arXiv subjects

Frédéric Magoulès

Publications and source records attributed to Frédéric Magoulès.

15 recordsLinked to original sources

arXiv2026

Pointwise Error Estimates for Numerical Physics-Informed Neural Networks

Physics-informed neural networks are often evaluated by residual losses sampled at finitely many points, which do not by themselves certify pointwise values of a partial differential equation solution. In this work, deterministic pointwise error intervals are developed for mesh-based, piecewise-linear numerical physics-informed neural networks. The proposed error estimation is given for a compatible field, which is the finite-element reconstruction of an admissible prediction on a mesh. The certifying residual is then obtained by applying the finite-dimensional numerical system to this compatible field. For compatible square linear systems, the pointwise error relative to the discrete target has an exact adjoint Green representation, and the computed signed error recovers the finite element solution exactly. Norm-based, inexact, localized, and randomized variants provide computable intervals when the exact correction computation is impractical. The extension from the discrete target to the continuous solution is supplied by comparison estimates. For a one-dimensional coercive reaction-diffusion class, this transfer layer is made fully computable by an explicit residual-based a posteriori estimator with querywise constants. The error bound derivation is extended to nonlinear residual systems with explicit Taylor remainders. Numerical experiments assess compatibility and calibration on manufactured examples, on a large-scale public three-dimensional elasticity benchmark, and on projected neural load families on the same benchmark.

math.NA↗

arXiv2024

Physics-Informed Graph-Mesh Networks for PDEs: A hybrid approach for complex problems

The recent rise of deep learning has led to numerous applications, including solving partial differential equations using Physics-Informed Neural Networks. This approach has proven highly effective in several academic cases. However, their lack of physical invariances, coupled with other significant weaknesses, such as an inability to handle complex geometries or their lack of generalization capabilities, make them unable to compete with classical numerical solvers in industrial settings. In this work, a limitation regarding the use of automatic differentiation in the context of physics-informed learning is highlighted. A hybrid approach combining physics-informed graph neural networks with numerical kernels from finite elements is introduced. After studying the theoretical properties of our model, we apply it to complex geometries, in two and three dimensions. Our choices are supported by an ablation study, and we evaluate the generalisation capacity of the proposed approach.

math.NA↗

arXiv2023

Distributed convergence detection based on global residual error under asynchronous iterations

Convergence of classical parallel iterations is detected by performing a reduction operation at each iteration in order to compute a residual error relative to a potential solution vector. To efficiently run asynchronous iterations, blocking communication requests are avoided, which makes it hard to isolate and handle any global vector. While some termination protocols were proposed for asynchronous iterations, only very few of them are based on global residual computation and guarantee effective convergence. But the most effective and efficient existing solutions feature two reduction operations, which constitutes an important factor of termination delay. In this paper, we present new, non-intrusive, protocols to compute a residual error under asynchronous iterations, requiring only one reduction operation. Various communication models show that some heuristics can even be introduced and formally evaluated. Extensive experiments with up to 5600 processor cores confirm the practical effectiveness and efficiency of our approach.

cs.DC↗

arXiv2023

Asynchronous iterations of HSS method for non-Hermitian linear systems

A general asynchronous alternating iterative model is designed, for which convergence is theoretically ensured both under classical spectral radius bound and, then, for a classical class of matrix splittings for $\mathsf H$-matrices. The computational model can be thought of as a two-stage alternating iterative method, which well suits to the well-known Hermitian and skew-Hermitian splitting (HSS) approach, with the particularity here of considering only one inner iteration. Experimental parallel performance comparison is conducted between the generalized minimal residual (GMRES) algorithm, the standard HSS and our asynchronous variant, on both real and complex non-Hermitian linear systems respectively arising from convection-diffusion and structural dynamics problems. A significant gain on execution time is observed in both cases.

math.NA↗

arXiv2023

Resilient asynchronous primal Schur method

This paper introduces the application of the asynchronous iterations theory within the framework of the primal Schur domain decomposition method. A suitable relaxation scheme is designed, which asynchronous convergence is established under classical spectral radius conditions. For the usual case where the local Schur complement matrices are not constructed, suitable splittings only based on explicitly generated matrices are provided. Numerical experiments are conducted on a supercomputer for both Poisson's and linear elasticity problems. The asynchronous Schur solver outperformed the classical conjugate-gradient-based one in case of compute node failures.

