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  37 is based on a data format and computational technique originally used in an out-of-core context  51. 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  24, 25, 23. 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  29.

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 Material physics

Participants: Olivier Coulaud, Pierre Esterie.

Due to the increase of available computer power, new applications in nano science and physics appear such as study of properties of new materials (photovoltaic materials, bio- and environmental sensors, ...), failure in materials, nano-indentation. Chemists, physicists now commonly perform simulations in these fields. These computations simulate systems up to billion of atoms in materials, for large time scales up to several nanoseconds. The larger the simulation, the smaller the computational cost of the potential driving the phenomena, resulting in low precision results. So, if we need to increase the precision, there are two ways to decrease the computational cost. In the first approach, we improve algorithms and their parallelization and in the second way, we will consider a multiscale approach.

A domain of interest is the material aging for the nuclear industry. The materials are exposed to complex conditions due to the combination of thermo-mechanical loading, the effects of irradiation and the harsh operating environment. This operating regime makes experimentation extremely difficult and we must rely on multi-physics and multi-scale modeling for our understanding of how these materials behave in service. This fundamental understanding helps not only to ensure the longevity of existing nuclear reactors, but also to guide the development of new materials for 4th generation reactor programs and dedicated fusion reactors. For the study of crystalline materials, an important tool is dislocation dynamics (DD) modeling. This multiscale simulation method predicts the plastic response of a material from the underlying physics of dislocation motion. DD serves as a crucial link between the scale of molecular dynamics and macroscopic methods based on finite elements; it can be used to accurately describe the interactions of a small handful of dislocations, or equally well to investigate the global behavior of a massive collection of interacting defects.

To explore i.e. to simulate these new areas, we need to develop and/or to improve significantly models, schemes and solvers used in the classical codes. In the project, we want to accelerate algorithms arising in those fields.

We will focus on the following topics

• The interaction between dislocations is long ranged ($O(1/r)$) and anisotropic, leading to severe computational challenges for large-scale simulations. In dislocation codes, the computation of interaction forces between dislocations is still the most CPU time consuming and has to be improved to obtain faster and more accurate simulations.
• In such simulations, the number of dislocations grows while the phenomenon occurs and these dislocations are not uniformly distributed in the domain. This means that strategies to dynamically construct a good load balancing are crucial to acheive high performance.
• From a physical and a simulation point of view, it will be interesting to couple a molecular dynamics model (atomistic model) with a dislocation one (mesoscale model). In such three-dimensional coupling, the main difficulties are firstly to find and characterize a dislocation in the atomistic region, secondly to understand how we can transmit with consistency the information between the two micro and meso scales.

4.2 Co-design of algorithms in scientific applications

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

6.1 New software

7.1 A block minimum residual norm subspace solver with partial convergence management for sequences of linear systems

We are concerned with the iterative solution of linear systems with multiple right-hand sides available one group after another with possibly slowly-varying left-hand sides. For such sequences of linear systems, we first develop a new block minimum norm residual approach that combines two main ingredients. The first component exploits ideas from GCRO-DR [SIAM J. Sci. Comput., 28(5) (2006), pp. 1651–1674], enabling to recycle information from one solve to the next. The second component is the numerical mechanism to manage the partial convergence of the right-hand sides, referred to as inexact breakdown detection in IB-BGMRES [Linear Algebra Appl., 419 (2006), pp. 265–285], that enables the monitoring of the rank deficiency in the residual space basis expanded block-wise. Secondly, for the class of block minimum norm residual approaches, that relies on a block Arnoldi-like equality between the11 search space and the residual space (e.g., any block GMRES or block GCRO variants), we introduce new search space expansion policies defined on novel criteria to detect the partial convergence. These novel detection criteria are tuned to the selected stopping criterion and targeted convergence threshold to best cope with the selected normwise backward error stopping criterion, enabling to monitor the computational effort while ensuring the final accuracy of each individual solution. Numerical experiments are reported to illustrate the numerical and computational features of both the new block Krylov solvers and the new search space block expansion polices.

