2025Activity 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 $R^{\mathrm{n‌}}$ 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 71​​​‌ is based on a‌ data format and computational‌​‌ technique originally used in​​ an out-of-core context 82​​​‌. 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 60,​​​‌ 61, 59.‌ 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​​​‌ 64.

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

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 73.‌ 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 $f^{\mathrm{2‌}}$ for BEM and ${\mathrm{f‌‌}}^3$ 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 Awards

• We‌ are delighted to welcome‌​‌ a new member to​​ the team. Clément Guillet​​​‌ has joined us as‌ an ISFP for the‌​‌ 2025 campaign.
• A new​​ release of Composyx has​​​‌ been released to the‌ public.
• The Concace Steering‌​‌ Committee approved the continuation​​ of the team, subject​​​‌ to the completion of‌ the ongoing Inria scientific‌​‌ evaluation process.

6.1 Latest software‌ developments

7.1 Error Estimates for‌ Sparse Tensor Products of‌​‌ B-spline Approximation Spaces

This​​ work introduces and analyzes​​​‌ B-spline approximation spaces defined‌ on general geometric domains‌​‌ obtained through a mapping​​ from a parameter domain.​​​‌ These spaces are constructed‌ as sparse-grid tensor products‌​‌ of univariate spaces in​​ the parameter domain and​​​‌ are mapped to the‌ physical domain via a‌​‌ geometric parametrization. Both the​​ univariate approximation spaces and​​​‌ the geometric mapping are‌ built using maximally smooth‌​‌ B-splines. We construct two​​ such spaces, employing either​​​‌ the sparse-grid combination technique‌ or the hierarchical subspace‌​‌ decomposition of sparse-grid tensor​​ products, and we prove​​​‌ their mathematical equivalence. Furthermore,‌ we derive approximation error‌​‌ estimates and inverse inequalities​​ that highlight the advantages​​​‌ of sparse-grid tensor products.‌ Specifically, under suitable regularity‌​‌ assumptions on the solution,​​ these spaces achieve the​​​‌ same approximation order as‌ standard tensor product spaces‌​‌ while using significantly fewer​​ degrees of freedom. Additionally,​​​‌ our estimates indicate that,‌ in the case of‌​‌ non-tensor-product domains, stronger regularity​​ assumptions on the solution-particularly​​​‌ concerning isotropic (non-mixed) derivatives-are‌ required to achieve optimal‌​‌ convergence rates compared to​​ sparse-grid methods defined on​​​‌ tensor-product domains.

For more‌ details on this work‌​‌ we refer to  48​​​‌.

7.2 On some​ orthogonalization schemes in Tensor​‌ Train format

In the​​ framework of tensor spaces,​​​‌ we consider orthogonalization algorithms​ to generate an orthogonal​‌ basis of a tensor​​ subspace from a set​​​‌ of linearly independent tensors.​ All variants, except for​‌ the Householder transformation, are​​ straightforward extensions of well-known​​​‌ algorithms in matrix computation​ to tensors. In particular,​‌ we experimentally study the​​ loss of orthogonality of​​​‌ six orthogonalization methods: Classical​ and Modified Gram-Schmidt with​‌ (CGS2, MGS2) and without​​ (CGS, MGS) re-orthogonalization, the​​​‌ Cholesky-QR, and the Householder​ transformation. To overcome the​‌ curse of dimensionality, we​​ represent tensors with a​​​‌ low-rank approximation using the​ Tensor Train (TT) formalism.​‌ Additionally, we introduce recompression​​ steps in the standard​​​‌ algorithm outline through the​ TT-round method at a​‌ prescribed accuracy. After describing​​ the structure and properties​​​‌ of the algorithms, we​ illustrate their loss of​‌ orthogonality with numerical experiments.​​ Although no formal proof​​​‌ exists at this time,​ we observe very clearly​‌ that the well-established properties​​ verified over decades of​​​‌ research by the round​ error analysis community in​‌ matrix computation appear to​​ extend to the case​​​‌ of low-rank tensors, with​ the unit round-off replaced​‌ by the TT-round accuracy.​​ The computational analysis for​​​‌ each orthogonalization scheme in​ terms of memory requirements​‌ and computational complexity, measured​​ as a function of​​​‌ the number of TT-round​ operations, which happens to​‌ be the most computationally​​ expensive operation, completes the​​​‌ study.

