Lise-Marie Imbert-Gerard

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Journals/Publications

• Imbert-Gerard, L., & Sylvand, G. (2024). Three types of quasi-Trefftz functions for the 3D convected Helmholtz equation: construction and theoretical approximation properties. .IMA Journal of Numerical Analysis.
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  Trefftz methods are numerical methods for the approximation of solutions to boundary and/or initial value problems. They are Galerkin methods with particular test and trial functions, which solve locally the governing partial differential equation (PDE). This property is called the Trefftz property. Quasi-Trefftz methods were introduced to leverage the advantages of Trefftz methods for problems governed by variable coefficient PDEs, by relaxing the Trefftz property into a so-called quasi-Trefftz property: test and trial functions are not exact solutions but rather local approximate solutions to the governing PDE. In order to develop quassi-Trefftz methods for aero-acoustics problems governed by the convected Helmholtz equation, the present work tackles the question of the definition, construction and approximation properties of three families of quasi-Trefftz functions: two based on generalizations on plane wave solutions, and one polynomial. [Journal_ref: ]
• Moiola, A., Stocker, P., & Imbert-Gérard, L. (2023). A SPACE–TIME QUASI-TREFFTZ DG METHOD FOR THE WAVE EQUATION WITH PIECEWISE-SMOOTH COEFFICIENTS. Mathematics of Computation, 92(341). doi:10.1090/MCOM/3786
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  Trefftz methods are high-order Galerkin schemes in which all discrete functions are elementwise solution of the PDE to be approximated. They are viable only when the PDE is linear and its coefficients are piecewiseconstant. We introduce a “quasi-Trefftz” discontinuous Galerkin (DG) method for the discretisation of the acoustic wave equation with piecewise-smooth material parameters: the discrete functions are elementwise approximate PDE solutions. We show that the new discretisation enjoys the same excellent approximation properties as the classical Trefftz one, and prove stability and high-order convergence of the DG scheme. We introduce polynomial basis functions for the new discrete spaces and describe a simple algorithm to compute them. The technique we propose is inspired by the generalised plane waves previously developed for time-harmonic problems with variable coefficients; it turns out that in the case of the time-domain wave equation under consideration the quasi-Trefftz approach allows for polynomial basis functions.
• Imbert-Gerard, L., Moiola, A., & Stocker, P. (2022). A space–time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients. Mathematics of Computation.
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  Trefftz methods are high-order Galerkin schemes in which all discrete functions are elementwise solution of the PDE to be approximated. They are viable only when the PDE is linear and its coefficients are piecewise-constant. We introduce a “quasi-Trefftz” discontinuous Galerkin (DG) method for the discretisation of the acoustic wave equation with piecewise-smooth material parameters: the discrete functions are elementwise approximate PDE solutions. We show that the new discretisation enjoys the same excellent approximation properties as the classical Trefftz one, and prove stability and high-order convergence of the DG scheme. We introduce polynomial basis functions for the new discrete spaces and describe a simple algorithm to compute them. The technique we propose is inspired by the generalised plane waves previously developed for time-harmonic problems with variable coefficients; it turns out that in the case of the time-domain wave equation under consideration the quasi-Trefftz approach allows for polynomial basis functions.
• Sylvand, G., & Imbert-gerard, L. (2021). A roadmap for Generalized Plane Waves and their interpolation properties. Numerische Mathematik, 149(1), 87-137. doi:10.1007/s00211-021-01220-9
• Imbert-Gerard, L., Vico, F., Greengard, L., & Ferrando, M. (2019). Integral equation methods for electrostatics, acoustics, and electromagnetics in smoothly varying, anisotropic media. SIAM Journal on Numerical Analysis, 57(3). doi:10.1137/18M1187039
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  We present a collection of well-conditioned integral equation methods for the solution of electrostatic, acoustic, or electromagnetic scattering problems involving anisotropic, inhomogeneous media. In the electromagnetic case, our approach involves a minor modification of a classical formulation. In the electrostatic or acoustic setting, we introduce a new vector partial differential equation, from which the desired solution is easily obtained. It is the vector equation for which we derive a well-conditioned integral equation. In addition to providing a unified framework for these solvers, we illustrate their performance using iterative solution methods coupled with the FFT-based technique of [F. Vico, L. Greengard, M. Ferrando, J. Comput. Phys., 323 (2016), pp. 191–203] to discretize and apply the relevant integral operators.
