intrinsic complexity of these digital treatments requires a holistic approach with collaborations of multidisciplinary teams capable of mastering all the scientific skills required for each component of this chain of expertise. To have a real impact on scientific progress and advances, these skills must include the efficient management of the massive number of compute nodes using programming paradigms with a high level of expressiveness, exploiting high-performance communications layers, effective management for intensive I / O, efficient scheduling mechanisms on platforms with a large number of computing units and massive I / O volumes, innovative and powerful numerical methods for analyzing volumes of data produced and efficient algorithms that can be integrated into applications representing recognized scientific challenges with high societal and economic impacts. The project we propose aims to consider each of these links in a consistent, coherent and consolidated way. For this purpose, we propose to develop a unified Execution Support (SE) for large-scale numerical simulation and the processing of large volumes of data. We identified four Application Challenges (DA) identified by the Nouvelle-Aquitaine region that we propose to carry over this unified support. We will finally develop four Methodological Challenges (CM) to evaluate the impact of the project. This project will make a significant contribution to the emerging synergy on the convergence between two yet relatively distinct domains, namely High Performance Computing (HPC) and the processing, management of large masses of data (Big Data); this project is therefore clearly part of the emerging field of High Performance Data Analytics (HPDA).

9.1 Promoting scientific activities

9.2 Teaching - Supervision - Juries

10.1 Major publications

• 1 articleE.Emmanuel Agullo, B.Bérenger Bramas, O.Olivier Coulaud, E.Eric Darve, M.Matthias Messner and T.Toru Takahashi. Task-Based FMM for Multicore Architectures.SIAM Journal on Scientific Computing3612014, 66-93HALDOI
• 2 articleE.Emmanuel Agullo, A.Alfredo Buttari, A.Abdou Guermouche, J.Julien Herrmann and A.Antoine Jego. Task-based parallel programming for scalable matrix product algorithms.ACM Transactions on Mathematical Software2023HAL
• 3 articleE.Emmanuel Agullo, L.Luc Giraud and L.Louis Poirel. Robust preconditioners via generalized eigenproblems for hybrid sparse linear solvers.SIAM Journal on Matrix Analysis and Applications4022019, 417–439HALDOI
• 4 miscE.Emily Bourne, P.Philippe Leleux, K.Katharina Kormann, C.Carola Kruse, V.Virginie Grandgirard, Y.Yaman Güçlü, M. J.Martin J Kühn, U.Ulrich Rüde, E.Eric Sonnendrücker and E.Edoardo Zoni. Solver comparison for Poisson-like equations on tokamak geometries.September 2022HAL
• 5 articleB.Bérenger Bramas, O.Olivier Coulaud and G.Guillaume Sylvand. Time-domain BEM for the wave equation on distributed-heterogeneous architectures: A blocking approach.Parallel Computing49July 2015, 66-82HALDOI
• 6 articleF.Fabien Casenave, A.Alexandre Ern and G.Guillaume Sylvand. Coupled BEM-FEM for the convected Helmholtz equation with non-uniform flow in a bounded domain.Journal of Computational Physics257AJanuary 2014, 627-644HALDOI
• 7 articleS.Siegfried Cools, E. F.Emrullah Fatih Yetkin, E.Emmanuel Agullo, L.Luc Giraud and W.Wim Vanroose. Analyzing the Effect of Local Rounding Error Propagation on the Maximal Attainable Accuracy of the Pipelined Conjugate Gradient Method.SIAM Journal on Matrix Analysis and Applications391March 2018, 426 - 450HALDOI
• 8 articleD. A.Daniele Antonio Di Pietro, P.Pierre Matalon, P.Paul Mycek and U.Ulrich Rüde. High-order multigrid strategies for HHO discretizations of elliptic equations.Numerical Linear Algebra with ApplicationsJune 2022HALDOI
• 9 articleL.Luc Giraud, Y.-F.Yan-Fei Jing and Y.Yanfei Xiang. A block minimum residual norm subspace solver with partial convergence management for sequences of linear systems.SIAM Journal on Matrix Analysis and Applications4322022, 710-739HALDOI
• 10 miscY.Yongseok Jang, J.Jerome Bonelle, C.Carola Kruse, F.Frank Hülsemann and U.Ulrich Ruede. Fast Linear Solvers for Incompressible CFD Simulations with Compatible Discrete Operator Schemes.April 2023HAL
• 11 articleM.Marc Odunlami, V.Vincent Le Bris, D.Didier Bégué, I.Isabelle Baraille and O.Olivier Coulaud. A-VCI: A flexible method to efficiently compute vibrational spectra.Journal of Chemical Physics14621June 2017HALDOI

