My research interests and fundamental contributions to academics and industry are in following domains:
- Absorbing boundary conditions for acoustic and electromagnetic problems
- Classical shape Optimization
- Automatic differentiation and higher order derivatives
- The topological gradient
- Mathematics of imagery and mathematical image processing
In the electromagnetic and acoustic fields, one needs to truncate the computational domain using an absorbing boundary conditions. I was fortunate to be one of the first contributors on a topic that is still very active. The method introduced in [A87] is the subject of recent interest by INRIA Bordeaux.
References: [ A94 , B89 , B90, T91 , D91]
Most effective methods in shape optimization require the gradient with respect to shape variations using one of the following methods:
- Start by differenting the continuous problem and then discretize the derivative,
- Start by discretizing the continuous problem and then compute the derivative
Mathematicians lean towards the first method using mathematical analysis tools. And engineers prefer the second method arguing that the optimization, which works on the discrete problem requires the gradient of that problem. In the context of the finite element method, I showed that the two approaches are equivalent, if the element is defined by mapping a reference element [A94, B89, B90]. As a corollary of this result, analytical and tedious calculation of the discrete gradient found in some publications, can be replaced by a very simple and elegant way via the continuous gradient. I have also shown that the error in computing the derivative of a cost function in a direction V, heavily depends on the regularity of the perturbation V. The standard V in Wm,∞ (Ω) acts as a constant multiplicative of the error and order of regularity m occurs in the order of the error. In most cases, we can succeed in obtaining an order 2k error on the calculation of the derivative when finite element of degree k is used [A94, B89, B90].
This research was largely driven by a Cifre agreement (Thesis of Michel Rochette T91), which was supported by the company Sfernice that manufactures resistive circuits with high accuracy. Patent [D91] describes the design of a universal resistance replacing a range of products. According to management of Sfernice, this patent is still responsible for half of the company's revenues.
I am interested in the shape optimization in the context of integral equations and have shown that the shape gradient can be written as a simple expression (often the product) of the direct adjoint solutions, if underlying partial differential equations are considered [A83]. In the literature, the direct derivation of the problem of integral equations defined on the boundary of the domain leads to tedious calculations. The case of Maxwell's equations was considered in partnership with France Telecom as a part of the thesis of Frédérique Millot [T92 ].
My interactions with industry helped me realize very early that reduced order model is of great interest for industrial applications [ A93, A94 , A97, A10a ] . In recent years, this topic has become a central concern of prestigious research centers.
While studying automatic differentiation for the calculation of gradient of a function defined by its program or its algorithm, I thought about its application to the calculation of higher order derivatives. We began by showing that for a large class of problems, including electromagnetics and acoustics, the solution is analytical as a function of problem parameters.
We then showed that order one derivative results generalize to higher order derivatives [ A93, A94 , A97.]
For the higher order derivatives, derive then discretize is equivalent to discretize then derive.
The higher order derivatives have the same order of error as the first order derivative (results may surprise!).
Note that the articles of that time were disappointed by the fact that second derivative did not satisfy Schwarz theorem. We had to choose the right mathematical framework. The approximation by Taylor polynomial was the first approach considered.
Then I highlighted, with Sandrine Bargiacchi (Thesis [T04a] , [A10a ]) that stationary iterative methods (such as Jacobi, Gauss-Seidel, Relaxation, etc.) are just Taylor's approximation of the solution with respect to a hidden parameter, which is a continuation parameter leading from the preconditioning system to the system to be solved.
It is the same for the eigenvalue problem, wherein the method of inverse powers; successive iterates are none other than the derivatives with respect to the eigenvalues. Non-stationary iterative methods such as conjugate gradient and GMRS project the linear system generated by the successive derivatives, called Krylov space space. Thus, Sandrine Bargiacchi proposed as part of his thesis, a generalization of the GMRES method for nonlinear problems [A10a] .
In case of penalty method of optimization under constraints, it shown that the solution x (ε), where ε is the penalization parameter, has an analytic extension at 0. Calculation of x(0) by Taylor polynomial in ε= ε0, where ε0 is small gives very good results [A02].