• Malhotra, D., Cerfon, A., O'Neil, M., & Imbert-Gérard, L. (2019). Taylor states in stellarators: A fast high-order boundary integral solver. Journal of Computational Physics, 397. doi:10.1016/j.jcp.2019.06.067
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  We present a boundary integral equation solver for computing Taylor relaxed states in non-axisymmetric solid and shell-like toroidal geometries. The computation of Taylor states in these geometries is a key element for the calculation of stepped pressure stellarator equilibria. The integral representation of the magnetic field in this work is based on the generalized Debye source formulation, and results in a well-conditioned second-kind boundary integral equation. The integral equation solver is based on a spectral discretization of the geometry and unknowns, and the computation of the associated weakly-singular integrals is performed with high-order quadrature based on a partition of unity. The resulting scheme for applying the integral operator is then coupled with an iterative solver and suitable preconditioners. Several numerical examples are provided to demonstrate the accuracy and efficiency of our method, and a direct comparison with the leading code in the field is reported.
• Despres, B., Imbert-Gerard, L., & Lafitte, O. (2017). Solutions to the cold plasma model at resonances. Journal de l'Ecole Polytechnique - Mathematiques, 4. doi:10.5802/jep.41
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  Little is known on the mathematical theory of hybrid and cyclotron solutions of Maxwell’s equations with the cold plasma dielectric tensor. These equations arise in magnetized plasmas to model an electromagnetic wave in a Tokamak. The solutions can behave extremely di erently from those in vacuum. This work contributes to the local theory of the hybrid resonance by means of an original representation formula based on special functions, a certain eikonal equation and with a careful treatment of the singularity.
• Greengard, L., & Imbert-Gérard, L. (2017). Pseudo-Spectral Methods for the Laplace-Beltrami Equation and the Hodge Decomposition on Surfaces of Genus One. Numerical Methods for Partial Differential Equations, 33(3). doi:10.1002/num.22131
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  The inversion of the Laplace-Beltrami operator and the computation of the Hodge decomposition of a tangential vector field on smooth surfaces arise as computational tasks in many areas of science, from computer graphics to machine learning to computational physics. Here, we present a high-order accurate pseudo-spectral approach, applicable to closed surfaces of genus one in three-dimensional space, with a view toward applications in plasma physics and fluid dynamics. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 941–955, 2017.
• Monk, P., & Imbert-Gérard, L. (2017). Numerical simulation of wave propagation in inhomogeneous media using Generalized Plane Waves. ESAIM: Mathematical Modelling and Numerical Analysis, 51(4). doi:10.1051/m2an/2016067
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  The Trefftz Discontinuous Galerkin (TDG) method is a technique for approximating the Helmholtz equation (or other linear wave equations) using piecewise defined local solutions of the equation to approximate the global solution. When coefficients in the equation (for example, the refractive index) are piecewise constant it is common to use plane waves on each element. However when the coefficients are smooth functions of position, plane waves are no longer directly applicable. In this paper we show how Generalized Plane Waves (GPWs) can be used in a modified TDG scheme to approximate the solution for piecewise smooth coefficients in two dimensions. GPWs are approximate solutions to the equation that reduce to plane waves when the medium through which the wave propagates is constant. We shall show how to modify the TDG sesquilinear form to allow us to prove convergence of the GPW based version. The new scheme retains the high order convergence of the original TDG scheme (when the solution is smooth) and also retains the same number of degrees of freedom per element (corresponding to the directions of the GPWs). Unfortunately it looses the advantage that only skeleton integrals need to be performed. Besides proving convergence, we provide numerical examples to test our theory.
• Ambikasaran, S., Borges, C., Imbert-Gerard, L., & Greengard, L. (2016). Fast, adaptive, high-order accurate discretization of the Lippmann-Schwinger equation in two dimensions. SIAM Journal on Scientific Computing, 38(3). doi:10.1137/15M102455X
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  We present a fast direct solver for two-dimensional scattering problems, where an incident wave impinges on a penetrable medium with compact support. We represent the scattered field using a volume potential whose kernel is the outgoing Green's function for the exterior domain. Inserting this representation into the governing partial differential equation, we obtain an integral equation of Lippmann-Schwinger type. The principal contribution here is the development of an automatically adaptive, high-order accurate discretization based on a quad-tree data structure which provides rapid access to arbitrary elements of the discretized system matrix. This permits the straightforward application of state-of-the-art algorithms for constructing compressed versions of the solution operator. These solvers typically require O(N3/2) work, where N denotes the number of degrees of freedom. We demonstrate the performance of the method for a variety of problems in both low and high frequency regimes.
• Imbert-Gérard, L. (2015). Interpolation properties of generalized plane waves. Numerische Mathematik, 131(4). doi:10.1007/s00211-015-0704-y
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  This paper is aimed at developing new shape functions adapted to the scalar wave equation with smooth (possibly vanishing) coefficients and investigates the numerical analysis of their interpolation properties. The interpolation is local, but high order convergence is shown with respect to the size of the domain considered. The new basis functions are then implemented in a numerical method to solve a scalar wave equation problem with a mixed boundary condition. The main theoretical result states that any given order of approximation can be achieved by an appropriate choice of parameters for the design of the shape functions. The convergence is studied with respect to the size of the domain, which is referred to in the literature as $$h$$h-convergence.
