Magique-3D is a joint project-team between Inria and the Department of Applied Mathematics (LMA) of the University of Pau, which is associated with CNRS. Gathering several researchers of different backgrounds in geophysics, physics, mathematics and scientific computing, MAGIQUE-3D team aims at developing sophisticated modeling tools, validating them in a rigorous way and applying them to real cases of geophysical interest. This project is intrinsically multi-disciplinary and is strongly related to the regional and national industrial environment. In particular, we develop strong collaborations with TOTAL but the topics studied can lead to applications other than petroleum engineering. During the period 2005-09, the research program of MAGIQUE-3D was mainly composed of two main topics that structured the original parts of the activities of the group. The first topic, entitled `Depth Imaging', was related to modeling of seismic wave propagation in complex geological structures, taking into account underlying physical phenomena. It has been defined jointly by working groups composed of members of MAGIQUE-3D and of its main industrial partner TOTAL in order to make sure that actual results of interest in the context of the oil industry could be reached. One usually tackles such problems by defining approximate models that either lead to less expensive numerical methods (for example by decreasing the number of unknowns by means of an approximation of the original equations), or to high-performance numerical methods applied to the full system, which leads to an accurate solution but implies a high computation cost. Both of these approaches have been considered in the project.

The second topic, that could be given the general title `Advanced modeling in wave propagation', was related to the realistic numerical simulation of complex three-dimensional geophysical phenomena and its comparison with real data recorded in the field. One of the main issues was the choice of the numerical method, which implicitly defines the subset of configurations that can be studied. Comparisons with recorded seismic data for real geological cases have been carried out and then, numerical algorithms have been optimized and implemented on parallel computers with a large number of processors and a large memory size, within the framework of message-passing programming. We have reached a maximum resolution in terms of the seismic frequencies that can be accurately simulated on currently available supercomputers.

During the period 2005-2009, MAGIQUE-3D has worked on the development of optimized software for the simulation of 3D phenomena in geophysics. The team tackled this question addressing different and complementary issues such as the development of new discretization schemes, the construction of new boundary conditions used to reduce the size of the computational domain, the porting of our software on GPU to speed up their performances. All the algorithms we have proposed are compatible with high resolution techniques. We now would like to continue working on the same subjects but also to apply our knowledge on the direct problem to the solution of inverse problems. It is now a natural goal for the team since we develop a significant research program with Total, in particular in the context of the research program DIP (Depth Imaging Partnership), where the solution of inverse problems has become a big challenge for oil industry.

**Inverse scattering problems.**The determination of the shape of an obstacle from its effects on known acoustic or electromagnetic waves is an important problem in many technologies
such as sonar, radar, geophysical exploration, medical imaging and nondestructive testing. This inverse obstacle problem (IOP) is difficult to solve, especially from a numerical viewpoint,
because it is ill-posed and nonlinear
. Moreover the precision in the reconstruction of the shape of
an obstacle strongly depends on the quality of the given far-field pattern (FFP) measurements: the range of the measurements set and the level of noise in the data. Indeed, the numerical
experiments (for example
,
,
,
) performed in the resonance region, that is, for a wavelength
that is approximately equal to the diameter of the obstacle, tend to indicate that in practice, and at least for simple shapes, a unique and reasonably good solution of the IOP can be often
computed using only one incident wave and
*full aperture*far-field data (FFP measured only at a limited range of angles), as long as the aperture is larger than

This plus the fact that from a mathematical viewpoint the FFP can be determined on the entire sphere
*analytic*function, we propose
,
a solution methodology to extend the range of FFP data when
measured in a limited aperture and not on the entire sphere

We would like also to consider electrical impedance tomography, which is a technique to recover spatial properties of the interior of an object from measurements of the potential of the boundary of the object (see by Liliana Borcea and by Martin Hanke and Martin Brühl). In shape identification problems, the measured quantities do not depend linearly on the shape of the obstacle. Most popular approaches describe the objects by appropriate parameterizations and compute the parameters by iterative schemes based on Newton-type methods which require to solve a collection of direct problems. We plan to begin with this kind of approaches since we already have an efficient solver for the direct problem and these iterative schemes are known to be very successful in many cases. Their main disadvantage is that they are expensive since they must solve a direct problem at each step. We hope that our solver will be sufficiently optimized to limit this disadvantage.

**Depth Imaging in the context of DIP.**The challenge of seismic imaging is to obtain the best representation of the subsurface from the solution of the full wave equation that is the
best mathematical model according to the time reversibility of its solution. The most used technique of imaging is RTM (Reverse Time Migration),
, which is an iterative process based on the solution of a
collection of wave equations. The high complexity of the propagation medium requires the use of advanced numerical methods, which allows one to solve several wave equations quickly and
accurately. The research program DIP has been defined by researchers of
Magique-3D and engineers of
Totaljointly. It has been created with the aim of gathering researchers of INRIA, with different backgrounds and the scientific programm will be
coordinated by
Magique-3D. In this context,
Magique-3D will contribute by working on the inverse problem and by continuing to develop new algorithms in order to improve the RTM.

**Tomography.**Seismic tomography allows one to describe the geometry and the physical characteristics of the heterogeneities inside the earth by analyzing the propagation speed of the
seismic waves. The last past ten years have known a lot of developments like the introduction of sensitivity kernels which complete the rai theory which is often used in short period
seismology. However the kernel sensitivity theory introduces very large matrices and the computations which are necessary to solve the inverse problem are very expensive. The idea would be
to represent the kernels by a reduced number of parameters by using appropriate methods of compression. The wavelets of Haar have been used by Chevrot and Zhao
but they do not seem to be optimal. We propose to address
this kind of issue by aiming at giving parcimonious representations of kernels of sensitivity.

