arXiv subjects

Hélène Barucq

Publications and source records attributed to Hélène Barucq.

14 recordsLinked to original sources

arXiv2026

Model order reduction of HDG discretizations for harmonic wave propagation

We propose and analyze a projection-based model reduction method for the rapid and reliable solution of time-harmonic acoustic wave propagation problems over a frequency band, with particular emphasis on seismic applications. The high-fidelity model is based on a hybridizable discontinuous Galerkin (HDG) discretization, formulated as a generalized saddle-point system coupling the volume and trace unknowns. The basis construction combines residual-based greedy sampling of the frequency parameter, enrichment with frequency derivatives of the HDG solution, and proper orthogonal decomposition (POD) compression. We establish the well posedness of both the high-fidelity and reduced problems and characterize their stability with inf-sup constants. We investigate the interaction between static condensation, inherent to the HDG discretization, and projection onto the reduced spaces, characterizing its implications for the reduced formulation and approximation errors. Since these operations do not commute in general, we derive a priori error estimates for both the trace and volume variables to quantify the gap between the two operations. We further identify a lifting condition on the volume reduced space that eliminates these gaps and yields a trace error estimate independent of the volume approximation error. Numerical experiments on the two-dimensional Marmousi model and a three-dimensional heterogeneous medium, with more than $10^{7}$ unknowns, illustrate the theoretical estimates and show that the reduced model achieves speedups of several orders of magnitude compared to the high-fidelity solver.

math.NA↗

arXiv2026

Numerical simulations of oscillations for axisymmetric solar backgrounds with differential rotation and gravity

Local helioseismology comprises of imaging and inversion techniques employed to reconstruct the dynamic and interior of the Sun from correlations of oscillations observed on the surface, all of which require modeling solar oscillations and computing Green's kernels. In this context, we implement and investigate the robustness of the Hybridizable Discontinuous Galerkin (HDG) method in solving the equation modeling stellar oscillations for realistic solar backgrounds containing acoustic attenuation, gravity, and differential rotation. While a common choice for modeling stellar oscillations is the Galbrun's equation, our working equations are derived from an equivalent variant, involving less regularity in its coefficients, working with Lagrangian displacement and pressure perturbation as unknowns. Under differential rotation and axisymmetric assumption, the system is solved in azimuthal decomposition with the HDG method. Compared to no-gravity approximations, the mathematical nature of the wave operator is now linked to the profile of the solar buoyancy frequency $N$ which encodes gravity, and leads to distinction into regions of elliptic or hyperbolic behavior of the wave operator at zero attenuation. While small attenuation is systematically included to guarantee theoretical well-posedness, the above phenomenon affects the numerical solutions in terms of amplitude and oscillation pattern, and requires a judicious choice of stabilization. We investigate the stabilization of the HDG discretization scheme, and demonstrate its importance to ensure the accuracy of numerical results, which is shown to depend on frequencies relative to $N$, and on the position of the Dirac source. As validations, the numerical power spectra reproduce accurately the observed effects of the solar rotation on acoustic waves.

astro-ph.SR↗

arXiv2026

Enriching continuous Lagrange finite element approximation spaces using neural networks

In this work, we present a study combining two approaches in the context of solving PDEs: the continuous finite element method (FEM) and more recent techniques based on neural networks. In recent years, physics-informed neural networks (PINNs) have become particularly interesting for rapidly solving PDEs, especially in high dimensions. However, their lack of accuracy can be a significant drawback in this context, hence the interest in combining them with FEM, for which error estimates are already known. The complete pipeline proposed here consists in modifying the classical FEM approximation spaces by taking information from a prior, chosen as the prediction of a neural network. On the one hand, this combination improves and certifies the prediction of neural networks, to obtain a fast and accurate solution. On the other hand, error estimates are proven, showing that such strategies outperform classical ones by a factor that depends only on the quality of the prior. We validate our approach with numerical results performed on parametric problems with 1D, 2D and 3D geometries. These experiments demonstrate that to achieve a given accuracy, a coarser mesh can be used with our enriched FEM compared to the standard FEM, leading to reduced computational time, particularly for parametric problems.

