Eric Darrigrand

Volume and surface integral equations for electromagnetic scattering by a dielectric body

We derive and analyze two equivalent integral formulations for the time-harmonic electromagnetic scattering by a dielectric object. One is a volume integral equation (VIE) with a strongly singular kernel and the other one is a coupled surface-volume system of integral equations with weakly singular kernels. The analysis of the coupled system is based on standard Fredholm integral equations, and it is used to derive properties of the volume integral equation.

Combining analytic preconditioner and Fast Multipole Method for the 3-D Helmholtz equation

The paper presents a detailed numerical study of an iterative solution to 3-D sound-hard acoustic scattering problems at high frequency considering the Combined Field Integral Equation (CFIE). We propose a combination of an OSRC preconditioning technique and a Fast Multipole Method which leads to a fast and efficient algorithm independently of both a frequency increase and a mesh refinement. The OSRC-preconditioned CFIE exhibits very interesting spectral properties even for trapping domains. Moreover, this analytic preconditioner shows highly-desirable advantages: sparse structure, ease of implementation and low additional computational cost. We first investigate the numerical behavior of the eigenvalues of the related integral operators, CFIE and OSRC-preconditioned CFIE, in order to illustrate the influence of the proposed preconditioner. We then apply the resolution algorithm to various and significant test-cases using a GMRES solver. The OSRC-preconditioning technique is combined to a Fast Multipole Method in order to deal with high-frequency 3-D cases. This variety of tests validates the effectiveness of the method and fully justifies the interest of such a combination.

The inverse electromagnetic scattering problem for screens

We consider the inverse electromagnetic scattering problem for perfectly conducting screens. We solve this problem by modifying the linear sampling method formulated in Cakoni and Colton (2003 Math. Methods Appl. Sci. 26 413–29) for the case of scattering obstacles with nonempty interior. Numerical examples are given for screens in 3.

Volume integral equations for electromagnetic scattering in two dimensions

We study the strongly singular volume integral equation that describes the scattering of time-harmonic electromagnetic waves by a penetrable obstacle. We consider the case of a cylindrical obstacle and fields invariant along the axis of the cylinder, which allows the reduction to two-dimensional problems. With this simplification, we can refine the analysis of the essential spectrum of the volume integral operator started in a previous paper (Costabel et?al., 2012) and obtain results for non-smooth domains that were previously available only for smooth domains. It turns out that in the transverse electric (TE) case, the magnetic contrast has no influence on the Fredholm properties of the problem. As a byproduct of the choice that exists between a vectorial and a scalar volume integral equation, we discover new results about the symmetry of the spectrum of the double layer boundary integral operator on Lipschitz domains.

The pVHL 172 isoform is not a tumor suppressor and up-regulates a subset of pro-tumorigenic genes including TGFB1 and MMP13

The von Hippel-Lindau (VHL) tumor suppressor gene is often deleted or mutated in ccRCC (clear cell renal cell carcinoma) producing a non-functional protein. The gene encodes two mRNA, and three protein isoforms (pVHL213, pVHL160 and pVHL172). The pVHL protein is part of an E3 ligase complex involved in the ubiquitination and proteasomal degradation of different proteins, particularly hypoxia inducible factors (HIF) that drive the transcription of genes involved in the regulation of cell proliferation, angiogenesis or extracellular matrix remodelling. Other non-canonical (HIF-independent) pVHL functions have been described. A recent work reported the expression of the uncharacterized protein isoform pVHL172 which is translated from the variant 2 by alternative splicing of the exon 2. This splice variant is sometimes enriched in the ccRCCs and the protein has been identified in the respective samples of ccRCCs and different renal cell lines. Functional studies on pVHL have only concerned the pVHL213 and pVHL160 isoforms, but no function was assigned to pVHL172. Here we show that pVHL172 stable expression in renal cancer cells does not regulate the level of HIF, exacerbates tumorigenicity when 786-O-pVHL172 cells were xenografted in mice. The pVHL172-induced tumors developed a sarcomatoid phenotype. Moreover, pVHL172 expression was shown to up regulate a subset of pro-tumorigenic genes including TGFB1, MMP1 and MMP13. In summary we identified that pVHL172 is not a tumor suppressor. Furthermore our findings suggest an antagonistic function of this pVHL isoform in the HIF-independent aggressiveness of renal tumors compared to pVHL213.The von Hippel-Lindau (VHL) tumor suppressor gene is often deleted or mutated in ccRCC (clear cell renal cell carcinoma) producing a non-functional protein. The gene encodes two mRNA, and three protein isoforms (pVHL213, pVHL160 and pVHL172). The pVHL protein is part of an E3 ligase complex involved in the ubiquitination and proteasomal degradation of different proteins, particularly hypoxia inducible factors (HIF) that drive the transcription of genes involved in the regulation of cell proliferation, angiogenesis or extracellular matrix remodelling. Other non-canonical (HIF-independent) pVHL functions have been described. A recent work reported the expression of the uncharacterized protein isoform pVHL172 which is translated from the variant 2 by alternative splicing of the exon 2. This splice variant is sometimes enriched in the ccRCCs and the protein has been identified in the respective samples of ccRCCs and different renal cell lines. Functional studies on pVHL have only concerned the pVHL213 and pVHL160 isoforms, but no function was assigned to pVHL172. Here we show that pVHL172 stable expression in renal cancer cells does not regulate the level of HIF, exacerbates tumorigenicity when 786-O-pVHL172 cells were xenografted in mice. The pVHL172-induced tumors developed a sarcomatoid phenotype. Moreover, pVHL172 expression was shown to up regulate a subset of pro-tumorigenic genes including TGFB1, MMP1 and MMP13. In summary we identified that pVHL172 is not a tumor suppressor. Furthermore our findings suggest an antagonistic function of this pVHL isoform in the HIF-independent aggressiveness of renal tumors compared to pVHL213.