For more details on this work we refer to  49.

7.2 Direct solution of larger coupled sparse/dense linear systems using low-rank compression on single-node multi-core machines in an industrial context

While hierarchically low-rank compression methods are now commonly available in both dense and sparse direct solvers, their usage for the direct solution of coupled sparse/dense linear systems has been little investigated. The solution of such systems is though central for the simulation of many important physics problems such as the simulation of the propagation of acoustic waves around aircrafts. Indeed, the heterogeneity of the jet flow created by reactors often requires a Finite Element Method (FEM) discretization, leading to a sparse linear system, while it may be reasonable to assume as homogeneous the rest of the space and hence model it with a Boundary Element Method (BEM) discretization, leading to a dense system. In an industrial context, these simulations are often operated on modern multicore workstations with fully-featured linear solvers. Exploiting their low-rank compression techniques is thus very appealing for solving larger coupled sparse/dense systems (hence ensuring a finer solution) on a given multicore workstation, and – of course – possibly do it fast. The standard method performing an efficient coupling of sparse and dense direct solvers is to rely on the Schur complement functionality of the sparse direct solver. However, to the best of our knowledge, modern fully- featured sparse direct solvers offering this functionality return the Schur complement as a non compressed matrix. In this paper, we study the opportunity to process larger systems in spite of this constraint. For that we propose two classes of algorithms, namely multi-solve and multi-factorization, consisting in composing existing parallel sparse and dense methods on well chosen submatrices. An experimental study conducted on a 24 cores machine equipped with 128 GiB of RAM shows that these algorithms, implemented on top of state-of-the-art sparse and dense direct solvers, together with proper low-rank assembly schemes, can respectively process systems of 9 million and 2.5 million total unknowns instead of 1.3 million unknowns with a standard coupling of compressed sparse and dense solvers.

For more details on this work we refer to  30, 31.

7.3 Study of the processor and memory power consumption of coupled sparse/dense solvers

In the aeronautical industry, aeroacoustics is used to model the propagation of acoustic waves in air flows enveloping an aircraft in flight. This for instance allows one to simulate the noise produced at ground level by an aircraft during the takeoff and landing phases, in order to validate that the regulatory environmental standards are met. Unlike most other complex physics simulations, the method resorts to solving coupled sparse/dense systems. In a previous study, we proposed two classes of algorithms for solving such large systems on a relatively small workstation (one or a few multicore nodes) based on compression techniques. The objective of this study is to assess whether the positive impact of the proposed algorithms on time to solution and memory usage translates to the energy consumption as well. Because of the nature of the problem, coupling dense and sparse matrices, and the underlying solutions methods, including dense, sparse direct and compression steps, this yields an interesting processor and memory power profile which we aim to analyze in details.

For more details on this work we refer to  28.

7.4 Task-based randomized singular value decomposition and multidimensional scaling

The multidimensional scaling (MDS) is an important and robust algorithm for representing individual cases of a dataset out of their respective dissimilarities. However, heuristics, possibly trading-off with robustness, are often preferred in practice due to the potentially prohibitive memory and computational costs of the MDS. The recent introduction of random projection techniques within the MDS allowed it to be become competitive on larger test cases. The goal of this manuscript is to propose a high-performance distributed-memory MDS based on random projection for processing data sets of even larger size (up to one million items). We propose a task-based design of the whole algorithm and we implement it within an efficient software stack including state-of-the-art numerical solvers, runtime systems and communication layers. The outcome is the ability to efficiently apply robust MDS to large data sets on modern supercomputers. We assess the resulting algorithm and software stack to the point cloud visualization for analyzing distances between sequencesin metabarcoding.

For more details on this work we refer to  19.