math.NA↗

arXiv2025

3D Modeling of Solar Oscillations with Hybridizable Discontinuous Galerkin Method

With increasing quantity and quality of solar observations, it becomes essential to account for three-dimensional heterogeneities in wave modeling for seismic data interpretation. In this context, we present a 3D solver of the time-harmonic adiabatic stellar oscillation equations without background flows on a domain consisting of the Sun and its photosphere. The background medium consists of 3D heterogeneities on top of a radial strongly-stratified standard solar model. The oscillation equations are solved with the Hybridizable Discontinuous Galerkin (HDG) method, considering a first-order formulation in terms of the vector displacement and the pressure perturbation. This method combines the high-order accuracy and the parallelism of DG methods while yielding smaller linear systems. These are solved with a direct solver, with block low-rank compression and mixed-precision arithmetic to reduce memory footprint. The trade-off between compression and solution accuracy is investigated, and our 3D solver is validated by comparing with resolution under axial symmetry for solar backgrounds. The capacity of the solver is illustrated with wave speed heterogeneities characteristic of two physical phenomena: active regions and convection. We show the importance of global 3D gravito-acoustic wave simulations, in particular when the amplitudes of the perturbations are strong and their effect on the wavefield cannot be estimated by linear approximations.

math.AP↗

arXiv2025

MOR-T L : A Novel Model Order Reduction Method for Parametrized Problems with Application to Seismic Wave Propagation

This paper presents an efficient strategy for constructing Reduced-Order Model (ROM) bases using Taylor polynomial expansions and Fr{é}chet derivatives with respect to model parameters. The proposed approach enables the construction of ROM bases with minimal additional computational cost. By exploiting Fr{é}chet derivatives -solution to the same problem with distinct right-hand sides -the method introduces a streamlined multiple-right-hand-side (RHS) strategy for ROM bases construction. This approach not only reduces overall computational expenses but also improves accuracy during model parameter updates. Numerical experiments on a two-dimensional wave problem demonstrate significant efficiency gains and enhanced performance, highlighting the potential of the proposed method to advance computational cost-effectiveness, particularly in seismic inversion applications.

math.NA↗

arXiv2024

Assembling algorithm for Green's tensors and absorbing boundary conditions for Galbrun's equation in radial symmetry

Solar oscillations can be modeled by Galbrun's equation which describes Lagrangian wave displacement in a self-gravitating stratified medium. For spherically symmetric backgrounds, we construct an algorithm to compute efficiently and accurately the coefficients of the Green's tensor of the time-harmonic equation in vector spherical harmonic basis. With only two resolutions, our algorithm provides values of the kernels for all heights of source and receiver, and prescribes analytically the singularities of the kernels. We also derive absorbing boundary conditions (ABC) to model wave propagation in the atmosphere above the cut-off frequency. The construction of ABC, which contains varying gravity terms, is rendered difficult by the complex behavior of the solar potential in low atmosphere and for frequencies below the Lamb frequency. We carry out extensive numerical investigations to compare and evaluate the efficiency of the ABCs in capturing outgoing solutions. Finally, as an application towards helioseismology, we compute synthetic solar power spectra that contain pressure modes as well as internal-gravity (g-) and surface-gravity (f-) ridges which are missing in simpler approximations of the wave equation. For purpose of validation, the location of the ridges in the synthetic power spectra are compared with observed solar modes.

astro-ph.SR↗

arXiv2024

Numerical investigation of stabilization in the Hybridizable Discontinuous Galerkin method for linear anisotropic elastic equation

This work is concerned with implementing the hybridizable discontinuous Galerkin (HDG) method to solve the linear anisotropic elastic equation in the frequency domain. First-order formulation with the compliance tensor and Voigt notation are employed to provide a compact description of the discretized problem and flexibility with highly heterogeneous media. We further focus on the question of optimal choices of stabilization in the definition of HDG numerical traces. For this purpose, we construct a hybridized Godunov-upwind flux for anisotropic elastic media possessing three distinct wavespeeds. This stabilization removes the need to choose a scaling factor, contrary to the identity and Kelvin-Christoffel based stabilizations which are popular choices in the literature. We carry out comparisons among these families for isotropic and anisotropic material, with constant background and highly heterogeneous ones, in two and three dimensions. These experiments establish the optimality of the Godunov stabilization which can be used as a reference choice for a generic material in which different types of waves propagate.