**Potential techniques Inversion: parallel Hybrid local/global optimization.**In many applications, acoustic and seismic inversion are not enough to reconstruct multiphase component
structures. Different potential techniques like electrical capacitance, resistivity, gravimetry and magnetometry are necessary. As potential techniques require the resolution of Poisson or
Laplace-like equations, huge linear systems need to be solved using very large multi-CPU/multi-GPU clusters. Today, finite volume/conjugate gradient solvers are running on 200 processors
for electrical capacitance and gravimetry problems at CINES/Montpellier supercomputing center as a proof of concept. The very promising results obtained lead us to run them on more than
2000 CPUs and perhaps 200 or 300 GPU clusters. By developing higher order versions we will be able to increase significantly the accuracy of the solutions and the speed of calculations. As
the inversion process is performed iteratively, it should be worthwhile to incorporate at the same time local (least square methods) and global (neighborhood/very fast simulated annealing)
optimization techniques. An acceptable model could then be taken as the new current model and at some degree, data compression will be used in order to compute an accurate sensitivity
matrix for this current model computed with local/global optimization. Then, using local/global optimization, purely sensitivy matrix based inversion could be used to accelerate all the
inversion processes. In the case of electrical capacitance tomography, the forward problem is accelerated by almost a factor of 100 when a GPU is preferred to a CPU. On a multiCPU/multiGPU,
an asynchronous strategy of communications between processors and copies of informations between host (CPU) and device (GPU) is retained and will be implemented more properly. We plan to
apply this to joint inversion at the regional and global earth scales. A collaboration with CAPS entreprises and GENCI has been approved in November 2009 for the multi GPU porting of a 3D
finite volume code implemented using MPI by Roland Martin. On a signe GPU an acceleration factor of 23 has been already obtained. This collaboration is under its way. We have the intention
to extend this to high order spectral element method in the context of AHPI ANR project in 2010 by taking the SPECFEM3D parallel code as a fundamental code that will be transformed into an
elliptic large system solver.

The main activities of Magique-3D in modeling are the derivation and the analysis of models that are based on mathematical physics and are suggested by geophysical problems. In particular, Magique-3D considers equations of interest for the oil industry and focus on the development and the analysis of numerical models which are well-adapted to solve quickly and accurately problems set in very large or unbounded domains as it is generally the case in geophysics.

**High-Order Schemes in Space and Time.**Using the full wave equation for migration implies very high computational burdens, in order to get high resolution images. Indeed, to improve the
accuracy of the numerical solution, one must considerably reduce the space step, which is the distance between two points of the mesh representing the computational domain. Obviously this
results in increasing the number of unknowns of the discrete problem. Besides, the time step, whose value fixes the number of required iterations for solving the evolution problem, is linked
to the space step through the CFL (Courant-Friedrichs-Levy) condition. The CFL number defines an upper bound for the time step in such a way that the smaller the space step is, the higher the
numbers of iterations (and of multiplications by the stiffness matrix) will be. The method that we proposed in
allows for the use of local time-step, adapted to the various
sizes of the cells and we recently extended it to deal with

We are also considering an alternative approach to obtain high-order schemes. The main idea is to apply first the time discretization thanks to the modified equation
technique and after to consider the space discretization. Our approach involves

Once we have performant

**Mixed hybrid finite element methods for the wave equation.**The new mixed-hybrid-like method for the solution of Helmholtz problems at high frequency we have built enjoys the three
following important properties: (1) unlike classical mixed and hybrid methods, the method we proposed is not subjected to an inf-sup condition. Therefore, it does not involve numerical
instabilities like the ones that have been observed for the DGM method proposed by Farhat and his collaborators
,
. We can thus consider a larger class of discretization spaces
both for the primal and the dual variables. Hence we can use unstructured meshes, which is not possible with DGM method (2) the method requires one to solve Helmholtz problems which are set
inside the elements of the mesh and are solved in parallel(3) the method requires to solve a system whose unknowns are Lagrange multipliers defined at the interfaces of the elements of the
mesh and, unlike a DGM, the system is hermitian and positive definite. Hence we can use existing numerical methods such as the gradient conjugate method. We intend to continue to work on this
subject and our objectives can be described following three tasks: (1) Follow the numerical comparison of performances of the new methods with the ones of DGM. We aim at considering high
order elements such as R16-4, R32-8, ...; (2)Evaluate the performance of the method in case of unstructured meshes. This analysis is very important from a practical point of view but also
because it has been observed that the DGM deteriorates significantly when using unstructured meshes; (3) Extend the method to the 3D case. This is the ultimate objective of this work since we
will then be able to consider applications.

Obviously the study we propose will contain a mathematical analysis of the method we propose. The analysis will be done in the same time and we aim at establishing a priori and a posteriori estimates, the last being very important in order to adopt a solution strategy based on adaptative meshes.

**Boundary conditions.**The construction of efficient absorbing conditions is very important for solving wave equations, which are generally set in unbounded or very large domains. The
efficiency of the conditions depends on the type of waves which are absorbed. Classical conditions absorb propagating waves but recently new conditions have been derived for both propagating
and evanescent waves in the case of flat boundaries. MAGIQUE-3D would like to develop new absorbing boundary conditions whose derivation is based on the full factorization of the wave
equation using pseudodifferential calculus. By this way, we can take the complete propagation phenomenon into account which means that the boundary condition takes propagating, grazing and
evanescent waves into account, and then the absorption is optimized. Moreover our approach can be applied to arbitrarily-shaped regular surfaces.

We intend to work on the development of interface conditions that can be used to model rough interfaces. One approach, already applied in electromagnetism , consists in using homogenization methods which describes the rough surface by an equivalent transmission condition. We propose to apply it to the case of elastodynamic equations written as a first-order system. In particular, it would be very interesting to investigate if the rigorous techniques that have been used in , can be applied to the theory of elasticity. This type of investigations could be a way for MAGIQUE-3D to consider medical applications where rough interfaces are often involved. Indeed, we would like to work on the modelling and the numerical simulation of ultrasonic propagation and its interaction with partially contacting interfaces, for instance bone/titanium in the context of an application to dentures, in collaboration with G. Haiat (University of Paris 7).

**Asymptotic modeling**.

In the context of wave propagation problems, we are investigating physical problems which involves multiple scales. Due to the presence of boundary layers (and/or thin layers, rough interfaces, geometric singularities), the direct numerical simulation (DNS) of these phenomenas involves a large numbers of degrees of freedom and high performance computing is required. The aim of this work is to develop credible alternatives to the DNS approach.

Performing a multi-scale asymptotic analysis, we derive approximate models whose solution can be computed for a low computational cost. We study these approximate models mathematically (well-posedness, uniform error estimates) and numerically (we compare the solution of these approximate models to the solution of the initial model computed with high performance computating).

We are mostly interested in the following problems.