10.2 Publications of the year

10.3 Cited publications

• 31 articleE.E. Agullo, O.O. Aumage, B.B. Bramas, O.O. Coulaud and S.S. Pitoiset. Bridging the gap between openMP and task-based runtime systems for the fast multipole method.IEEE Transactions on Parallel and Distributed Systems28102017DOIback to text
• 32 articleE.Emmanuel Agullo, B.Bérenger Bramas, O.Olivier Coulaud, E.Eric Darve, M.Matthias Messner and T.Toru Takahashi. Task-Based FMM for Multicore Architectures.SIAM Journal on Scientific Computing3612014, 66-93HALDOIback to text
• 33 articleE.Emmanuel Agullo, B.Berenger Bramas, O.Olivier Coulaud, E.Eric Darve, M.Matthias Messner and T.Toru Takahashi. Task-based FMM for heterogeneous architectures.Concurrency and Computation: Practice and Experience289jun 2016, 2608--2629URL: http://doi.wiley.com/10.1002/cpe.3723DOIback to text
• 34 articleE.Emmanuel Agullo, S.Siegfried Cools, E.Emrullah Fatih-Yetkin, L.Luc Giraud, N.Nick Schenkels and W.Wim Vanroose. On soft errors in the conjugate gradient method: sensitivity and robust numerical detection.SIAM Journal on Scientific Computing426November 2020HALDOIback to text
• 35 articleE.Emmanuel Agullo, E.Eric Darve, L.Luc Giraud and Y.Yuval Harness. Low-Rank Factorizations in Data Sparse Hierarchical Algorithms for Preconditioning Symmetric Positive Definite Matrices.SIAM Journal on Matrix Analysis and Applications394October 2018, 1701-1725HALback to text
• 36 techreportE.Emmanuel Agullo, M.Marek Felšöci and G.Guillaume Sylvand. A comparison of selected solvers for coupled FEM/BEM linear systems arising from discretization of aeroacoustic problems: literate and reproducible environment.RT-0513Inria Bordeaux Sud-OuestJune 2021, 100HALback to text
• 37 articleE.Emmanuel Agullo, L.Luc Giraud and Y.-F.Y-F Jing. Block GMRES method with inexact breakdowns and deflated restarting.SIAM Journal on Matrix Analysis and Applications3542014, 1625--1651back to text
• 38 articleE.Emmanuel Agullo, L.Luc Giraud and L.Louis Poirel. Robust preconditioners via generalized eigenproblems for hybrid sparse linear solvers.SIAM Journal on Matrix Analysis and Applications4022019, 417--439HALDOIback to text
• 39 techreportP.Pierre Blanchard, O.Olivier Coulaud and E.Eric Darve. Fast hierarchical algorithms for generating Gaussian random fields.8811Inria Bordeaux Sud-OuestDecember 2015HALback to textback to text
• 40 phdthesisP.Pierre Blanchard. Fast hierarchical algorithms for the low-rank approximation of matrices, with applications to materials physics, geostatistics and data analysis.Bordeaux2017, URL: https://tel.archives-ouvertes.fr/tel-01534930back to text
• 41 techreportS.Steffen Börm, L.Lars Grasedyck and W.Wolfgang Hackbusch. Hierarchical Matrices.2003, 1--173back to text
• 42 articleA.Alfredo Buttari, J.Julien Langou, J.Jakub Kurzak and J.Jack Dongarra. Parallel tiled QR factorization for multicore architectures.Concurrency and Computation: Practice and Experience20132008, 1573--1590back to text
• 43 articleE.Erin Carson, N. J.Nicholas J. Higham and S.Srikara Pranesh. Three-Precision GMRES-Based Iterative Refinement for Least Squares Problems.SIAM Journal on Scientific Computing426January 2020, A4063--A4083DOIback to text
• 44 articleF.Fabien Casenave, A.Alexandre Ern and G.Guillaume Sylvand. Coupled BEM-FEM for the convected Helmholtz equation with non-uniform flow in a bounded domain.Journal of Computational Physics257A23 pages, 9 figuresJanuary 2014, 627-644HALDOIback to text
• 45 bookA.Andrzej Cichocki, R.Rafal Zdunek, A. H.Anh Huy Phan and S.-i.Shun-ichi Amari. Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation.Wiley2009back to text
• 46 articleS.Siegfried Cools, E. F.Emrullah Fatih Yetkin, E.Emmanuel Agullo, L.Luc Giraud and W.Wim Vanroose. Analyzing the Effect of Local Rounding Error Propagation on the Maximal Attainable Accuracy of the Pipelined Conjugate Gradient Method.SIAM Journal on Matrix Analysis and Applications391March 2018, 426 - 450HALDOIback to text
• 47 techreportO.Olivier Coulaud, A. A.Alain A. Franc and M.Martina Iannacito. Extension of Correspondence Analysis to multiway data-sets through High Order SVD: a geometric framework.RR-9429Inria Bordeaux - Sud-Ouest ; InraeNovember 2021HALback to text
• 48 techreportO.Olivier Coulaud, L.Luc Giraud and M.Martina Iannacito. On some orthogonalization schemes in Tensor Train format.RR-9491Inria Bordeaux - Sud-OuestNovember 2022HALback to text