For more details​ on this work we​‌ refer to  18.​​

7.3 A note on​​​‌ TT-GMRES for the solution​ of parametric linear systems​‌

We study the solution​​ of linear systems with​​​‌ tensor product structure using​ the Generalized Minimal RESidual​‌ (GMRES) algorithm. To manage​​ the computational complexity of​​​‌ high dimensional problems our​ approach relies on low-rank​‌ tensor representation, focusing specifically​​ on the Tensor Train​​​‌ format. We implement and​ experimentally study the TT-GMRES​‌ algorithm. Our analysis bridges​​ the heuristic methods proposed​​​‌ for TT-GMRES by Dolgov​ in [Russian J. Numer.​‌ Anal. Math. Modelling, 28​​ (2013), pp. 149–172] and​​​‌ the theoretical framework of​ inexact GMRES by Simoncine​‌ and Szyld [SIAM J.​​ Sci. Comput. 25 (2003),​​​‌ pp. 454–477]. This approach​ is particularly relevant in​‌ a scenario where a​​ $(d+\mathrm{1‌})$-dimensional problem arises​ from concatenating a sequence​‌ of $d$-dimensional problems,​​ as in the case​​​‌ of a parametric linear​ operator or parametric right-hand-side​‌ formulation. Thus, we provide​​ backward error bounds that​​​‌ link the accuracy of​ the computed $(\mathrm{d‌}+1)$-dimensional​​ solution to the numerical​​​‌ quality of the extracted​ $d$-dimensional solutions. This​‌ facilitates the prescription of​​ a convergence threshold ensuring​​​‌ that the $d$-dimensional​ solutions extracted from the​‌ $(d+\mathrm{1})$-dimensional result have​​​‌ the desired accuracy once​ the solver converges. We​‌ illustrate these results with​​ academic examples across varying​​​‌ dimensions and sizes. Our​ experiments indicate that the​‌ TT-GMRES retains the theoretical​​ rounding error properties observed​​​‌ in matrix-based GMRES.

For​ more details on this​‌ work we refer to​​  17.

7.4 Solving​​ eigenvalue problems in high​​​‌ dimensions using contour integration‌ and Tensor Train format‌​‌

In high-dimensional settings, solving​​ eigenvalue problems is hindered​​​‌ by the curse of‌ dimensionality, particularly when only‌​‌ a subset of eigenpairs​​ within a prescribed spectral​​​‌ interval is sought. In‌ this work, we investigate‌​‌ an adaptation of the​​ FEAST algorithm, originally developed​​​‌ for symmetric eigenproblems based‌ on contour integration, to‌​‌ computations where both operators​​ and vectors are represented​​​‌ in the Tensor Train‌ (TT) format. This representation‌​‌ drastically reduces memory and​​ computational demands. We introduce​​​‌ an adaptive scheme for‌ determining the projection subspace‌​‌ dimension by incorporating a​​ rank-revealing Modified Gram–Schmidt procedure​​​‌ with pivoting tailored to‌ TT-vectors. A perturbation-based analysis‌​‌ provides explicit bounds on​​ the attainable residual accuracy,​​​‌ from which we derive‌ a robust stopping criterion‌​‌ for the proposed TT-FEAST​​ algorithm. Moreover, we design​​​‌ a continuation strategy that‌ gradually refines convergence and‌​‌ rounding tolerances to effectively​​ control memory growth during​​​‌ iterations. To demonstrate the‌ effectiveness of TT-FEAST as‌​‌ a viable alternative to​​ existing high-dimensional eigensolvers when​​​‌ a few eigenvalues are‌ required, we present numerical‌​‌ experiments on problems up​​ to twelve dimensions, including​​​‌ the Laplacian and a‌ vibrational Hamiltonian operator.

For‌​‌ more details on this​​ work we refer to​​​‌  44.

7.5 A‌ Tensor Train solver for‌​‌ the Magnetic Moment Method​​

In this work, we​​​‌ focus on enhancing the‌ computational efficiency of the‌​‌ Magnetic Moment Method (MMM)​​ using low-rank tensor representations,​​​‌ specifically the Tensor Train‌ (TT) formats. By transforming‌​‌ the dense linear system​​ generated by MMM into​​​‌ TT format, we achieve‌ significant reductions in both‌​‌ computational cost and memory​​ usage. Furthermore, when the​​​‌ problem structure allows—such as‌ in cases with regular‌​‌ grids and binary-compatible sizes—the​​ Quantized Tensor Train (QTT)​​​‌ format is employed to‌ exploit additional compression through‌​‌ quantization, leading to even​​ larger performance gains. The​​​‌ proposed methods are tested‌ on several problems involving‌​‌ a ferromagnetic part with​​ a regular mesh. For​​​‌ simple cases, our approach‌ demonstrates superior performance compared‌​‌ to traditional techniques. As​​ the problem complexity increases,​​​‌ the TT- and QTT-based‌ methods remaincompetitive, maintaining efficiency‌​‌ while addressing the added​​ computational challenges.