Eddy current modeling in the context of electrothermic applications for the design of electromagnetic devices in collaboration with laboratories Ampre, Laplace, INRIA Team MC2, IRMAR, and F.R.S.-FNRS;

ultrasonic wave propagation through bone-titanium media in medicine in collaboration with INRIA Team MC2, and MSME;

asymptotic modeling of multi perforate plates in turbo reactors in collaboration with Cerfacs, INSA-Toulouse, Onera and Snecma in the framework of the ANR APAM.

**Nonlinear problems in fluid dynamics.**In order to model heat transfers, fluid-solid interactions, in particular landslides and tsunamis induced by earthquakes, tremors induced by fluid
motions in volcanoes, sharp solid-to-fluid transitions in some planets, it is of crucial importance to develop efficient parallel solvers on multicore/multi-processor supercomputing
platforms. High order finite volumes introducing compact schemes or spectral-like integrations as well as high order finite elements and their related high order boundary conditions are
needed to take into account, at the same time, discontinuities in geological structures, sharp variations and shocks in fluid velocities and properties (density, pressure and temperature),
and the coupling between both codes. Discrete Galerkin techniques, spectral finite volumes or finite-volume techniques should be taken into account in compact schemes in order to reduce
drastically the memory storage involved and compute larger models. Viscous compressible and incompressible codes need to be solved using non-conforming meshes between solid and fluid, and
large linear systems need to be solved on very huge multi-CPU/multi-GPU supercomputers. Moving meshes close to the interface between solids and fluids should be taken into account by dynamic
or adaptive remeshing. Furthermore we developped PML for the full compressible Navier-Stokes system of equations
using finite-differences discretization in curvilinear
coordinates and we are planning to extend PML conditions to both compressible and incompressible viscous flows in the context of high order finite volumes or Discontinuous galerkin
methods.

Another direction that we would like to consider would be the use of solitons in nonlinear problems. Indeed, a soliton is an interesting tool for modeling and explaining some nonlinear phenomena. For example tsumanis are sometimes explained by the emergence of solitons created by earth tremor. Strain solitons can be also used to explain the propagation of breaking in solids . Therefore it would be interesting to investigate more this issue.

A tremendous increase of the sustained power of supercomputers has occurred in the last few years, in particular with the first `petaflops' machines that have been built in the USA and also with new technology such as general-purpose computing on graphics cards (so-called `GPU computing'). Nowadays, one has access to powerful numerical methods that, when implemented on supercomputers, make it possible to simulate both forward and inverse seismic wave propagation problems in complex three-dimensional (3D) structures. Moreover, very spectacular progress in computer science and supercomputer technology is amplified by recent advances in High Performing Computing (HPC) both from a software and hardware point of view. One can in this respect say that HPC should make it possible in the near future to perform large-scale calculations and inversion of geophysical data for models and distributed data volumes with a resolution impossible to reach in the past. Our group has for instance already run simulations in parallel on 150,000 processor core, obtaining an excellent sustained performance level and almost perfect performance scaling .

We will therefore work on three HPC issues in the next few years. The first will be very large scale inversion of seismic model based on sensitivity kernels. In the context of a collaboration with TOTAL and also with Prof. Jeroen Tromp at Princeton University (USA), we will use adjoint simulations and sensitivity kernels to solve very-large scale inverse problems for seismology and for oil industry models, for instance deep offshore regions and/or complex foothills regions or sedimentary basins. The second issue is Graphics Processing Unit (GPU) computing: in the context of a collaboration with Prof. Gordon Erlebacher (Florida State University, USA) and Dr. Dominik Göddeke (Technical University of Dortmund, Germany) we have modified our existing seismic wave propagation software packages to port them to GPU computing in order to reach speedup factors of about 20x to 30x on GPU clusters (for instance at GENCI/CEA CCRT in Bruyères-le-Châtel, France). The third issue is porting our software packages to Symmetric Multi Processors (SMP) massive multicore computing to take advantage of future processors, which will have a large number of cores on petaflops or exaflops machine. In the context of a collaboration with Prof. Jesús Labarta and Prof. Rosa Badia from the Barcelona Supercomputing Center (Catalonia, Spain) we will use their 'StarSs' programming environment to take advantage of multicore architectures while keeping a flexible software package relatively simple to modify for geophysicists that may not be computer-programming experts.

The main objective of modern seismic processing is to find the best representation of the subsurface that can fit the data recorded during the seismic acquisition survey. In this context, the seismic wave equation is the most appropriate mathematical model. Numerous research programs and related publications have been devoted to this equation. An acoustic representation is suitable if the waves propagate in a fluid. But the subsurface does not contain fluids only and the acoustic representation is not sufficient in the general case. Indeed the acoustic wave equation does not take some waves into account, for instance shear waves, turning waves or the multiples that are generated after several reflections at the interfaces between the different layers of the geological model. It is then necessary to consider a mathematical model that is more complex and resolution techniques that can model such waves. The elastic or viscoelastic wave equations are then reference models, but they are much more difficult to solve, in particular in the 3D case. Hence, we need to develop new high-performance approximation methods.

Reflection seismics is an indirect measurement technique that consists in recording echoes produced by the propagation of a seismic wave in a geological model. This wave is created artificially during seismic acquisition surveys. These echoes (i.e., reflections) are generated by the heterogeneities of the model. For instance, if the seismic wave propagates from a clay layer to sand, one will observe a sharp reflected signal in the seismic data recorded in the field. One then talks about reflection seismics if the wave is reflected at the interface between the two media, or talks about seismic refraction if the wave is transmitted along the interface. The arrival time of the echo enables one to locate the position of this transition, and the amplitude of the echo gives information on some physical parameters of the two geological media that are in contact. The first petroleum exploration surveys were performed at the beginning of the 1920's and for instance, the Orchard Salt Dome in Texas (USA) was discovered in 1924 by the seismic-reflection method.

We already applied our techniques to the study of strong ground motion and associated seismic risk in the Los Angeles basin area. This region consists of a basin of great dimension (more
than 100 km

We wish to improve these studies of seismic risk in densely populated areas by considering other regions of the world, for example the Tokyo basin, the area of Kobe or the Mexico City region. We also plan to generalize this type of calculations to the knowledge and modeling of site effects, i.e. of the local amplification of the response of the ground to seismic excitation. The study of such effects is an important observation in urban areas to be able to anticipate the damage to constructions and, if necessary, to plan the organization of search and rescue operations. It is also a significant element of the definition of paraseismic standards. Site effects can be determined experimentally, but that requires the installation of stations for a sufficient period of time to record a few tens of seismic events. Numerical modeling makes it possible to avoid this often long and difficult experimentation, assuming of course that one has good knowledge of the geological structure of the subsurface in the studied area. We thus propose in the Magique-3Dproject to use the numerical techniques mentioned above for instance to quantify the effects of topographic variations in the structure.