math.AP↗

arXiv2022

Imaging individual active regions on the Sun's far side with improved helioseismic holography

Helioseismic holography is a useful method to detect active regions on the Sun's far side and improve space weather forecasts. We aim to improve helioseismic holography by using a clear formulation of the problem, an accurate forward solver in the frequency domain, and a better understanding of the noise properties. Building on the work of Lindsey et al., we define the forward- and backward-propagated wave fields (ingression and egression) in terms of a Green's function. This Green's function is computed using an accurate forward solver in the frequency domain. We analyse overlapping segments of 31 hr of SDO/HMI dopplergrams, with a cadence of 24 hr. Phase shifts between the ingression and the egression are measured and averaged to detect active regions on the far side. The phase maps are compared with direct EUV intensity maps from STEREO/EUVI. We confirm that medium-size active regions can be detected on the far side with high confidence. Their evolution (and possible emergence) can be monitored on a daily time scale. Seismic maps averaged over 3 days provide an active region detection rate as high as 75% and a false discovery rate only as low as 7%, for active regions with areas above one thousandth of an hemisphere. For a large part, these improvements can be attributed to the use of a complete Green's function (all skips) and to the use of all observations on the front side (full pupil). Improved helioseismic holography enables the study of the evolution of medium-size active regions on the Sun's far side.

astro-ph.SR↗

arXiv2021

Polynomial-reproducing spline spaces from fine zonotopal tilings

Given a point configuration A, we uncover a connection between polynomial-reproducing spline spaces over subsets of conv(A) and fine zonotopal tilings of the zonotope Z(V) associated to the corresponding vector configuration. This link directly generalizes a known result on Delaunay configurations and naturally encompasses, due to its combinatorial character, the case of repeated and affinely dependent points in A. We prove the existence of a general iterative construction process for such spaces. Finally, we turn our attention to regular fine zonotopal tilings, specializing our previous results and exploiting the adjacency graph of the tiling to propose a set of practical algorithms for the construction and evaluation of the associated spline functions.

math.NA↗

arXiv2020

Full Reciprocity-Gap Waveform Inversion in the frequency domain, enabling sparse-source acquisition

The quantitative reconstruction of sub-surface Earth properties from the propagation of waves follows an iterative minimization of a misfit functional. In marine seismic exploration, the observed data usually consist of measurements of the pressure field but dual-sensor devices also provide the normal velocity. Consequently, a reciprocity-based misfit functional is specifically designed, and defines the Full Reciprocity-gap Waveform Inversion (FRgWI ) method. This misfit functional provides additional features compared to the more traditional least-squares approaches with, in particular, that the observational and computational acquisitions can be different. Therefore, the positions and wavelets of the sources from which the measurements are acquired are not needed in the reconstruction procedure and, in fact, the numerical acquisition (for the simulations) can be arbitrarily chosen. Based on three-dimensional experiments, FRgWI is shown to behave better than Full Waveform Inversion (FWI) in the same context. Then, it allows for arbitrary numerical acquisitions in two ways: when few measurements are given, a dense numerical acquisition (compared to the observational one) can be used to compensate. On the other hand, with a dense observational acquisition, a sparse computational one is shown to be sufficient, for instance with multiple-point sources, hence reducing the numerical cost. FRgWI displays accurate reconstructions in both situations and appears more robust with respect to cross-talk than the least-squares shot-stacking.

physics.geo-ph↗

arXiv2019

Eigenvector Model Descriptors for Solving an Inverse Problem of Helmholtz Equation: Extended Materials