Fast multipole method and microlocal discretization for the 3-D Helmholtz equation

A numerical solution of the boundary integral equation for the exterior Helmholtz problem in three dimensions, leads to the solution of a dense linear system. In order to have a well conditioned system, we consider the Despres integral equations (see J. Electromag. Waves and Appl., vol.13, p.1553-68, 1999). An iterative method is given by Despres to solve this system. If Niter is the number of the iterations, the complexity of this resolution is of order Niter k/sup 4/, where k is the wave number. In order to speed up the iterative solution of the system, we have considered the coupling of two methods, the microlocal discretization method and the fast multipole method (FMM). The microlocal discretization method of T. Abboud, J.-C. Nedelec and B. Zhou (1995), enables one to consider a new system whose size is of order k/sup 2/3//spl times/k/sup 2/3/ instead of k/sup 2//spl times/k/sup 2/ for convex geometries. However, due to the geometrical approximation of the surface, the fine mesh of the standard case is still considered. Another method, the fast multipole method, is one of the most efficient and robust methods used to speed up the calculation of matrix-vector products, with a cost of order k/sup 3/ instead of k/sup 4/ for the one level FMM. In this paper, a coupling of these two methods is presented. It enables one to reduce the CPU time very efficiently for large wave number, with a complexity of order k/sup 3/+Niter k/sup 4/3/.

Convergence of Krylov subspace solvers with Schwarz preconditioner for the exterior Maxwell problem

The consideration of an integral representation as an exact boundary condition for the finite element resolution of wave propagation problems in exterior domain induces algorithmic difficulties. In this paper, we are interested in the resolution of an exterior Maxwell problem in 3D. As a first step, we focus on the justification of an algorithm described in literature, using an interpretation as a Schwarz method. The study of the convergence indicates that it depends significantly on the thickness of the domain of computation. This analysis suggests the use of the finite element term of Schwarz method as a preconditioner for use of Krylov iterative solvers. An analytical study of the case of a spherical perfect conductor indicates the efficiency of such approach. The consideration of the preconditioner suggested by the Schwarz method leads to a superlinear convergence of the GMRES predicted by the analytical study and verified numerically.

Coupling of a Fast Multipole Method and a Microlocal Discretization for Integral Equations of Electromagnetism

A numerical solution of the integral equations for the 3-D exterior problem of electromagnetism, leads to the solution of dense linear systems. Those systems have generally a bad conditionement and a size that strongly increases with the frequency. We then consider the Despres’s Integral Equations to have a well conditioned system ([5], [8]). If N iter is the number of iterations, the classical complexity of the iterative solution is O(N iter κ 4), where κ, is the wave number. In order to speed up the solution of the system, we have considered the coupling of two methods, the microlocal discretization method and the fast multipole method (FMM). The microlocal discretization method according to Abboud et al. [1], enables one to consider new systems whose size is O(κ 2/3 x κ 2/3) instead of O(κ 2 × κ 2) thanks to a coarse discretization of the unknown for convex geometries. However, due to the geometrical approximation of the surface, the fine mesh of a classical solution is still considered. An other method, the fast multipole method ([6], [7], [4]), is one of the most efficient and robust methods used to speed up iterative solutions. Using a multilevel algorithm, it leads to the cost O(N iter κ 2 ln κ 2) instead of O(N iter κ 4). In this paper, the coupling of both methods, using a multilevel algorithm, enables one to reduce the CPU time efficiently for large wave numbers, with the complexity O(κ 8⁄3 ln κ 2 + N iter κ 4⁄3) (see also [2]).