The problems of seismic imaging can be related to non destructive testing, in particular medical imaging. For instance, the rheumatologist are now trying to use the propagation of ultrasounds in the body as a noninvasive way to diagnose osteoporosis. Then, the bones can be regarded as elastodynamic or poroelastic media while the muscles and the marrow can be regarded as acoustic media. Hence the computational codes we use for seismic imaging could be applied to such a problem.

The Magique-3D project is based (in part) on existing software packages, which are already validated, portable and robust. The SPECFEM3D software package, developed by Dimitri Komatitsch and his colleagues in collaboration with Jeroen Tromp and his colleagues at the California Institute of Technology and at Princeton University (USA), and which is still actively maintained by Dimitri Komatitsch and his colleagues, allows the precise modeling of seismic wave propagation in complex three-dimensional geological models. Phenomena such as anisotropy, attenuation (i.e., anelasticity), fluid-solid interfaces, rotation, self-gravitation, as well as crustal and mantle models can be taken into account. The software is written in Fortran95 with MPI message-passing on parallel machines. It won the Gordon Bell Prize for best performance of the Supercomputing'2003 conference. In 2006, Dimitri Komatitsch established a new collaboration with the Barcelona Supercomputing Center (Spain) to work on further optimizing the source code to prepare it for very large runs on future petaflops machines to solve either direct or inverse problems in seismology. Optimizations have focused on improving load balancing, reducing the number of cache misses and switching from blocking to non-blocking MPI communications to improve performance on very large systems. Because of its flexibility and portability, the code has been run successfully on a large number of platforms and is used by more than 150 academic institutions around the world. In November 2008 this software package was again among the six finalists of the pretigious Gordon Bell Prize of the SuperComputing'2008 conference in the USA for a calculation performed in parallel on 150,000 processor cores, reaching a sustained performance level of 0.16 petaflops.

This software, written in FORTRAN 90, simulates the propagation of acoustic waves in heterogeneous 2D and 3D media. It is based on an Interior Penalty Discontinuous Galerkin Method (IPDGM).
The 2D version of the code has been implemented in the Reverse Time Migration (RTM) software of
Total in the framework of the PhD. thesis of Caroline Baldassari and the 3D version should be implemented soon. The 2D code allows for the use of
meshes composed of cells of various order (
**High-Order Schemes in Space and Time**which permits not only the use of different time-step, but also to adapt the order of the time-discretization to the order of each cells (

The main competitors of Hou10ni are codes based on Finite Differences, Spectral Element Method or other Discontinuous Galerkin Methods (such as the ADER schemes). During her PhD. thesis,
Caroline Baldassari compared the solution obtained by Hou10ni to the solution obtained by a Finite Difference Method and by a Spectral Element Method (SPECFEM). To evaluate the accuracy of the
solutions, we have compared it to analytical solutions provided by the codes Gar6more (see below). The results of these comparisons is: a) that Hou10ni outperforms the Finite Difference Methods
both in terms of accuracy and of computational burden and b) that its performances are similar to Spectral Element Methods. Since Hou10ni allows for the use of meshes based on tetraedrons,
which are more appropriate to mesh complex topographies, and for the

This code computes the analytical solution of problems of waves propagation in two layered 3D media such as- acoustic/acoustic- acoustic/elastodynamic- acoustic/porous- porous/porous,based on the Cagniard-de Hoop method.

See also the web page
http://

The main objective of this code is to provide reference solutions in order to validate numerical codes. They have been already used by J. Tromp and C. Morency to validate their
code of poroelastic wave propagation
. They are freely distributed under a CECILL licence and can be
downloaded on the website
http://

ACM: J.2

AMS: 34B27 35L05 35L15 74F10 74J05

Programming language: Fortran 90

The determination of the shape of an obstacle from its effects on known acoustic or electromagnetic waves is an important problem in many technologies such as sonar, radar, geophysical exploration, medical imaging and nondestructive testing. This inverse obstacle problem (IOP) is difficult to solve, especially from a numerical viewpoint, because it is ill-posed and nonlinear. Its investigation requires as a prerequisite the fundamental understanding of the theory for the associated direct scattering problem, and the mastery of the corresponding numerical solution methods.

In this work, we are interested in retrieving the shape of an elastic obstacle from the knowledge of some scattered far-field patterns, and assuming certain characteristics
of the surface of the obstacle. The corresponding direct elasto-acoustic scattering problem consists in the scattering of time-harmonic acoustic waves by an elastic obstacle

where

This boundary value problem has been investigated mathematically and results pertaining to the existence, uniqueness and regularity can be found in
and the references therein, among others. We propose a solution
methodology based on a regularized Newton-type method for solving the IOP. The proposed method is an extension of the regularized Newton algorithm developed for solving the case where only
Helmholtz equation is involved, that is the acoustic case by impenetrable scatterers
. The direct elasto-acoustic scattering problem defines an
operator

We propose a solution methodology based on a regularized Newton-type method to solve this inverse obstacle problem. At each Newton iteration, we solve the forward problem using a finite element solver based on discontinuous Galerkin approximations, and equipped with high-order absorbing boundary conditions. We have first characterized the Fréchet derivatives of the scattered field. They are solution to the same boundary value problem as the direct problem with other transmission conditions. This work has been presented both in FACM11 and in WAVES 2011. A paper has been submitted.

In order to improve the subsoil images in regions which are not well covered by a dense seismic array or can not be well retrieved by using seismic imaging techniques alone (salty dome
regions like in the Gulf of Mexico for instance), we have been developping new gravity imaging techniques using supercomputing. In regions like Ghana or the Chicxulub crater located in
Yucatan plate (Mexico), 3D sensitivity kernels are calculated for gravity potential data sets measured over 2000 up to 10000 locations randomly distributed in space. The density anomaly
computational domain covers a 250 km

We investigate analytically the asymptotic behavior of high-order spurious prolate spheroidal modes induced by a second-order local approximate DtN absorbing boundary condition (DtN2) when employed for solving high-frequency acoustic scattering problems. We prove that these reflected modes decay exponentially in the high frequency regime. This theoretical result demonstrates the great potential of the considered absorbing boundary condition for solving efficiently exterior high-frequency Helmholtz problems. In addition, this exponential decay proves the superiority of DtN2 over the widely used Bayliss-Gunsburger-Turkel absorbing boundary condition. This work has been accepted for publication in Progress In Electromagnetics Research B. .