This work​​​‌ is part of Amine‌ Zekri's PhD thesis and‌​‌ is carried out in​​ the context of the​​​‌ TensoVim ANR project. For‌ more details on this‌​‌ work we refer to​​  25.

7.6 Generalized​​​‌ Golub–Kahan Bidiagonalization for Nonsymmetric‌ Saddle-Point Systems

The generalized‌​‌ Golub–Kahan bidiagonalization has been​​ used to solve saddle-point​​​‌ systems where the leading‌ block is symmetric and‌​‌ positive definite. We extend​​ this iterative method for​​​‌ the case where the‌ symmetry condition no longer‌​‌ holds. We do so​​ by relying on the​​​‌ known connection the algorithm‌ has with the conjugate‌​‌ gradient method and following​​ the line of reasoning​​​‌ that adapts the latter‌ into the full orthogonalization‌​‌ method. We propose appropriate​​ stopping criteria based on​​​‌ the residual and an‌ estimate of the energy‌​‌ norm for the error​​ associated with the primal​​​‌ variable. Numerical comparison with‌ GMRES highlights the advantages‌​‌ of our proposed strategy​​​‌ regarding its low memory​ requirements and the associated​‌ implications.

For more details​​ on this work we​​​‌ refer to  20.​

7.7 A note on​‌ the partial convergence management​​ for the solution of​​​‌ symmetric linear systems with​ multiple right-hand sides

We​‌ consider the solution of​​ large sparse symmetric linear​​​‌ systems with multiple right-hand​ sides available simultaneously. Based​‌ on the partial convergence​​ detection and management, described​​​‌ in IB-BGMRES [Linear Algebra​ Appl., 419 (2006), pp.​‌ 265-285] and the breakdown-free​​ idea discussed in [BIT​​​‌ Numer. Math., 57 (2017),​ pp. 379-403], the block​‌ conjugate residual and block​​ conjugate gradient methods with​​​‌ partial convergence management are​ proposed. It enable to​‌ select the directions to​​ use for extending the​​​‌ search space from one​ iteration to the next​‌ by choosing the directions​​ that contribute the most​​​‌ to the residual norms.​ We illustrate the numerical​‌ and computational benefits of​​ these two novel block​​​‌ conjugate direction variants on​ a set of simple​‌ academic examples enabling reproducible​​ experiments.

For more details​​​‌ on this work we​ refer to  47.​‌

7.8 A Scalable and​​ Parameter-Robust Preconditioner for a​​​‌ Second Gradient of Dilation​ Regularization Applied to a​‌ Mechanics Problem

We propose​​ a rigorous analysis of​​​‌ the second gradient of​ dilation regularization as it​‌ is used in many​​ geomechanical applications. In this​​​‌ method, a new primal​ unknown, modeling the trace​‌ of the displacement gradient,​​ is introduced leading to​​​‌ a saddle-point system. The​ resulting balance equations depend​‌ on parameters that range​​ on several orders of​​​‌ magnitude. We prove the​ well-posedness of the continuous​‌ and discrete problems using​​ parameter dependent norms. This​​​‌ allows the definition of​ a robust block preconditioner​‌ for the discrete problem​​ with respect to the​​​‌ parameter variations and to​ mesh refinement. Numerical results​‌ confirm our theoretical findings.​​

For more details on​​​‌ this work we refer​ to  49.

7.9​‌ Neural network preconditioning: a​​ case study for the​​​‌ solution of the parametric​ Helmholtz equation

This work​‌ presents a hybrid numerical​​ approach for solving linear​​​‌ systems arising from the​ discretization of the two-dimensional​‌ parametric Helmholtz equation. A​​ convolutional neural network based​​​‌ on the U-Net architecture​ is trained in an​‌ unsupervised manner to approximate​​ the inverse of the​​​‌ discretized Helmholtz operator, using​ a loss function involving​‌ the residual norm of​​ the linear system. The​​​‌ trained network is used​ as a nonlinear preconditioner​‌ within the Flexible GMRES​​ (FGMRES) algorithm. Numerical experiments​​​‌ show that while the​ neural network is not​‌ accurate enough to act​​ as a standalone solver,​​​‌ it significantly improves the​ convergence of FGMRES when​‌ employed as a preconditioner.​​ The neural preconditioner demonstrates​​​‌ robust performance and generalization​ capabilities with respect to​‌ variations in the velocity​​ field and the domain​​​‌ size. Comparisons with classical​ algebraic preconditioners based on​‌ sparsified LU factorizations indicate​​ superior efficiency of the​​​‌ neural approach under equivalent​ conditions. We believe that​‌ the proposed method is​​ not tied to a​​​‌ specific neural architecture and​ can be extended to​‌ other parametric PDEs.