7.5 Guix-HPC Activity Report 2020-2021

Guix-HPC is a collaborative effort to bring reproducible software deployment to scientific workflows and high-performance computing (HPC). Guix-HPC builds upon the GNU Guix software deployment tool and aims to make it a better tool for HPC practitioners and scientists concerned with reproducible research. This report highlights key achievements of Guix-HPC between our previous report a year ago and today, February 2022. This report highlights developments on GNU Guix proper, but also downstream on Guix-Jupyter, the Guix Workflow Language, upstream with Software Heritage integration, as well as experience reports on end-to-end reproducible research article authoring pipelines.

For more details on this work we refer to  19.

7.6 Decentralized in-order execution of a sequential task-based code for shared-memory architectures

The hardware complexity of modern machines makes the design of adequate pro- gramming models crucial for jointly ensuring performance, portability, and productivity in high- performance computing (HPC). Sequential task-based programming models paired with advanced runtime systems allow the programmer to write a sequential algorithm independently of the hard- ware architecture in a productive and portable manner, and let a third party software layer —the runtime system— deal with the burden of scheduling a correct, parallel execution of that algorithm to ensure performance. Many HPC algorithms have successfully been implemented following this paradigm, as a testimony of its effectiveness. Developing algorithms that specifically require fine-grained tasks along this model is still considered prohibitive, however, due to per-task management overhead , forcing the programmer to resort to a less abstract, and hence more complex “task+X” model. We thus investigate the possibility to offer a tailored execution model, trading dynamic mapping for efficiency by using a decentralized, conservative in-order execution of the task flow, while preserving the benefits of relying on the sequential task-based programming model. We propose a formal specification of the execution model as well as a prototype implementation, which we assess on a shared-memory multicore architecture with several synthetic workloads. The results show that under the condition of a proper task mapping supplied by the programmer, the pressure on the runtime system is significantly reduced and the execution of fine-grained task flows is much more efficient.

For more details on this work we refer to  40.

7.7 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  60.

7.8 The backward stable variants of GMRES in variable accuracy

In the context where the representation of the data is decoupled from the arithmetic used to process them, we investigate the backward stability of two backward-stable implementations of the GMRES method, namely the so-called Modified Gram-Schmidt (MGS) and the Householder variants. Considering data may be compressed to alleviate the memory footprint, we are interested in the situation where the leading part of the rounding error is related to the data representation. When the data representation of vectors introduces componentwise perturbations, we show that the existing backward stability analyses of MGS-GMRES and Householder-GMRES still apply. We illustrate this backward stability property in a practical context where an agnostic lossy compressor is employed and enables the reduction of the memory requirement to store the orthonormal Arnoldi basis or the Householder reflectors. Although technical arguments of the theoretical backward stability proofs do not readily apply to the situation where only the normwise relative perturbations of the vector storage can be controlled, we show experimentally that the backward stability is maintained; that is, the attainable normwise backward error is of the same order as the normwise perturbations induced by the data storage. We illustrate it with numerical experiments in two practical different contexts. The first one corresponds to the use of an agnostic compressor where vector compression is controlled normwise. The second one arises in the solution of tensor linear systems, where low-rank tensor approximations based on Tensor-Train is considered to tackle the curse of dimensionality.

For more details on this work we refer to 20.

7.9 A robust GMRES algorithm in Tensor Train format

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. For more details on this work we refer to 21.

7.10 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.

For more details on this work we refer to 22.

7.11 High-order multigrid strategies for HHO discretizations of elliptic equations

This study compares various multigrid strategies for the fast solution of elliptic equations discretized by the Hybrid High-Order method. Combinations of h-, p-and hp-coarsening strategies are considered, combined with diverse intergrid transfer operators. Comparisons are made experimentally on 2D and 3D test cases, with structured and unstructured meshes, and with nested and non-nested hierarchies. Advantages and drawbacks of each strategy are discussed for each case to establish simplified guidelines for the optimization of the time to solution.