math.NA↗

arXiv2023

Asynchronous multiplicative coarse-space correction

This paper introduces the multiplicative variant of the recently proposed asynchronous additive coarse-space correction method. Definition of an asynchronous extension of multiplicative correction is not straightforward, however, our analysis allows for usual asynchronous programming approaches. General asynchronous iterative models are explicitly devised both for shared or replicated coarse problems and for centralized or distributed ones. Convergence conditions are derived and shown to be satisfied for M-matrices, as also done for the additive case. Implementation aspects are discussed, which reveal the need for non-blocking synchronization for building the successive right-hand-side vectors of the coarse problem. Optionally, a parameter allows for applying each coarse solution a maximum number of times, which has an impact on the algorithm efficiency. Numerical results on a high-speed homogeneous cluster confirm the practical efficiency of the asynchronous two-level method over its synchronous counterpart, even when it is not the case for the underlying one-level methods.

math.NA↗

arXiv2023

Physics-Informed Graph Convolutional Networks: Towards a generalized framework for complex geometries

Since the seminal work of [9] and their Physics-Informed neural networks (PINNs), many efforts have been conducted towards solving partial differential equations (PDEs) with Deep Learning models. However, some challenges remain, for instance the extension of such models to complex three-dimensional geometries, and a study on how such approaches could be combined to classical numerical solvers. In this work, we justify the use of graph neural networks for these problems, based on the similarity between these architectures and the meshes used in traditional numerical techniques for solving partial differential equations. After proving an issue with the Physics-Informed framework for complex geometries, during the computation of PDE residuals, an alternative procedure is proposed, by combining classical numerical solvers and the Physics-Informed framework. Finally, we propose an implementation of this approach, that we test on a three-dimensional problem on an irregular geometry.

cs.LG↗

arXiv2023

Enhancing the Global/Local Coupling Method: An Asynchronous Parallel Framework

A novel approach is being developed to introduce a parallel asynchronous implementation of non-intrusive global-local coupling. This study examines scenarios involving numerous patches, including those covering the entire structure. By leveraging asynchronous, the method aims to minimize reliance on communication, handle failures effectively, and address load imbalances. Detailed insights into the methodology are presented, accompanied by a demonstration of its performance through an academic case study.

cs.DC↗

arXiv2023

Accurate Coarse Residual for Two-Level Asynchronous Domain Decomposition Methods

Recently, asynchronous coarse-space correction has been achieved within both the overlapping Schwarz and the primal Schur frameworks. Both additive and multiplicative corrections have been discussed. In this paper, we address some implementation drawbacks of the proposed additive correction scheme. In the existing approach, each coarse solution is applied only once, leaving most of the iterations of the solver without coarse-space information while building the right-hand side of the coarse problem. Moreover, one-sided routines of the Message Passing Interface (MPI) standard were considered, which introduced the need for a sleep statement in the iterations loop of the coarse solver. This implies a tuning of the sleep period, which is a non-discrete quantity. In this paper, we improve the accuracy of the coarse right-hand side, which allowed for more frequent corrections. In addition, we highlight a two-sided implementation which better suits the asynchronous coarse-space correction scheme. Numerical experiments show a significant performance gain with such increased incorporation of the coarse space.

math.NA↗

arXiv2022

Asynchronous global-local non-invasive coupling for linear elliptic problems

This paper presents the first asynchronous version of the Global/Local non-invasive coupling, capable of dealing efficiently with multiple, possibly adjacent, patches. We give a new interpretation of the coupling in terms of primal domain decomposition method, and we prove the convergence of the relaxed asynchronous iteration. The asynchronous paradigm lifts many bottlenecks of the Global/Local coupling performance. We illustrate the method on several linear elliptic problems as encountered in thermal and elasticity studies.