For​​ more details on this​​ work we refer to​​​‌  46.

7.10 Memory-and‌ compute-optimized geometric multigrid GMGPolar‌​‌ for curvilinear coordinate representations​​ -Applications to fusion plasma​​​‌

Tokamak fusion reactors are‌ actively studied as a‌​‌ means of realizing energy​​ production from plasma fusion.​​​‌ However, due to the‌ substantial cost and time‌​‌ required to construct fusion​​ reactors and run physical​​​‌ experiments, numerical experiments are‌ indispensable for understanding plasma‌​‌ physics inside tokamaks, supporting​​ the design and engineering​​​‌ phase, and optimizing future‌ reactor designs. Geometric multigrid‌​‌ methods are optimal solvers​​ for many problems that​​​‌ arise from the discretization‌ of partial differential equations.‌​‌ It has been shown​​ that the multigrid solver​​​‌ GMGPolar solves the 2D‌ gyrokinetic Poisson equation in‌​‌ linear complexity and with​​ only small memory requirements​​​‌ compared to other state-of-the-art‌ solvers. In this paper,‌​‌ we present a completely​​ refactored and object-oriented version​​​‌ of GMGPolar which offers‌ two different matrix-free implementations.‌​‌ Among other things, we​​ leverage the Sherman-Morrison formula​​​‌ to solve cyclic tridiagonal‌ systems from circular line‌​‌ solvers without additional fill-in​​ and we apply reordering​​​‌ to optimize cache access‌ of circular and radial‌​‌ smoothing operations. With the​​ Give approach, memory requirements​​​‌ are further reduced and‌ speedups of four to‌​‌ seven are obtained for​​ usual test cases. For​​​‌ the Take approach, speedups‌ of 16 to 18‌​‌ can be attained.

For​​ more details on this​​​‌ work we refer to‌  50.

7.11 Complexity‌​‌ analysis and scalability of​​ a matrix-free extrapolated geometric​​​‌ multigrid solver for curvilinear‌ coordinates representations from fusion‌​‌ plasma applications

Tokamak fusion​​ reactors are promising alternatives​​​‌ for future energy production.‌ Gyrokinetic simulations are important‌​‌ tools to understand physical​​ processes inside tokamaks and​​​‌ to improve the design‌ of future plants. In‌​‌ gyrokinetic codes such as​​ Gysela, these simulations involve​​​‌ at each time step‌ the solution of a‌​‌ gyrokinetic Poisson equation defined​​ on disk-like cross sections.​​​‌ The authors of [14,15]‌ proposed to discretize a‌​‌ simplified differential equation using​​ symmetric finite differences derived​​​‌ from the resulting energy‌ functional and to use‌​‌ an implicitly extrapolated geometric​​ multigrid scheme tailored to​​​‌ problems in curvilinear coordinates.‌ In this article, we‌​‌ extend the discretization to​​ a more realistic partial​​​‌ differential equation and demonstrate‌ the optimal linear complexity‌​‌ of the proposed solver,​​ in terms of computation​​​‌ and memory. We provide‌ a general framework to‌​‌ analyze floating point operations​​ and memory usage of​​​‌ matrix-free approaches for stencil-based‌ operators. Finally, we give‌​‌ an efficient matrix-free implementation​​ for the considered solver​​​‌ exploiting a task-based multithreaded‌ parallelism which takes advantage‌​‌ of the disk-shaped geometry​​ of the problem. We​​​‌ demonstrate the parallel efficiency‌ for the solution of‌​‌ problems of size up​​ to 50 million unknowns.​​​‌

For more details on‌ this work we refer‌​‌ to  23.

7.12​​ Fault-tolerant numerical iterative algorithms​​​‌ at scale

This work‌ investigates how to protect‌​‌ numerical iterative algorithms from​​ all types of errors​​​‌ that can strike at‌ scale: fail-stop errors (a.k.a.‌​‌ failures) and silent errors,​​ striking both as computation​​​‌ errors and memory bit-flips.‌ We combine various techniques:‌​‌ detectors for computation errors,​​​‌ checksums for memory errors,​ and checkpoint/restart for failures.​‌ The objective is to​​ minimize the expected time​​​‌ per iteration of the​ algorithm. We design a​‌ hierarchical pattern that combines​​ and interleaves all these​​​‌ fault-tolerance mechanisms, and we​ determine the optimal periodic​‌ pattern that achieves this​​ objective. We instantiate these​​​‌ results for the performance​ analysis of the Preconditioned​‌ Conjugate Gradient (PCG) algorithm:​​ we report several scenarios​​​‌ where the optimal pattern​ dramatically decreases the overhead​‌ due to error mitigation.​​

For more details on​​​‌ this work we refer​ to  24.