The modeling of wave propagation problems using finite element methods usually requires the truncation of the computational domain around the scatterer of interest. Absorbing boundary condition are classically considered in order to avoid spurious reflections. In this paper, we investigate some properties of the Dirichlet to Neumann map posed on a spheroidal boundary in the context of the Helmholtz equation. We focus on the impedance coefficients defining the DtN condition and we aim at establishing suitable properties in order to propose an accurate numerical method for their computation. Then, we state the well-posedness of the corresponding mixed problem and propose a variational formulation adapted to a finite element discretization. This work has been submitted.

The new method involving

The Interior Penalty Discontinuous Galerkin Method , , we use in the IPDGFEM code requires the introduction of a penalty parameter. Except for regular quadrilateral or cubic meshes, the optimal value of this parameter is not explicitely known. Moreover, the condition number of the resulting stiffness matrix is an increasing function of this parameter, but the precise behaviour has not been explicited neither. We have carried out a theoretical and numerical study of the CFL condition for quadrilateral and cubic meshes, which is presented in the Phd thesis of Cyril Agut . These results were also presented at the peer-reviewed conference Waves 2011 (Vancouver, Canada, July 2011).

The numerical simulation of wave propagation is generally performed by truncating the propagation medium. Absorbing boundary conditions are then needed. We construct a new family of absorbing boundary conditions from the factorization of the wave equation formulated as a first order system. Using the method of M.E. Taylor, we show that we can generate an infinite number of boundary conditions which can not be obtained via the Niremberg’s factorization method. The conditions can be applied on arbitrarily-shaped surfaces and involve second-order derivatives. We then propose a reduced formulation of the wave equation using an auxiliary unknown which is defined on the regular surface only. The reduced problem allows one to easily include the boundary conditions inside the variational formulation. The corresponding boundary value problem remains well-posed in suitable Hilbert spaces and we give a demonstration in a framework that is suitable to applications. We then study the long-time behavior of the wave field and we show that it tends to 0 as time tends to infinity. This provides a weak stability result that should be completed in the second part of this work. We have then decided to improve the stability result by performing a quantitative study of the energy. We have then shown that the energy is exponentially decaying if the obstacle is star-shaped and the external boundary is convex. This work has been published as INRIA Resarch Reports , and two papers are submitted. We have next addressed the issue of enriching these ABCs by representing evanescent and damping waves. This has given rise to a work for the Hemlholtz equation and we have shown that the enriched ABCs performed better than standard ABCs. The xetension to the acoustic wave has led to new conditions involvin fractional derivatives. To the bast of our knowledge, it is the first time that fractional derivatives have been used for optimizing the performance of ABCs. These new results have been presented in two seminars (University of Bordeaux I and University of Genova) and in two conferences (FACM11 and WAVES 2011). A paper has been published for the case of evanescent waves and two papers are in preparation. All these results are presented in the PhD thesis of Véronique Duprat .

In order to justify the use of our code IPDGFEM for the Reverse Time Migration, we have carried out a performance analysis of the Interior Penalty Discontinuous Galerkin method and of the Spectral Element Method. This analysis, which shows that IPDG performs as well as SEM, has been presented in .

Another aspect of the work concerns the design of local time-stepping algorithms. The local-time stepping strategy proposed in allows for high-order time schemes where the time scheme is adapted to the various space step of the mesh. However, when the mesh contains both low-order and high-order cells, this method not allows for the adaptation of the order of the time-scheme to the order of the cells. We have then presented a new local time-stepping algorithm where both the order of the scheme and the time step vary in the different parts of the mesh. This method has been presented in and at the peer-reviewed conferences Waves 2011 (Vancouver, Canada, July 2011) and DD20 (Domain Decompostion, San Diego, USA, February 2011).

The local-time stepping algorithm is not adapted to handle dissipation terms. A method has been proposed in , but it is based on an Adams-Bashworth scheme and it requires the storage of additional unknowns. We can not use this scheme for the simulation of seismic waves in very large heterogeneous domains due to memory limitation. We are now working on the design of alternative schemes which would not require the introduction of the auxiliary unknowns. This one of the topics of the PhD. thesis of Florent Ventimiglia.

Acoustic engineers use approximate heuristic models to deal with multiperforated plates in liners and in combustion chambers of turbo-engines. These models were suffering from a lack of mathematical justifications and were consequently difficult to improve. Performing an asymptotic analysis (the small parameter is the radius of the perforations), we have justified these models and proposed some improvement. Our theoretical results have been compared to numerical simulations performed at CERFACS (M'Barek Fares) and to acoustical experiments realized at ONERA (Estelle Piot). Two papers are in preparation.

The following results rely on several problematics developed in section
, item
**Asymptotic modeling**.

We present a numerical treatment of rounded and sharp corners in the modeling of 2D electrostatic fields in . This work deals with numerical techniques to compute electrostatic fields in devices with rounded corners in 2D situations. The approach leads to the solution of two problems: one on the device where rounded corners are replaced by sharp corners and the other on an unbounded domain representing the shape of the rounded corner after an appropriate rescaling. Both problems are solved using different techniques and numerical results are provided to assess the efficiency and the accuracy of the techniques.

