Aurelien Citrain defended his PhD thesis entitled:

Hybrid finite element methods for seismic wave simulation: coupling of
Discontinuous Galerkin and Spectral Element discretizations

To solve wave equations in heterogeneous media with finite elements with
a reasonable numerical cost, we couple the Discontinuous Galerkin method (DGm) with
Spectral Elements method (SEm). We use hybrid meshes composed of tetrahedra and
structured hexahedra. The coupling is carried out starting from a mixed-primal DG
formulation applied on a hybrid mesh composed of a hexahedral macro-element and a
sub-mesh composed of tetrahedra. The SEm is applied in the macro-element paved with
structured hexahedrons and the coupling is ensured by the DGm numerical fluxes applied
on the internal faces of the macro-element common with the tetrahedral mesh. The sta-
bility of the coupled method is demonstrated when time integration is performed with a
Leap-Frog scheme. The performance of the coupled method is studied numerically and it
is shown that the coupling reduces numerical costs while keeping a high level of accuracy.
It is also shown that the coupled formulation can stabilize the DGm applied in areas that
include Perfectly Matched Layers.

Pau, December 16, 2019

Elvira Shishenina defended his PhD thesis entitled:

Space-Time Discretization of Elasto-Acoustic Wave Equation in Polynomial Trefftz-DG Bases

Discontinuous Finite Element Methods (DG FEM) have proven flexibility and accuracy for solving wave problems in complex media. However, they require a large number of degrees of freedom, which increases the corresponding computational cost compared with that of continuous finite element methods. Among the different variational approaches to solve boundary value problems, there exists a particular family of methods, based on the use of trial functions in the form of exact local solutions of the governing equations. The idea was first proposed by Trefftz in 1926, and since then it has been further developed and generalized. A Trefftz-DG variational formulation applied to wave problems reduces to surface integrals that should contribute to decreasing the computational costs.
Trefftz-type approaches have been widely used for time-harmonic problems, while their implementation for time-dependent simulations is still limited. The feature of Trefftz-DG methods applied to time-dependent problems is in the use of space-time meshes. Indeed, standard DG methods lead to the construction of a semi-discrete system of ordinary differential equations in time which are integrated by using an appropriate scheme. But Trefftz-DG methods applied to wave problems lead to a global matrix including time and space discretizations which is huge and sparse. This significantly hampers the deployment of this technology for solving industrial problems.
In this work, we develop a Trefftz-DG framework for solving mechanical wave problems including elasto-acoustic equations. We prove that the corresponding formulations are well-posed and we address the issue of solving the global matrix by constructing an approximate inverse obtained from the decomposition of the global matrix into a block-diagonal one. The inversion is then justified under a CFL-type condition. This idea allows for reducing the computational costs but its accuracy is limited to small computational domains. According to the limitations of the method, we have investigated the potential of Tent Pitcher algorithms following the recent works of Gopalakrishnan et al. It consists in constructing a space-time mesh made of patches that can be solved independently under a causality constraint. We have obtained very promising numerical results illustrating the potential of Tent Pitcher in particular when coupled with a Trefftz-DG method involving only surface terms. In this way, the space-time mesh is composed of elements which are 3D objects at most. It is also worth noting that this framework naturally allows for local time-stepping which is a plus to increase the accuracy while decreasing the computational burden.

Pau, December 7, 2018

Florian Faucher defended his PhD thesis entitled:

Contributions to Seismic Full Waveform Inversion for Time Harmonic Wave Equations: Stability Estimates, Convergence Analysis, Numerical Experiments involving Large Scale Optimization Algorithms

In this project, we investigate the recovery of subsurface Earth parameters. We consider the seismic imaging as a large scale iterative minimization problem, and deploy the Full Waveform Inversion (FWI) method, for which several aspects must be treated. The reconstruction is based on the wave equations because the characteristics of the measurements indicate the nature of the medium in which the waves propagate. First, the natural heterogeneity and anisotropy of the Earth require numerical methods that are adapted and efficient to solve the wave propagation problem. In this study, we have decided to work with the harmonic formulation, i.e., in the frequency domain. Therefore, we detail the mathematical equations involved and the numerical discretization used to solve the wave equations in large scale situations.

The inverse problem is then established in order to frame the seismic imaging. It is a nonlinear and ill-posed inverse problem by nature, due to the limited available data, and the complexity of the subsurface characterization. However, we obtain a conditional Lipschitz-type stability in the case of piecewise constant model representation. We derive the lower and upper bound for the underlying stability constant, which allows us to quantify the stability with frequency and scale. It is of great use for the underlying optimization algorithm involved to solve the seismic problem. We review the foundations of iterative optimization techniques and provide the different methods that we have used in this project. The Newton method, due to the numerical cost of inverting the Hessian, may not always be accessible. We propose some comparisons to identify the benefits of using the Hessian, in order to study what would be an appropriate procedure regarding the accuracy and time. We study the convergence of the iterative minimization method, depending on different aspects such as the geometry of the subsurface, the frequency, and the parametrization. In particular, we quantify the frequency progression, from the point of view of optimization, by showing how the size of the basin of attraction evolves with frequency.

Following the convergence and stability analysis of the problem, the iterative minimization algorithm is conducted via a multi-level scheme where frequency and scale progress simultaneously. We perform a collection of experiments, including acoustic and elastic media, in two and three dimensions. The perspectives of attenuation and anisotropic reconstructions are also introduced. Finally, we study the case of Cauchy data, motivated by the dual sensors devices that are developed in the geophysical industry. We derive a novel cost function, which arises from the stability analysis of the problem. It allows elegant perspectives where no prior information on the acquisition set is required.

Pau, November 29, 2017

Marie Bonnasse-Gahot defended his PhD thesis entitled:

High order discontinuous Galerkin methods for time-harmonic elastodynamics

The main objective of this work is the design of an efficient numerical strategy to solve the Helmholtz equation in highly heterogeneous media. We propose a methodology based on coarse meshes and high order polynomials together with a special quadrature scheme to take into account fine scale heterogeneities. The idea behind this choice is that high order polynomials are known to be robust with respect to the pollution effect and therefore, efficient to solve wave problems in homogeneous media. In this work, we are able to extend so-called “asymptotic error-estimate” derived for problems homogeneous media to the case of heterogeneous media. These results are of particular interest because they show that high order polynomials bring more robustness with respect to the pollution effect even if the solution is not regular, because of the fine scale heterogeneities. We propose special quadrature schemes to take int account fine scale heterogeneities. These schemes can also be seen as an approximation of the medium parameters. If we denote by h the finite-element mesh step and by e the approximation level of the medium parameters, we are able to show a convergence theorem which is explicit in terms of h, e and f, where f is the frequency. The main theoretical results are further validated through numerical experiments. 2D and 3D geophysica benchmarks have been considered. First, these experiments confirm that high-order finite-elements are more efficient to approximate the solution if they are coupled with our multiscale strategy. This is in agreement with our results about the pollution effect. Furthermore, we have carried out benchmarks in terms of computational time and memory requirements for 3D problems. We conclude that our multiscale methodology is able to greatly reduce the computational burden compared to the standard finite-element method

Pau, December 15, 2015

Lionel Boillot defended his PhD thesis entitled:

Contributions to the mathematical modeling and to the parallel algorithmic for the optimization of an elastic wave propagator in anisotropic media

Continue reading