7.13​‌ A filtered multilevel Monte​​ Carlo method for estimating​​​‌ the expectation of cell-centered​ 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 numerical simulators​‌ have inconsistent dimensions across​​ the multilevel hierarchy. This​​​‌ requires the introduction of​ grid transfer operators borrowed​‌ from multigrid methods. By​​ adapting mathematical tools from​​​‌ 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. We then propose​​​‌ filtered MLMC (F-MLMC) estimators​ based on a filtering​‌ mechanism similar to the​​ smoothing process of multigrid​​​‌ methods, and we show​ that 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.​​​‌ Next, the conclusions of​ the spectral analysis are​‌ experimentally verified with a​​ one-dimensional illustration. Finally, the​​​‌ proposed 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 F-MLMC​ estimator compared to both​‌ a crude MC and​​ an unfiltered MLMC estimator.​​​‌

For more details on​ this work we refer​‌ to  15.

7.14​​ Multilevel Monte Carlo methods​​​‌ for ensemble variational data​ assimilation

Ensemble variational data​‌ assimilation relies on ensembles​​ of forecasts to estimate​​​‌ the background error covariance​ matrix B. The ensemble​‌ can be provided by​​ an Ensemble of Data​​​‌ Assimilations (EDA), which runs​ independent perturbed data assimilation​‌ and forecast steps. The​​ accuracy of the ensemble​​​‌ estimator of B is​ strongly limited by the​‌ small ensemble size that​​ is needed to keep​​​‌ the EDA computationally affordable.​ We investigate here the​‌ potential of the multilevel​​ Monte Carlo (MLMC) method,​​​‌ a type of multifidelity​ Monte Carlo method, to​‌ improve the accuracy of​​ the standard Monte-Carlo estimator​​​‌ of B while keeping​ the computational cost of​‌ ensemble generation comparable. MLMC​​ exploits the availability of​​​‌ a range of discretization​ grids, thus shifting part​‌ of the computational work​​ from the original assimilation​​​‌ grid to coarser ones.​ MLMC differs from the​‌ mere averaging of statistical​​ estimators, as it ensures​​ that no bias from​​​‌ the coarse resolution grids‌ is introduced in the‌​‌ estimation. The implications for​​ ensemble variational data assimilation​​​‌ systems based on EDAs‌ are discussed. Numerical experiments‌​‌ with a quasi-geostrophic model​​ demonstrate the potential of​​​‌ the approach, as MLMC‌ yields more accurate background‌​‌ error covariances and reduced​​ analysis error. The challenges​​​‌ involved in cycling a‌ multilevel variational data assimilation‌​‌ system are identified and​​ discussed.

For more details​​​‌ on this work we‌ refer to  19.‌​‌

7.15 Convergence analysis of​​ overlapping domain decomposition preconditioners​​​‌ for nonlinear problems

Numerical‌ simulations of nonlinear partial‌​‌ differential equations often involve​​ solving large nonlinear systems,​​​‌ for which Newton's method‌ is widely employed due‌​‌ to its fast convergence​​ near the solution. However,​​​‌ its performance can deteriorate‌ in the presence of‌​‌ strong nonlinearities or poor​​ initial guesses. Nonlinear overlapping​​​‌ domain decomposition methods, such‌ as RASPEN and Substructured‌​‌ RASPEN (SRASPEN), have proven​​ effective in addressing these​​​‌ challenges. Because SRASPEN reduces‌ the problem size by‌​‌ restricting computations to a​​ substructure, it does not​​​‌ update the solution outside‌ the substructure, so that‌​‌ no natural initial guesses​​ for the nonlinear local​​​‌ solution exists that might‌ lead to additional inner‌​‌ subdomain nonlinear iterations or​​ even prevent the local​​​‌ solvers to converge. In‌ this study, we analyze‌​‌ the convergence of RASPEN.​​ We show how domain​​​‌ decomposition improves the convergence‌ rate of the Newton's‌​‌ method by highlighting the​​ key role of the​​​‌ substructure on the global‌ error contraction. Moreover, our‌​‌ analysis provides insight into​​ an inexpensive modification to​​​‌ SRASPEN that mitigates the‌ lack of iterations outside‌​‌ the substructure. The proposed​​ variant significantly reduces computational​​​‌ cost while improving overall‌ efficiency compared to existing‌​‌ techniques in the literature.​​ Numerical experiments confirm the​​​‌ computational performance and robustness‌ of the improved SRASPEN,‌​‌ establishing it as a​​ reliable approach for solving​​​‌ large-scale nonlinear systems.