cs.DC↗

arXiv2021

Interactive simulation for easy decision-making in fluid dynamics

A conventional study of fluid simulation involves different stages including conception, simulation, visualization, and analysis tasks. It is, therefore, necessary to switch between different software and interactive contexts which implies costly data manipulation and increases the time needed for decision making. Our interactive simulation approach was designed to shorten this loop, allowing users to visualize and steer a simulation in progress without waiting for the end of the simulation. The methodology allows the users to control, start, pause, or stop a simulation in progress, to change global physical parameters, to interact with its 3D environment by editing boundary conditions such as walls or obstacles. This approach is made possible by using a methodology such as the Lattice Boltzmann Method (LBM) to achieve interactive time while remaining physically relevant. In this work, we present our platform dedicated to interactive fluid simulation based on LBM. The contribution of our interactive simulation approach to decision making will be evaluated in a study based on a simple but realistic use case.

cs.CE↗

arXiv2021

Power Consumption Analysis of Parallel Algorithms on GPUs

Due to their highly parallel multi-cores architecture, GPUs are being increasingly used in a wide range of computationally intensive applications. Compared to CPUs, GPUs can achieve higher performances at accelerating the programs' execution in an energy-efficient way. Therefore GPGPU computing is useful for high performance computing applications and in many scientific research fields. In order to bring further performance improvements, GPU clusters are increasingly adopted. The energy consumed by GPUs cannot be neglected. Therefore, an energy-efficient time scheduling of the programs that are going to be executed by the parallel GPUs based on their deadline as well as the assigned priorities could be deployed to face their energetic avidity. For this reason, we present in this paper a model enabling the measure of the power consumption and the time execution of some elementary operations running on a single GPU using a new developed energy measurement protocol. Consequently, using our methodology, energy needs of a program could be predicted, allowing a better task scheduling.

cs.DC↗

arXiv2021

Parallel Sub-Structuring Methods for solving Sparse Linear Systems on a cluster of GPU

The main objective of this work consists in analyzing sub-structuring method for the parallel solution of sparse linear systems with matrices arising from the discretization of partial differential equations such as finite element, finite volume and finite difference. With the success encountered by the general-purpose processing on graphics processing units (GPGPU), we develop an hybrid multiGPUs and CPUs sub-structuring algorithm. GPU computing, with CUDA, is used to accelerate the operations performed on each processor. Numerical experiments have been performed on a set of matrices arising from engineering problems. We compare C+MPI implementation on classical CPU cluster with C+MPI+CUDA on a cluster of GPU. The performance comparison shows a speed-up for the sub-structuring method up to 19 times in double precision by using CUDA.

math.NA↗

arXiv2020

On the existence of optimal shapes in architecture

We consider shape optimization problems for elasticity systems in architecture. A typical question in this context is to identify a structure of maximal stability close to an initially proposed one. We show the existence of such an optimally shaped structure within classes of bounded Lipschitz domains and within wider classes of bounded uniform domains with boundaries that may be fractal. In the first case the optimal shape realizes the infimum of a given energy functional over the class, in the second case it realizes the minimum. As a concrete application we discuss the existence of maximally stable roof structures under snow loads.

math.AP↗

arXiv2020

Optimal absorption of acoustical waves by a boundary

In the aim to find the simplest and most efficient shape of a noise absorbing wall to dissipate the acoustical energy of a sound wave, we consider a frequency model described by the Helmholtz equation with a damping on the boundary. The well-posedness of the model is shown in a class of domains with d-set boundaries (N -- 1 $\le$ d < N). We introduce a class of admissible Lipschitz boundaries, in which an optimal shape of the wall exists in the following sense: We prove the existence of a Radon measure on this shape, greater than or equal to the usual Lebesgue measure, for which the corresponding solution of the Helmholtz problem realizes the infimum of the acoustic energy defined with the Lebesgue measure on the boundary. If this Radon measure coincides with the Lebesgue measure, the corresponding solution realizes the minimum of the energy. For a fixed porous material, considered as an acoustic absorbent, we derive the damping parameters of its boundary from the corresponding time-dependent problem described by the damped wave equation (damping in volume).

math.AP↗