• 49 phdthesisA.Aurelien Falco. Combler l'écart entre $$-Matrices et méthodes directes creuses pour la résolution de systèmes linéaires de grandes tailles.Université de BordeauxJune 2019HALback to text
• 50 articleA. A.Alain A. Franc, P.Pierre Blanchard and O.Olivier Coulaud. Nonlinear mapping and distance geometry.Optimization Letters1422020, 453-467HALDOIback to text
• 51 bookN.Nicolas Gillis. Nonnegative Matrix Factorization.Society for Industrial and Applied MathematicsJanuary 2020DOIback to text
• 52 techreportL.Luc Giraud, Y.-F.Yan-Fei Jing and Y.Yanfei Xiang. A block minimum residual norm subspace solver for sequences of multiple left and right-hand side linear systems.RR-9393Inria Bordeaux Sud-OuestFebruary 2021, 60HALback to text
• 53 articleL.Lars Grasedyck, W.Wolfgang Hackbusch and B.Bericht Nr. An Introduction to Hierachical ( H - ) Rank and TT - Rank of Tensors with Examples.Computational Methods in Applied Mathematics113292011, 291--304back to text
• 54 articleB.Brian Gunter and R.Robert Van De Geijn. Parallel out-of-core computation and updating of the QR factorization.ACM Transactions on Mathematical Software (TOMS)3112005, 60--78back to text
• 55 bookW.Wolfgang Hackbusch. Hierarchical Matrices: Algorithms and Analysis.Springer Publishing Company, Incorporated2015back to text
• 56 articleN.Nathan Halko, P.-G. G.Per-Gunnar G. Martinsson and J. A.Joel A. Tropp. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions.SIAM Review5322011, 217--288URL: http://arxiv.org/abs/0909.4061DOIback to text
• 57 articleT. G.Tamara G. Kolda and B. W.Brett W. Bader. Tensor Decompositions and Applications.SIAM Review513aug 2009, 455--500URL: http://epubs.siam.org/doi/abs/10.1137/07070111XDOIback to text
• 58 articleC.Carola Kruse, V.Vincent Darrigrand, N.Nicolas Tardieu, M.Mario Arioli and U.Ulrich Rüde. Application of an iterative Golub-Kahan algorithm to structural mechanics problems with multi-point constraints.Adv. Model. Simul. Eng. Sci.712020, 45URL: https://doi.org/10.1186/s40323-020-00181-2DOIback to text
• 59 articleC.Carola Kruse, M.Masha Sosonkina, M.Mario Arioli, N.Nicolas Tardieu and U.Ulrich Rüde. Parallel solution of saddle point systems with nested iterative solvers based on the Golub-Kahan Bidiagonalization.Concurr. Comput. Pract. Exp.33112021, URL: https://doi.org/10.1002/cpe.5914DOIback to text
• 60 articleV.Vincent Le Bris, M.Marc Odunlami, D.Didier Bégué, I.Isabelle Baraille and O.Olivier Coulaud. Using computed infrared intensities for the reduction of vibrational configuration interaction bases.Phys. Chem. Chem. Phys.22132020, 7021-7030URL: http://dx.doi.org/10.1039/D0CP00593BDOIback to text
• 61 phdthesisB.Benôit Lizé. Résolution directe rapide pour les éléments finis de frontière en électromagnétisme et acoustique : $$-Matrices. Parallélisme et applications industrielles.Université Paris-Nord - Paris XIIIJune 2014HALback to text
• 62 articleP.-G.Per-Gunnar Martinsson and J.Joel Tropp. Randomized Numerical Linear Algebra: Foundations & Algorithms.2020, URL: http://arxiv.org/abs/2002.01387back to text
• 63 articleM.Marc Odunlami, V.Vincent Le Bris, D.Didier Bégué, I.Isabelle Baraille and O.Olivier Coulaud. A-VCI: A flexible method to efficiently compute vibrational spectra.The Journal of Chemical Physics14621june 2017, 214108URL: http://aip.scitation.org/doi/10.1063/1.4984266DOIback to text
• 64 articleI. V.I. V. Oseledets. Tensor-Train Decomposition.SIAM Journal on Scientific Computing335January 2011, 2295--2317URL: https://doi.org/10.1137/090752286DOIback to text
• 65 phdthesisL.Louis Poirel. Algebraic domain decomposition methods for hybrid (iterative/direct) solvers.Université de BordeauxNovember 2018HALback to text
• 66 articleJ.-R.Jean-René Poirier, O.Olivier Coulaud and O.Oguz Kaya. Fast BEM Solution for 2-D Scattering Problems Using Quantized Tensor-Train Format.IEEE Transactions on Magnetics563March 2020, 1-4HALDOIback to text
• 67 phdthesisG.Guillaume Sylvand. La méthode multipôle rapide en électromagnétisme. Performances, parallélisation, applications.Ecole des Ponts ParisTechJune 2002HALback to text
• 68 techreportN.Nicolas Venkovic, P.Paul Mycek, L.Luc Giraud and O.Olivier Le Maitre. Recycling Krylov subspace strategies for sequences of sampled stochastic elliptic equations.RR-9425Inria Bordeaux - Sud OuestOctober 2021HALback to text