This‌ work is part of‌​‌ Ettaouchi El Mehdi's PhD​​ thesis and is carried​​​‌ out in collaboration with‌ Nicolas Tardieu (EDF). For‌​‌ more details on this​​ work we refer to​​​‌  21

7.16 Robustness and‌ reliability of state-space, frame-based‌​‌ modeling for thermoacoustics

The​​ Galerkin modal expansion is​​​‌ a well-known method used‌ to develop reduced order‌​‌ models for thermoacoustics. A​​ known issue is the​​​‌ appearance of Gibbs-type oscillations‌ on velocity fluctuations at‌​‌ the interface between subdomains​​ and at boundary conditions.​​​‌ Recent work of Laurent‌ et al. (2019) and‌​‌ Laurent et al. (2021)​​ have shown that it​​​‌ is possible to overcome‌ this issue by using‌​‌ an over-completed frame, instead​​ of a Galerkin modal​​​‌ basis. However, the low-order‌ modeling based on this‌​‌ frame modal expansion may​​ generate spurious modes. In​​​‌ this paper, the origin‌ of these non-physical modes‌​‌ is identified and a​​ method is proposed to​​​‌ automatically remove them from‌ the outcome. By preventing‌​‌ any interaction between the​​ physical and non-physical components,​​​‌ the proposed methodology drastically‌ improves the robustness and‌​‌ reliability of the frame​​ modal expansion modeling for​​​‌ thermoacoustics.

For more details‌ on this work we‌​‌ refer to  16.​​​‌

7.17 Juxtaposing the fourth​ order vibrational operator perturbation​‌ theory CVPT(4) and the​​ adaptive VCI (A-VCI): Accuracy,​​​‌ vibrational resonances and polyads​ of C2H4 and C2D4​‌

Ab initio prediction of​​ anharmonic vibrational spectra produces​​​‌ an increasing computational overhead​ for larger molecules, requesting​‌ a balance between an​​ accuracy and resources. Two​​​‌ complementary fundamental quantum mechanical​ approaches, the perturbative and​‌ variational, have various strong​​ and weak features, depending​​​‌ on a specific target​ problem. The vibrational perturbation​‌ theory (VPT) treats weak​​ couplings and strong resonances​​​‌ separately, relying on somewhat​ artificial criteria. In contrast,​‌ the more precise but​​ computationally intense variational configuration​​​‌ interaction (VCI) method treats​ all couplings in universal​‌ manner. The active ongoing​​ development of approaches to​​​‌ solving vibrational problems requires​ an update of comparative​‌ benchmarks, helping to choose​​ the best theoretical tools​​​‌ for a particular target.​ In this work, the​‌ performance of two particular​​ modern implementations of these​​​‌ methods was juxtaposed: the​ second and fourth order​‌ operator canonical perturbation theory​​ CVPT(2,4) and a recently​​​‌ proposed adaptive vibrational configuration​ interaction method (A-VCI). Two​‌ practically important C2H4 and​​ C2D4 molecules and an​​​‌ accurate CCSD(T)/cc-pVQZ four-body sextic​ normal mode PES were​‌ employed for benchmarking. The​​ comprehensive picture of vibrational​​​‌ resonances and the polyad​ quantum number was revealed.​‌ A new quadratic resonance​​ criterion is proposed and​​​‌ its efficiency in elucidating​ polyad structures is demonstrated.​‌ A striking observation was​​ made that CVPT(2) often​​​‌ produces better predictions of​ fundamental frequencies, while CVPT(4)​‌ demonstrates an excellent level​​ of correlation with A-VCI​​​‌ results for both fundamentals​ and two-quanta states.

For​‌ more details on this​​ work we refer to​​​‌  22

7.18 Approximation Algorithms​ for Scheduling with/without Deadline​‌ Constraints where Rejection Costs​​ are Proportional to Processing​​​‌ Times