For more details on this work we refer to  44.

8.1 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.
• IFPEN for the PhD of Aboul-Karim Mohamed El Maarouf,

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

9.1 European initiatives

9.2 National initiatives

10.1 Promoting scientific activities

10.2 Teaching - Supervision - Juries

10.3 Popularization

11.1 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, 66-93
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• 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 Software2023
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• 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, 417–439
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• 4 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, 66-82
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• 5 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, 627-644
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• 6 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 - 450
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• 7 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 2022
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• 8 articleA.Alain Franc, P.Pierre Blanchard and O.Olivier Coulaud. Nonlinear mapping and distance geometry.Optimization Letters1422020, 453-467
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• 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, 710-739
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• 10 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 2017
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11.2 Publications of the year

11.3 Cited publications

• 23 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 Systems28102017
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• 24 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, 66-93
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• 25 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.3723
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• 26 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 2020
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• 27 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, 1701-1725
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• 28 techreportE.Emmanuel Agullo, M.Marek Felšöci, A.Amina Guermouche, H.Hervé Mathieu, G.Guillaume Sylvand and B.Bastien Tagliaro. Study of the processor and memory power consumption of coupled sparse/dense solvers.RR-9463Inria Bordeaux Sud-OuestFebruary 2022, 17
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• 29 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, 100
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• 30 inproceedingsE.Emmanuel Agullo, M.Marek Felšöci and G.Guillaume Sylvand. Direct solution of larger coupled sparse/dense linear systems using low-rank compression on single-node multi-core machines in an industrial context.IPDPS 2022 - 36th IEEE International Parallel and Distributed Processing SymposiumLyon, FranceIEEEMay 2022, 11
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• 31 techreportE.Emmanuel Agullo, M.Marek Felšöci and G.Guillaume Sylvand. Direct solution of larger coupled sparse/dense linear systems using low-rank compression on single-node multi-core machines in an industrial context.RR-9453Inria Bordeaux Sud-OuestFebruary 2022, 25
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• 32 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--1651
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• 33 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--439
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• 34 techreportP.Pierre Blanchard, O.Olivier Coulaud and E.Eric Darve. Fast hierarchical algorithms for generating Gaussian random fields.8811Inria Bordeaux Sud-OuestDecember 2015
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• 35 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-01534930
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• 36 techreportS.Steffen Börm, L.Lars Grasedyck and W.Wolfgang Hackbusch. Hierarchical Matrices.2003, 1--173
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• 37 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--1590
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• 38 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--A4083
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• 39 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, 627-644
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• 40 techreportC.Charly Castes, E.Emmanuel Agullo, O.Olivier Aumage and E.Emmanuelle Saillard. Decentralized in-order execution of a sequential task-based code for shared-memory architectures.RR-9450Inria Bordeaux - Sud Ouest2022, 30URL: https://hal.inria.fr/hal-03547334
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• 41 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.Wiley2009
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• 42 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 - 450
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• 43 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 2021
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• 44 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 2022
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• 45 phdthesisA.Aurelien Falco. Combler l'écart entre $$-Matrices et méthodes directes creuses pour la résolution de systèmes linéaires de grandes tailles.Université de BordeauxJune 2019
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• 46 articleA. A.Alain A. Franc, P.Pierre Blanchard and O.Olivier Coulaud. Nonlinear mapping and distance geometry.Optimization Letters1422020, 453-467
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• 47 bookN.Nicolas Gillis. Nonnegative Matrix Factorization.Society for Industrial and Applied MathematicsJanuary 2020
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• 48 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, 60
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• 49 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, 710-739
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• 50 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--304
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• 51 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--78
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• 52 bookW.Wolfgang Hackbusch. Hierarchical Matrices: Algorithms and Analysis.Springer Publishing Company, Incorporated2015
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• 53 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/0909.4061
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• 54 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/10.1137/07070111X
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