• Imbert-Gérard, L. (2015). Well-posedness and generalized plane waves simulations of a 2D mode conversion model. Journal of Computational Physics, 303. doi:10.1016/j.jcp.2015.09.033
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  Certain types of electro-magnetic waves propagating in a plasma can undergo a mode conversion process. In magnetic confinement fusion, this phenomenon is very useful to heat the plasma, since it permits to transfer the heat at or near the plasma center. This work focuses on a mathematical model of wave propagation around the mode conversion region, from both theoretical and numerical points of view. It aims at developing, for a well-posed equation, specific basis functions to study a wave mode conversion process. These basis functions, called generalized plane waves, are intrinsically based on variable coefficients. As such, they are particularly adapted to the mode conversion problem. The design of generalized plane waves for the proposed model is described in detail. Their implementation within a discontinuous Galerkin method then provides numerical simulations of the process. These first 2D simulations for this model agree with qualitative aspects studied in previous works.
• Després, B., & Imbert-Gérard, L. (2014). A generalized plane-wave numerical method for smooth nonconstant coefficients. IMA Journal of Numerical Analysis, 34(3). doi:10.1093/imanum/drt030
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  We propose an original method based on generalized plane waves and approximated coefficients for the numerical approximation of the Helmholtz equation with a smooth nonconstant coefficient. This is justified by a high-order convergence estimate rate. Our motivation stems from Maxwell's equations with Hermitian dielectric tensor ε which are used to model reflectometry in fusion plasma. Simplified models split them into two different propagation modes. Some numerical results are presented in dimensions 1 and 2. © 2013 The authors 2013. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
• Weder, R., Després, B., & Imbert-Gérard, L. (2014). Hybrid resonance of Maxwell's equations in slab geometry. Journal des Mathematiques Pures et Appliquees, 101(5). doi:10.1016/j.matpur.2013.10.001
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  Hybrid resonance is a physical mechanism for the heating of a magnetic plasma. In our context hybrid resonance is a solution of the time harmonic Maxwell's equations with smooth coefficients, where the dielectric tensor is a non-diagonal Hermitian matrix. The main part of this work is dedicated to the construction and analysis of a mathematical solution of the hybrid resonance with the limit absorption principle. We prove that the limit solution is singular: it consists of a Dirac mass at the origin plus a principal value and a smooth square integrable function. The formula obtained for the plasma heating is directly related to the singularity. © 2013 Elsevier Masson SAS.

Presentations

• Imbert-Gerard, L. (2021). A discussion around the magnetic differential equation. PPPL theory seminar. online: Princeton Plasma Physics Lab.
• Imbert-Gerard, L. (2021). Bits and pieces of an introduction to Stellarator design The Magnetic Differential Equation. INRIA Cage team seminar. online: INRIA Cage team - Laboratoire Jacques Louis Lions - Sorbonne universite.
• Imbert-Gerard, L. (2021). Bits and pieces of an introduction to Stellarator design The Magnetic Differential Equation. Numerical Methods in Plasma Physics seminar. online: Max Planck Institute for Plasma Physics - Garching.
• Imbert-Gerard, L. (2021). Magnetic coordinates. PPPL Graduate Summer School. online: Princeton Plasma Physics Lab.
• Imbert-Gerard, L. (2021). The Magnetic Differential Equation. PPPL Graduate Summer School. online: Princeton Plasma Physics Lab.
• Imbert-Gerard, L. (2021). Wave propagation in inhomogeneous media: A new family of Generalized Plane Waves. SIAM annual meeting. online: SIAM.
• Imbert-Gerard, L. (2021). Wave propagation in inhomogeneous media: An introduction to Generalized Plane Waves. Applied Mathematics and Statistics Colloquium. online: Colorado school of Mines - Department of Applied Mathematics and Statistics.
• Imbert-Gerard, L. (2021). Wave propagation in inhomogeneous media: An introduction to quasi-Trefftz methods. numerical analysis and PDE Seminar (Nice). online: University of Nice - Mathematiques.
• Imbert-Gerard, L., & Sylvand, G. (2021). Three quasi-Trefftz bases for the 3D convected Helmholtz equation. European Congress on Computational Methods in Applied Sciences and Engineering. online.
• Imbert-Gerard, L., Paul, E., & Wright, A. (2021). 3D magnetic fields and stellarator design: Challenges and Models. World Congress of Computational Mechanics. online.
• Imbert-Gerard, L., Paul, E., & Wright, A. (2021). An Introduction to Stellarators: a periodic transport equation. SIAM Conference on Computational Science and Engineering. online: SIAM.