Realistic numerical simulations of seimic wave propagation are complicated to handle because they must be performed in strongly heterogeneous media. Two different scales must then be taken into account. Indeed, the medium heterogeneities are very small compared to the characteristic dimensions of the propagation medium. To get acccurate numerical solutions, engineers are then forced to use meshes that match the finest scale representing the heterogeneities. Meshing the whole domain with the fine grid leads then to huge linear systems and the computational cost of the numerical method is then very high. It would be thus very interesting to dispose of a numerical method allowing to represent the heterogenities of the medium accurately while computing on a coarse grid. This is the challenge of multiscale approaches like homoegenization or upscaling. In this work, we use an operator-based upscaling method. Operator-based upscaling methods were first developed for elliptic flow problems (see ) and then extended to hyperbolic problems (see , , ). Operator-based upscaling method consists in splitting the solution into a coarse and a fine part. The coarse part is defined on a coarse mesh while the fine part is computed on a fine mesh. In order to speed up calculations, artificial boundary conditions (ABC) are imposed. By enforcing suitable ABCs on the boundary of every cells of the coarse mesh, calculations on the fine grid can be carried out locally. The coarse part is next computed globally on the coarse mesh. Operator-based upscaling methods were so far developed in joint with standard finite element discretisation strategy. In this work, we investigate the idea of combining an operator based upscaling method with discontinous Galerkin finite element methods(DGFEM). To begin with, we have used use the interior penalty method as presented in for elliptic problems and in , for the wave equation. This is a quite natural way of addressing this issue because we can use a software package that has been already developed in the team. The first results that we have obtained seem to indicate that an DG operator based uspcaling method could be interesting essentially in case of staionnary problems. Nevertheless, the numerical analysis of the discretized problem must be continued. This work has been initiated during the internship of Theophile Chaumont-Frelet who was a fourth year engineer student at Rouen INSA. A paper dealing with the case of the Laplace operator will be submitted soon.

In the framework of our collaboration with Total, we are implementing a Discontinuous Galerkin formulation of the first order elastodynamic wave equations in the plateform Diva which is developed by Total. We consider the formulation proposed in for isotropic media. During her post-doc, Caroline Baldassi has implemented a three dimensional code with Perfectly Matched Layers for this formulation. Jérôme Luquel has implemented the 2D version of this code during his internship. In the framework of the internship of Marie Bonnasse and the PhD thesis of Lionel Boillot, we have extended the formulation to Vertical Transverse Isotropic and Tilted Transverse Isotropic media in both 2D and 3D. The introduction of Absorbing Boundary Conditions or of PML is still an open problem for these types of media. It is one of the topics of the PhD thesis of Lionel Boillot.

The version of the code that we are using assumed that the properties of the media (density, velocity,...) are constant on each cells of the mesh. Discontinuous Galerkin methods allow for considering more general configurations, where these properties vary as polynomial functions inside each cells. Hence, it is not necessary to define the interfaces between the different media before constructing the mesh. The discontinuities are taken into account directly inside each cells. Moreover, we are able to consider smoothly varying media. In the framework of the internship of Vanessa Mattesi, we have implemented polynomial velocities in a Discontinuous Galerkin formulation. We have compared the results obtained with this method to the one obtained with piecewise constant properties. We have observed that the new formulation was more accurate and that it allowed for a simpler construction of the mesh. However, these gains do not counterbalance the increase of the computational induced by the new method. We have then concluded that considering piecewise constant properties was more appropriate to model seismic wave propagation.

We have designed a new and efficient solution methodology for solving high-frequency Helmholtz problems. The proposed method is a least-squares based technique that employs variable bases of plane waves at the element level of the domain partition. A local wave tracking strategy is adopted for the selection of the basis at the regional/element level. More specifically, for each element of the mesh partition, a basis of plane waves is chosen so that one of the plane waves in the basis is oriented in the direction of the propagation of the field inside the considered element. The determination of the direction of the field inside the mesh partition is formulated as a minimization problem. Since the problem is nonlinear, we apply Newton's method to determine the minimum. The computation of Jacobians and Hessians that arise in the iterations of the Newtonâs method is based on the exact characterization of the Fréchet derivatives of the field with respect to the propagation directions. Such a characterization is crucial for the stability, fast convergence, and computational efficiency of the Newton algorithm. These results are part of the Master thesis of Sharang Chaudhry (student à CSUN).

Depth Imaging Partnership (DIP)

Period: 2010 January - 2012 december, Management: INRIA Bordeaux Sud-Ouest, Amount: 3600000 euros. 50 000 euros have been devoted to hire an associate engineer (from Oct. 2010 to Sept. 2011).

Optimisation de codes pour la migration terrestre d'ondes élastiques.

Period: 2010 January - 2011 December, Management: INRIA Bordeaux Sud-Ouest, Amount: 60000 euros.

Schémas en temps d'ordre élevé pour la simulation d'ondes élastiques en milieux fortement hétérogènes par des méthodes DG.

Period: 2010 November - 2013 October, Management: INRIA Bordeaux Sud-Ouest, Amount: 150000 euros.

Propagateurs d'ondes élastiques en milieux anisotropes

Period: 2011 November - 2014 October, Management: INRIA Bordeaux Sud-Ouest, Amount: 150000 euros.

In the context of the Associate Team MAGIC.

Period: 2009 January - 2011 December, Total Amount: 15000 USD

The PhD fellowship of Elodie Estecahandy is partially (50%) financed by the Conseil Régional d'Aquitaine.

The PhD fellowship of Vanessa Mattesi is partially (50%) financed by the Conseil Régional d'Aquitaine.

The PhD fellowship of Cyril Agut is financed by the Conseil Général des Pyrénées Atlantiques.

ANR AHPI The endeavour of this project is to develop some methodology for modelling and solving certain inverse problems using tools from harmonic and complex analysis. These problems pertain to deconvolution issues, identification of fractal dimension for Gaussian fields, and free boundary problems for propagation and diffusion phenomena. The target applications concern radar detection, clinical investigation of the human body (e.g. to diagnose osteoporosis from X-rays or epileptic foci from electro/magneto encephalography), seismology, and the computation of free boundaries of plasmas subject to magnetic confinement in a tokamak. Such applications share as a common feature that they can be modeled through measurements of some transform (Fourier, Fourier-Wigner, Riesz) of an initial signal. Its non-local character generates various uncertainty principles that make all of these problems ill-posed. The techniques of harmonic analysis, as developed in each case below, form the thread and the mathematical core of the proposal. They are intended, by and large, to regularize the inverse issues under consideration and to set up constructive algorithms on structured models. These should be used to initialize numerical techniques based on optimization, which are more flexible for modelling but computationally heavy and whose convergence often require a good initial guess. In this context, the development of wavelet analysis in electrical engineering, as well as signal and image processing or singularity detection, during the last twenty years, may serve as an example. However, many other aspects of Fourier analysis are at work in various scientific fields. We believe there is a strong need to develop this interaction that will enrich both Fourier analysis itself and its fields of application, all the more than in France the scientific communities may be more separate than in some other countries.