We study the seismic inverse problem for the recovery of subsurface properties in acoustic media. In order to reduce the ill-posedness of the problem, the heterogeneous wave speed parameter to be recovered is represented using a limited number of coefficients associated with a basis of eigenvectors of a diffusion equation, following the regularization by discretization approach. We compare several choices for the diffusion coefficient in the partial differential equations, which are extracted from the field of image processing. We first investigate their efficiency for image decomposition (accuracy of the representation with respect to the number of variables and denoising). Next, we implement the method in the quantitative reconstruction procedure for seismic imaging, following the Full Waveform Inversion method, where the difficulty resides in that the basis is defined from an initial model where none of the actual structures is known. In particular, we demonstrate that the method is efficient for the challenging reconstruction of media with salt-domes. We employ the method in two and three-dimensional experiments and show that the eigenvector representation compensates for the lack of low frequency information, it eventually serves us to extract guidelines for the implementation of the method.

math.AP↗

arXiv2018

Signal and noise in helioseismic holography

Helioseismic holography is an imaging technique used to study heterogeneities and flows in the solar interior from observations of solar oscillations at the surface. Holograms contain noise due to the stochastic nature of solar oscillations. We provide a theoretical framework for modeling signal and noise in Porter-Bojarski helioseismic holography. The wave equation may be recast into a Helmholtz-like equation, so as to connect with the acoustics literature and define the holography Green's function in a meaningful way. Sources of wave excitation are assumed to be stationary, horizontally homogeneous, and spatially uncorrelated. Using the first Born approximation we calculate holograms in the presence of perturbations in sound-speed, density, flows, and source covariance, as well as the noise level as a function of position. This work is a direct extension of the methods used in time-distance helioseismology to model signal and noise. To illustrate the theory, we compute the hologram intensity numerically for a buried sound-speed perturbation at different depths in the solar interior. The reference Green's function is obtained for a spherically-symmetric solar model using a finite-element solver in the frequency domain. Below the pupil area on the surface, we find that the spatial resolution of the hologram intensity is very close to half the local wavelength. For a sound-speed perturbation of size comparable to the local spatial resolution, the signal-to-noise ratio is approximately constant with depth. Averaging the hologram intensity over a number $N$ of frequencies above 3 mHz increases the signal-to-noise ratio by a factor nearly equal to the square root of $N$. This may not be the case at lower frequencies, where large variations in the holographic signal are due to the individual contributions of the long-lived modes of oscillation.

astro-ph.SR↗

arXiv2016

Computational helioseismology in the frequency domain: acoustic waves in axisymmetric solar models with flows

Local helioseismology has so far relied on semi-analytical methods to compute the spatial sensitivity of wave travel times to perturbations in the solar interior. These methods are cumbersome and lack flexibility. Here we propose a convenient framework for numerically solving the forward problem of time-distance helioseismology in the frequency domain. The fundamental quantity to be computed is the cross-covariance of the seismic wavefield. We choose sources of wave excitation that enable us to relate the cross-covariance of the oscillations to the Green's function in a straightforward manner. We illustrate the method by considering the 3D acoustic wave equation in an axisymmetric reference solar model, ignoring the effects of gravity on the waves. The symmetry of the background model around the rotation axis implies that the Green's function can be written as a sum of longitudinal Fourier modes, leading to a set of independent 2D problems. We use a high-order finite-element method to solve the 2D wave equation in frequency space. The computation is `embarrassingly parallel', with each frequency and each azimuthal order solved independently on a computer cluster. We compute travel-time sensitivity kernels in spherical geometry for flows, sound speed, and density perturbations under the first Born approximation. Convergence tests show that travel times can be computed with a numerical precision better than one millisecond, as required by the most precise travel-time measurements. The method presented here is computationally efficient and will be used to interpret travel-time measurements in order to infer, e.g., the large-scale meridional flow in the solar convection zone. It allows the implementation of (full-waveform) iterative inversions, whereby the axisymmetric background model is updated at each iteration.

astro-ph.SR↗

arXiv2008

True amplitude one-way propagation in heterogeneous media

This paper deals with the numerical analysis of two one-way systems derived from the general complete modeling proposed by M.V. De Hoop. The main goal of this work is to compare two different formulations in which a correcting term allows to improve the amplitude of the numerical solution. It comes out that even if the two systems are equivalent from a theoretical point of view, nothing of the kind is as far as the numerical simulation is concerned. Herein a numerical analysis is performed to show that as long as the propagation medium is smooth, both the models are equivalent but it is no more the case when the medium is associated to a quite strongly discontinuous velocity.

math.NA↗