Section: Scientific Foundations

Inverse Problems

• 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 [52] . 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  [68] , [74] , [63] , [64] ) 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 $\pi$. For smaller apertures the reconstruction of the shape of an obstacle becomes more difficult and nearly impossible for apertures smaller than $\pi /4$.

  This plus the fact that from a mathematical viewpoint the FFP can be determined on the entire sphere $S1$ from its knowledge on a subset of $S1$ because it is an analytic function, we propose [44] , [46] a solution methodology to extend the range of FFP data when measured in a limited aperture and not on the entire sphere $S1$. It is therefore possible to solve the IOP numerically when only limited aperture measurements are available. The objective of Magique-3D  is to extend this work to 3D problems of acoustic scattering.

  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  [48] by Liliana Borcea and  [49] 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), [47] , 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 Total jointly. 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  [51] 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.