The project was created in july 2007. Meetings were organized twice a year, alternatively in Orléans, Bordeaux, Sophia and Pau Collaborations have began with the Bordeaux team on the use of bandelet formalism for the seismic inversion and a post-doc, hired in october 2008, had in charge to analyze with us the feasibility of this approach. We have worked on the approximation of seismic propagators involving Fourier integral operators by considering different approaches. From November 2010 to November 2011, we have hired an associate engineer who has worked with us on the development of a software for the gravimetric inversion.

Joint project with BCAM (Basque Center of Applied Mathematics) funded by the Conseil Régional d'Aquitaine and the Basque Government in the framework of the Aquitaine-Euskadi Call. Total Amount: 14 000 euros.

Program: Fonds commun de coopération Aquitaine/Euskadi

Project acronym: AKELARRE

Project title: Méthodes numériques innovantes et logiciels performants pour la simulation de la propagation des ondes électromagnétiques en milieux complexes

Duration: février 2011 - février 2013

Coordinator: Hélène Barucq

Other partners: BCAM (Basque Center of Applied Mathematics), Spain

Abstract: This project brings together the complementary skills in the field of wave propagation of two research teams which are respectively located in Pau and Bilbao. The main objective of this collaboration is to develop innovative numerical methods and to implement powerful software for the simulation of electromagnetic waves in complex media. These waves play an important role in many industrial applications and the development of such software is of great interest for many industrial enterprises located in the region. Theoretical and practical issues are considered. In particular, we focus on the mathematical analysis of boundary conditions that play a crucial role for accurate numerical simulations of waves.

Title: Advance Modelling in Geophysics

INRIA principal investigator: Hélène Barucq

International Partner:

Institution: California State University at Northridge (United States)

Laboratory: Department of Mathematics

Duration: 2006 - 2011

See also:
http://

The main objective of this three-year research program is the design of an efficient solution methodology for solving Helmholtz problems in heterogeneous domains, a key step for solving the inversion in complex tectonics. The proposed research program is based upon the following four pillars:

1. The design, implementation, and the performance assessment of a new hybrid mixed type method (HMM) for solving Helmholtz problems. 2. The construction of local nonreflecting boundary conditions to equip HMM when solving exterior high-frequency Helmholtz problems. 3. The design of an efficient numerical procedure for full-aperture reconstruction of the acoustic far-field pattern (FFP) when measured in a limited aperture. 4. The characterization of the Fr飨et derivative of the elasto-acoustic scattered field with respect to the shape of a given elastic scatterer.

Chokri Bekkey spent one week in Magique-3D in April 2011.

Yingxiang Xu, Post-doctoral student at Genova University, spent one week in Magique-3D in May 2011.

Robert Kotiuga, Professor at Boston University, spent one month as invited Professor in Magique-3D in September 2011.

Mounir Tlemcani spent two weeks in Magique-3Din September 2011 .

Mohamed Lakhdar Hadji (University of Annaba, Algeria) spent one month in Magique-3D in December 2011.

Jewoo Yoo, PhD student at Seoul University (Korea), is visiting us from october 2011 to February 2012.

In the framework of the Aquitaine/Euskadi programm, four scientists from the BCAM visited Magique 3D:

Alejandro Pozo, PhD student, spent two weeksin Magique-3Din october 2011.

Cristi Cazacu, PhD student, spent two weeks in Magique-3Din october 2011.

Aurora Marica, Post-Doctoral student, spent one week in Magique-3Din november 2011.

Javier Escartin, PhD student, spent two weeks in Magique-3Din december 2011 .

Depth Imaging Partnership Magique-3D maintains active collaborations with Total . In the context of depth imaging and with the collaboration of Henri Calandra from Total , Magique-3D coordinates research activities dealing with the development of high-performance numerical methods for solving wave equations in complex media. This project involves French academic researchers in mathematics, computing and in geophysics, and is funded by Total . At the end of 2011, four PhD students have defended their PhD dealing with contributions to new numerical imaging methods that are based on the solution of the full wave equation. Two were working in Magique-3D and in November 2011, a Ph.D. student has been hired in Magique-3D. Moreover, four internships have been realized in Magique3D. Always in the framework of DIP, Magique-3D has started a collaboration with Prof.Changsoo Shin who is an expert of Geophysics and works at the Department of Energy resources engineering (College of Engineering, Seoul National University). At that moment, Jewoo Yoo, who is a first year PhD student advised by Prof.Changsoo Shin, is visiting Magique-3D during four months.

To our knowledge, this network is the first in the French research community to establish links between industrial and academic researchers in the context of a long-term research program managed by an INRIA team.

Hélène Barucq is vice-chair of the Inria evaluation committee.

Hélène Barucq and Julien Diaz were editors of the special issue for Waves 2009 in Communications in Computational Physics.

Hélène Barucq and Julien Diaz were members of the scientific committee of Waves 2011 (Vancouver, Canada).

Julien Diaz is elected member of the Inria evaluation committee.

Sébastien Tordeux is elected member of the 26th section of the CNU.

Julien Diaz

Master : Introduction aux phénomènes de propagation d’ondes, 20 h, M2, Université de Pau, France

Victor Péron:

Licence :

Analyse 1, 39h, L1, Université de Pau et des Pays de l'Adour (UPPA), France;

Géométrie et calcul intégral, 48h , L2, UPPA, France;

Introduction à la variable complexe, 19h , L3, UPPA, France.

Master:

Analyse Numérique Fondamentale, 39h, M1, UPPA, France

Analyse Avancée Master Enseignement, 45h, M1, UPPA, France

Sébastien Tordeux

Master : Analyse numérique fondamental, 24h, M1, Université de Pau, France

Master : Introduction aux phénomènes de propagation d’ondes, 20h, M2, Université de Pau, France

PhD : Cyril Agut, Schémas numériques d'ordre élevé en espace et en temps pour l'équation des ondes, Université de Pau et des Pays de l'Adour, December 13th 2011, Hélène Barucq and Julien Diaz

PhD : Véronique Duprat, Conditions aux limites absorbantes enrichies pour l'équation des ondes acoustiques et l'équation d'Helmholtz, Université de Pau et des Pays de l'Adour, December 6th 2011, Hélène Barucq and Julien Diaz

PhD : Jonathan Gallon, Propagation automatique de Surface nD - Filtrage et traitement de la sismique avant stack , Université de Pau et des Pays de l'Adour, April 26th 2011, Hélène Barucq and Bruno Jobard