This work lead to the foundation of the company CADOE ( Calcul Adaptatif par Dérivées d’Ordre Elevé), which now has 40 employees, in 1994. Later CADOE was acquired by ANSYS, INC.
We have introduced a natural framework to anchor model reduction techniques on solid foundations. Today, these techniques are the subject of very active research in prestigious laboratories. Our team has played a pioneering role in this field. Our publications, which were ahead, are being referred now.
References: [A01].
Up to now, we worked on shape optimization assuming that the topology of the optimal geometry is known. When we look for complex shapes, we can not work with a fixed topology. The topological gradient answers the question: what will happen when you create a small hole in a shape? If geometry is defined by its characteristic function (it takes 1 inside the object and 0 outside), clearly we are dealing with a 0-1 optimization problem. The fundamental idea of topological gradient is to estimate the variation of the cost function when switching from 1 to 0 or from 0 to 1 in a small area. In the conventional differential calculus, a directional derivative make perturbation of small amplitude. Here the size of the perturbation is small.
Most of contributions in topology optimization are related to the minimization of the potential energy of mechanical structures with a Neumann (free boundary) condition on the unknown part of the boundary. Real life problems in fluid mechanics and electromagnetism involve Dirichlet condition on boundary of obstacles. We derived analytical expressions for the topological gradient, in dimension 2 and 3 with Dirichlet conditions, for the Helmholtz equation (thesis B. Samet [T04b] and article [A03]), equations of linear elasticity [A00 ], the Stokes equation and the Navier-Stokes ( PhD Thesis, S. Amstutz). Numerically, we built effective and efficient algorithms. In many cases, we obtain a satisfactory solution at the first iteration. This applies particularly to inverse problems and image processing [A07B, A09b , A09c, A10b].
Our methods are based on crack detection technique [A05]. The basic idea is to process an image of size n in O (n log2 n ) operations using an crack identification technique.
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Crack detection by boundary measurement and Inpainting: As part of our collaboration with the Lamsin, Tunisia, I was interested in the problem of crack detection from boundary measurements. I didn't know if this problem has a real application because the Laplacian inverse problem is difficult to solve, but I have found one: reconstruction of missing parts of an image (inpainting). The cracks show the contours of the image to be reconstructed.[A05].
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Image restoration, classification and segmentation: (See poster) The image restoration is also based on the crack detection by topological gradient. The basic idea is to detect cracks at the edges of the image. A diffusive technique (heat equation) is then used to smooth the image part devoid of cracks. We apply topological gradient for the minimization of the H1 seminorm of the standard image. The problem is simpler than the case of inpainting, as data is available at each point in the field and cracks are easy to find [ A06b , A06a A06e , A07a, A07d, A08a, A09A]. Again, the computation time is reduced to O (n log2 n) operations which allowed us to process video sequences. We treat problems of segmentation and classification using the same technique: crack detection identifies the interfaces between the components of an image [A07a, A08a, A09A]. We also successfully apply this technique to the detection of motion in a sequence of images.
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Real time image processing: All these methods are based on the same core of calculation. We start with a constant conductivity to calculate the state and adjoint state. The topological gradient is calculated for the detection of cracks. On the elements where we detected the presence of a crack are given a very low value of conductivity, c. The solution to this last problem gives the image solution. As c is constant or near constant, spectral methods are used as preconditioner to the conjugate gradient method. We do an iteration and iteration requires only O (n log2 n) operations, where n is the size of the image. Numerical results show a complexity of O(n) [ A06e ].
Despite our recent interest in imaging and image processing, our work is now recognized through publications [ A06b , A06a A06e , A07a, A07d, A08a, A09A], invitations to conferences [ C04a, C04e, C04f, C04h, C05b, C05d, C06b, C06C, C06D, C06e , C07a,C07b, C07c] and research projects (Project ACINIM/IMEG, ANR Project ADDISA). Our goal for the coming years is to take lead in real-time processing of dynamic images for medical applications, multimedia and robotics.