In this work,​ we address two offline​‌ job scheduling problems, where​​ jobs can either be​​​‌ processed on a limited​ supply of energy-efficient machines​‌ on the edge, or​​ offloaded to an unlimited​​​‌ supply of energy-inefficient machines​ on the cloud (called​‌ rejected in our context).​​ The goal is to​​​‌ minimize the total energy​ consumed in processing all​‌ tasks. We consider a​​ first scheduling problem with​​​‌ no due date (or​ deadline) constraints, and we​‌ formulate it as a​​ scheduling problem with rejection,​​​‌ where the cost of​ rejecting a job is​‌ directly proportional to its​​ processing time. We introduce​​​‌ a novel 5/4(1+ε) approximation​ algorithm BEKP by associating​‌ it with a Multiple​​ Subset Sum problem for​​​‌ this version. Our algorithm​ is an improvement over​‌ the existing literature, which​​ provides a (3/2 -1/2m)​​​‌ approximation for scenarios with​ arbitrary rejection costs. In​‌ the second scheduling problem,​​ jobs have due date​​​‌ (or deadline) constraints, and​ the goal is to​‌ minimize the weighted number​​ of late jobs. In​​​‌ our context, if a​ job is late, it​‌ is offloaded (rejected) to​​ an energy-inefficient machine on​​​‌ the cloud, which incurs​ a cost directly proportional​‌ to its processing time​​ of the job. We​​​‌ position this problem in​ the literature, and introduce​‌ a novel $(\mathrm{1}-{(m-1)}^m/‌m^m)$-approximation‌ algorithm MDP for this‌​‌ version, where we got​​ our inspiration from an​​​‌ algorithm for the interval‌ selection problem with a‌​‌ $(1-{\mathrm{m}}^m/{(\mathrm{m‌}+1)}^{\mathrm{m‌}})$ approximation ratio for‌​‌ arbitrary rejection costs. We​​ evaluate and discuss the​​​‌ effectiveness of our approaches‌ through a series of‌​‌ experiments, comparing them to​​ existing algorithms.

For more​​​‌ details on this work‌ we refer to  14‌​‌

7.19 Guix-HPC Activity Report​​ 2023-2024

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 tools and‌​‌ aims to make them​​ useful for HPC practitioners​​​‌ and scientists concerned with‌ dependency graph control and‌​‌ customization and, uniquely, reproducible​​ research. This report—our seventh​​​‌ report!—highlights key achievements of‌ Guix-HPC between our previous‌​‌ report a year ago​​ and today, February 2025.​​​‌ This year was marked‌ by exciting developments for‌​‌ HPC and reproducible workflows.​​ Significant advances were made​​​‌ in integrating Guix into‌ the complex software landscape‌​‌ of HPC, taking the​​ roles of software manager,​​​‌ workflow execution engine, backend‌ for generating container images,‌​‌ or provider for the​​ complete operating system layer.​​​‌ Support for reproducing computations‌ from the past was‌​‌ also much improved. And,​​ as usual, we have​​​‌ been using Guix for‌ research, and teaching other‌​‌ researchers how to get​​ started.

For more details​​​‌ on this work we‌ refer to  42.‌​‌

8.1 European initiatives‌

8.2​ National initiatives

9.1 Promoting scientific activities​

9.2 Teaching -‌ Supervision - Juries -‌​‌ Educational and pedagogical outreach​​

• Post graduate level/Master:
  • E.​​​‌ Agullo: Numerical algorithms 20h,‌ advanced Numerical Numerical Algebra‌​‌ 8h, and Implementation of​​ HPC dense linear algebra​​​‌ kernels 8h, at Bordeaux‌ INP (ENSEIRB-MatMeca).
  • L. Giraud:‌​‌ Introduction to intensive computing​​ and related programming tools​​​‌ 20h, INSA Toulouse; Advanced‌ numerical linear algebra 10h,‌​‌ ENSEEIHT Toulouse.
  • C. Kruse:​​ Iterative methods in linear​​​‌ algebra, 28h, ENSEEIHT Toulouse.‌
  • P. Mycek: Multifidelity methods‌​‌ 14h, ModIA (cursus en​​ alternance, INSA/N7), Toulouse.

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 Computing361​​2014, 66-93HAL​​​‌DOI
• 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 Applications40​‌22019, 417–439​​HALDOI
• 4 misc​​​‌E.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 Computing49​​​‌July 2015, 66-82‌HALDOI
• 6 article‌​‌F.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 Physics257‌​‌AJanuary 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 articleV.Vincent​​​‌ Darrigrand, A.Andrei‌ Dumitrasc, C.Carola‌​‌ Kruse and U.Ulrich​​ Rüde. Inexact inner–outer​​​‌ Golub–Kahan bidiagonalization method: A‌ relaxation strategy.Numerical‌​‌ Linear Algebra with Applications​​December 2022HALDOI​​​‌
• 9 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​‌
• 10 articleA.Alain​​ Franc, P.Pierre​​​‌ Blanchard and O.Olivier‌ Coulaud. Nonlinear mapping‌​‌ and distance geometry.​​Optimization Letters142​​​‌2020, 453-467HAL‌DOI
• 11 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 Applications43​​22022, 710-739​​​‌HALDOI
• 12 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​‌
• 13 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