PhD in progress : Lionel Boillot, Propagateurs optimisés pour les ondes élastiques en milieux anisotropes, May 2011, Hélène Barucq and Julien Diaz

PhD in progress : Élodie Estecahandy, Sur la rśolution de problèmes de diffraction inverses avec des angles d'ouverture réduits, October 2010, Hélène Barucq and Rabia Djellouli

PhD in progress : Jérôme Luquel, RTM en milieu hétérogène par équations d'ondes élastiques, November 2011, Hélène Barucq and Julien Diaz

PhD in progress : Vanessa Mattesi, détection des hétérogéenéeités en acoustique et élastodynamique, October 2011, Hélène Barucq and Sébastien Tordeux

PhD in progress : Florent Ventimiglia, Schémas d'ordre élevé et pas de temps local pour les ondes élastiques en milieux hétérogènes, November 2010, Hélène Barucq and Julien Diaz

Cyril Agut

C. Agut, J.Diaz
*Stability analysis of the interior penalty discontinuous Galerkin method for the wave equation*, 10th International conference on mathematical and numerical aspects of waves, WAVES
2011, Jul. 27, 2011, Vancouver, Canada,
http://

Caroline Baldassari

C. Baldassari, H. Barucq, H. Calandra, J. Diaz and F. Ventimiglia
*Hybrid local-time stepping strategy for the Reverse Time Migration*, 10th International Conference on Mathematical and Numerical Aspects of Waves, WAVES 2011, July 25-29, 2011,
Vancouver, Canada,
http://

Hélène Barucq

H. Barucq, J. Diaz and V. Duprat
*Enriched Absorbing Boundary Conditions for Acoustic Waves*, Eighth Annual Conference on Frontiers in Applied and Computational Mathematics, FACM 2011, Jun. 9, 2011, Newark, USA,
http://

H. Barucq,
*Analyse mathématique de conditions aux limites absorbantes enrichies pour l'équation des ondes acoustiques*, Université de Genève, 9 mars 2011.

H. Barucq,
*Nouvelles conditions aux limites absorbantes pour des équations d'ondes*, Université Bordeaux I, 23 juin 2011.

Véronique Duprat

H. Barucq, J. Diaz and V. Duprat
*Complete factorization of the wave equation for the construction of absorbing boundary conditions involving a fractional derivative*, Tenth International Conference on Mathematical
and Numerical Aspects of Waves, WAVES 2011, July 25-29, 2011, Vancouver, Canada,
http://

Julien Diaz

C. Baldassari, H. Barucq, and J. Diaz
*Explicit
$hp$-Adaptive Time Scheme for the Wave Equation*, 20th International Conference on Domain Decomposition Methods, DD20, Feb. 7-11, 2011, San Diego, USA,
http://

C. Agut and J. Diaz
*Stability Analysis of an Interior Penalty Discontinuous Galerkin discretization of the wave equation*, Seminar of the Basque Center for Applied Mathematics (BCAM), May 4, 2011,
Bilbao, Spain,
http://

Élodie Estecahandy

H. Barucq, R. Djellouli and É. Estecahandy
*Analysis of the Fréchet differentiability with Respect to Lipschitz Domains for an Elasto-Acoustic Scattering Problem*, Eighth Annual Conference on Frontiers in Applied and
Computational Mathematics, FACM 2011, Jun. 9-11, 2011, Newark, USA,
http://

H. Barucq, R. Djellouli and É. Estecahandy
*Characterization of the Fréchet derivative of the elastoacoustic field with respect to Lipschitz domains*, Tenth International Conference on Mathematical and Numerical Aspects of
Waves, WAVES 2011, July 25-29, 2011, Vancouver, Canada,
http://

Victor Péron

V. Péron
*Electromagnetical Field in Biological Cells*, Séminaire de Mathématiques et de leurs Applications, LMAP, Pau, France, Feb. 2011.

F. Buret, M. Dauge, P. Dular, L. Krähenbühl, V. Péron, R. Perrussel, C. Poignard, D. Voyer
*Eddy currents and corner singularities*, Compumag 2011, Jul. 2011, Sydney, Australia,
http://

F. Buret, M. Dauge, P. Dular, L. Krähenbühl, V. Péron, R. Perrussel, C. Poignard, D. Voyer
*2D electrostatic problems with rounded corners*, COMPUMAG 2011, Jul. 2011, Sydney, Australia,
http://

M. Duruflé, V. Péron, C. Poignard
*Thin Layer Models for Electromagnetism*, WAVES 2011, Jul. 2011, Vancouver, Canada,
http://

M. Dauge, P. Dular, L. Krähenbühl, V. Péron, R. Perrussel, C. Poignard
*Impedance condition close to a corner in eddy-current problems*, ACOMEN 2011, Nov. 2011, Liège, Belgium,
http://

Sébastien Tordeux

S. Tordeux,
*Matching of Asymptotic Expansions for eigenvalues problem with two cavities linked by a small hole*, GDR Chant, Vienne, Autriche, 2011.

S. Tordeux,
*Parois perforées et multiperforées en acoustique*, Polariton 2011, CIRM, Marseilles

S. Tordeux,
*Perforated and multiperforated plates in linear acoustic*, Second International Workshop on Multiphysics, Multiscale and Optimization Problems 2011, University of the Basque Country,
Bilbao

S. Tordeux,
*Matching of Asymptotic Expansions for an eigenvalue problem with two cavities linked by a small hole*, Institute of Computational Mathematics and Mathematical Geophysics, Novosibirsk
State University, Russie, 2011.

S. Tordeux,
*Matching of Asymptotic Expansions for an eigenvalue problem with two cavities linked by a small hole*, Sobolev Institute of Mathematics, Novosibirsk State University, Russie,
2011.

S. Tordeux,
*Perforated and multiperforated plates in linear acoustic*, BCAM, Bilbao, Espagne, 2011.

S. Tordeux,
*Self-adjoint curl operator*, Anglet, journées Bordeaux-Pau-Toulouse, 2011

S. Tordeux,
*Parois multiperforées en acoustique*, Journée APAM, INSA-Toulouse, 2011

E. Piot, S. Tordeux,
*Modèles de parois perforées et multiperforées en acoustique*, sémianire MODANT, Grenoble, 2011

S. Tordeux,
*Modélisation multi-échelle des antennes microrubans*, Journée Modélisation et Calcul, Université de Reims, 2011