• 59 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​DOIback to text​‌
• 60 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​back to text
• 61​‌ 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
• 62 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 Computing426​‌November 2020HALDOI​​back to text
• 63​​​‌ 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 Applications394​‌October 2018, 1701-1725​​HALback to text​​​‌
• 64 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-0513​Inria Bordeaux Sud-OuestJune​‌ 2021, 100HAL​​back to text
• 65​​​‌ 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 Applications354​​​‌2014, 1625--1651back​ to text
• 66 article​‌E.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 Applications402​‌2019, 417--439HAL​​DOIback to text​​
• 67 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​​​‌
• 68 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
• 69 techreportS.‌Steffen Börm, L.‌​‌Lars Grasedyck and W.​​Wolfgang Hackbusch. Hierarchical​​​‌ Matrices.2003,‌ 1--173back to text‌​‌
• 70 unpublishedJ.Jérémy​​ Briant, P.Paul​​​‌ Mycek, M.Mayeul‌ Destouches, O.Olivier‌​‌ Goux, S.Serge​​ Gratton, S.Selime​​​‌ Gürol, E.Ehouarn‌ Simon and A. T.‌​‌Anthony T. Weaver.​​ A filtered multilevel Monte​​​‌ Carlo method for estimating‌ the expectation of discretized‌​‌ random fields.November​​ 2023, working paper​​​‌ or preprintHALDOI‌back to text
• 71‌​‌ 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 Experience20‌​‌132008, 1573--1590​​back to text
• 72​​​‌ 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
• 73‌​‌ 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 Physics‌​‌257A23 pages,​​ 9 figuresJanuary 2014​​​‌, 627-644HALDOI‌back to text
• 74‌​‌ 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.Wiley​​2009back to text​​​‌
• 75 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 - 450HAL​​DOIback to text​​​‌
• 76 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-9429​​​‌Inria Bordeaux - Sud-Ouest‌ ; InraeNovember 2021‌​‌HALback to text​​
• 77 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
• 78 articleA.​​​‌ A.Alain A. Franc‌, P.Pierre Blanchard‌​‌ and O.Olivier Coulaud​​​‌. Nonlinear mapping and​ distance geometry.Optimization​‌ Letters1422020​​, 453-467HALDOI​​​‌back to text
• 79​ bookN.Nicolas Gillis​‌. Nonnegative Matrix Factorization​​.Society for Industrial​​​‌ and Applied MathematicsJanuary​ 2020DOIback to​‌ text
• 80 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
• 81 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 Mathematics11329​​2011, 291--304back​​​‌ to text
• 82 article​B.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)311​​​‌2005, 60--78back​ to text
• 83 book​‌W.Wolfgang Hackbusch.​​ Hierarchical Matrices: Algorithms and​​​‌ Analysis.Springer Publishing​ Company, Incorporated2015back​‌ to text
• 84 article​​N.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 Review532​​2011, 217--288URL:​​​‌ http://arxiv.org/abs/0909.4061DOIback to​ text
• 85 articleT.​‌ G.Tamara G. Kolda​​ and B. W.Brett​​​‌ W. Bader. Tensor​ Decompositions and Applications.​‌SIAM Review513​​aug 2009, 455--500​​​‌URL: http://epubs.siam.org/doi/abs/10.1137/07070111XDOIback​ to text
• 86 article​‌C.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.7​​​‌12020, 45​URL: https://doi.org/10.1186/s40323-020-00181-2DOIback​‌ to text
• 87 article​​C.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
• 88 article​​​‌V.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.2213​‌2020, 7021-7030URL:​​ http://dx.doi.org/10.1039/D0CP00593BDOIback to​​​‌ text
• 89 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 XIII​​​‌June 2014HALback​ to text
• 90 article​‌P.-G.Per-Gunnar Martinsson and​​ J.Joel Tropp.​​​‌ Randomized Numerical Linear Algebra:​ Foundations & Algorithms.​‌2020, URL: http://arxiv.org/abs/2002.01387​​back to text
• 91​​ 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
• 92 articleI.​​​‌ V.I. V. Oseledets‌. Tensor-Train Decomposition.‌​‌SIAM Journal on Scientific​​ Computing335January​​​‌ 2011, 2295--2317URL:‌ https://doi.org/10.1137/090752286DOIback to‌​‌ text
• 93 phdthesisL.​​Louis Poirel. Algebraic​​​‌ domain decomposition methods for‌ hybrid (iterative/direct) solvers.‌​‌Université de BordeauxNovember​​ 2018HALback to​​​‌ text
• 94 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 Magnetics56​​​‌3March 2020,‌ 1-4HALDOIback‌​‌ to text
• 95 phdthesis​​G.Guillaume Sylvand.​​​‌ La méthode multipôle rapide‌ en électromagnétisme. Performances, parallélisation,‌​‌ applications.Ecole des​​ Ponts ParisTechJune 2002​​​‌HALback to text‌
• 96 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 2021HAL​​back to text