Yann Favennec

The adjoint method coupled with the modal identification method for nonlinear model reduction

In order to obtain reduced models (RMs) from original detailed ones, the Modal Identification Method (MIM) has been developed for several years for linear and then for nonlinear systems. This identification is performed through the resolution of an inverse problem of parameter estimation. So far, the MIM was used with the Finite Difference Method (FDM) for the computation of the gradient of the functional to be minimized. This leads to important computation times. In order to save it up, the Adjoint Method (AM) has been coupled with the MIM to compute the gradient. All the equations related to the reduced model, the adjoint equations, the gradient and the optimization algorithm are clearly expressed. A test case involving a 3D nonlinear transient heat conduction problem is proposed. The accuracy of the identified RM is shown and the comparison between the proposed AM and the classical FDM shows the drastic reduction of computation time.

Numerical model reduction of 2D steady incompressible laminar flows: Application on the flow over a backward-facing step

The numerical solution of most fluid mechanics problems usually needs such a fine mesh that the associated computational times become non-negligible parts in any design process. In order to couple numerical modelling schemes with inversion or control algorithms, the size of such models needs to be highly reduced. The identification method is a way to build low-order models that fit with the original ones. The laminar flow over a backward-facing step is used as a test case. Presented solutions are found to be in good agreement with experimental and numerical results found in the literature.

Optical tomography reconstruction algorithm with the finite element method: An optimal approach with regularization tools

Optical tomography is mathematically treated as a non-linear inverse problem where the optical properties of the probed medium are recovered through the minimization of the errors between the experimental measurements and their predictions with a numerical model at the locations of the detectors. According to the ill-posed behavior of the inverse problem, some regularization tools must be performed and the Tikhonov penalization type is the most commonly used in optical tomography applications. This paper introduces an optimized approach for optical tomography reconstruction with the finite element method. An integral form of the cost function is used to take into account the surfaces of the detectors and make the reconstruction compatible with all finite element formulations, continuous and discontinuous. Through a gradient-based algorithm where the adjoint method is used to compute the gradient of the cost function, an alternative inner product is employed for preconditioning the reconstruction algorithm. Moreover, appropriate re-parameterization of the optical properties is performed. These regularization strategies are compared with the classical Tikhonov penalization one. It is shown that both the re-parameterization and the use of the Sobolev cost function gradient are efficient for solving such an ill-posed inverse problem.

Specular reflection treatment for the 3D radiative transfer equation solved with the discrete ordinates method

The contribution of this paper relies in the development of numerical algorithms for the mathematical treatment of specular reflection on borders when dealing with the numerical solution of radiative transfer problems. The radiative transfer equation being integro-differential, the discrete ordinates method allows to write down a set of semi-discrete equations in which weights are to be calculated. The calculation of these weights is well known to be based on either a quadrature or on angular discretization, making the use of such method straightforward for the state equation. Also, the diffuse contribution of reflection on borders is usually well taken into account. However, the calculation of accurate partition ratio coefficients is much more tricky for the specular condition applied on arbitrary geometrical borders. This paper presents algorithms that calculate analytically partition ratio coefficients needed in numerical treatments. The developed algorithms, combined with a decentered finite element scheme, are validated with the help of comparisons with analytical solutions before being applied on complex geometries.

Comparison between the Discretized Continuous Gradient and the Discrete Gradient when Dealing with Nonlinear Parabolic Problems

ABSTRACT This article presents a comparison of the discrete gradient, the discretized continuous gradient, and the intermediate gradients when dealing with inverse problems coupled with nonlinear parabolic direct problems. The article shows that the best strategy resides in differentiating the direct model after all are performed, but before the linearization procedure.

Hessian and Fisher Matrices For Error Analysis in Inverse Heat Conduction Problems

This article deals with the evaluation of the impact of some errors on the prediction when an inverse heat conduction problem (IHCP) is solved. This evaluation is performed through an analysis of the cost function Hessian to be minimized. More precisely, this article presents a comparison between the computations of a cost function Hessian and of the related Fisher matrix. The Fisher matrix is an approximation of the Hessian matrix. It needs only the solution of the sensitivity problems. On the other hand, the computation of the exact Hessian needs the solution of the Euler equation along with the second-order adjoint problems.

Finite elements parameterization of optical tomography with the radiative transfer equation in frequency domain

Optical tomography is a technique of probing semi-transparent media with the help of light sources. In this method, the spatial distribution of the optical properties inside the probed medium is reconstructed by minimizing a cost function based on the errors between the measurements and the predictions of a numerical model of light transport (also called forward/direct model) within the medium at the detectors locations. Optical tomography with finite elements methods involves generally continuous formulations where the optical properties are constant per mesh elements. This study proposes a numerical analysis in the parameterization of the finite elements space of the optical properties in order to improve the accuracy and the contrast of the reconstruction. Numerical tests with noised data using the same algorithm show that continuous finite elements spaces give better results than discontinuous ones by allowing a better transfer of the information between the whole computational nodes of the inversion. It is seen that the results are more accurate when the number of degrees of freedom of the finite element space of the optical properties (number of unknowns) is lowered. This shows that reducing the number of unknowns decreases the ill-posed nature of the inverse problem, thus it is a promising way of regularizing the inversion.

A Modified Cost Function Based Optical Tomography Reconstruction Algorithm Optimized for Finite-Elements Methods

In optical tomography, the optical properties of the medium under investigation are obtained through the minimization of an objective function. Generally, this function is expressed as a discrete sum of the square of the errors between measurements and predictions at the detectors. This paper introduces a continuous form of the objective function by taking the integral of the errors. The novelty is that the surfaces of the detectors are taken into account in the reconstruction and a compatibility is obtained for all finite element formulations (continuous and discontinuous). Numerical tests are used to compare the reconstructions with both objective functions. It is seen that the integral approach leads to low values of objective functions those reconstructions may be affected by rounding errors. Scaling of the objective function and its gradient shows that both methods give comparable accuracy with an advantage to the continuous approach where the integral acts as a filter of noise.© 2011 ASME

High performance computation of radiative transfer equation using the finite element method

Abstract This article deals with an efficient strategy for numerically simulating radiative transfer phenomena using distributed computing. The finite element method alongside the discrete ordinate method is used for spatio-angular discretization of the monochromatic steady-state radiative transfer equation in an anisotropically scattering media. Two very different methods of parallelization, angular and spatial decomposition methods, are presented. To do so, the finite element method is used in a vectorial way. A detailed comparison of scalability, performance, and efficiency on thousands of processors is established for two- and three-dimensional heterogeneous test cases. Timings show that both algorithms scale well when using proper preconditioners. It is also observed that our angular decomposition scheme outperforms our domain decomposition method. Overall, we perform numerical simulations at scales that were previously unattainable by standard radiative transfer equation solvers.

Space-Dependent Sobolev Gradients as a Regularization for Inverse Radiative Transfer Problems

Diffuse optical tomography problems rely on the solution of an optimization problem for which the dimension of the parameter space is usually large. Thus, gradient-type optimizers are likely to be used, such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, along with the adjoint-state method to compute the cost function gradient. Usually, the -inner product is chosen within the extraction procedure (i.e., in the definition of the relationship between the cost function gradient and the directional derivative of the cost function) while alternative inner products that act as regularization can be used. This paper presents some results based on space-dependent Sobolev inner products and shows that this method acts as an efficient low-pass filter on the cost function gradient. Numerical results indicate that the use of Sobolev gradients can be particularly attractive in the context of inverse problems, particularly because of the simplicity of this regularization, since a single additional diffusion equation is to be solved, and also because the quality of the solution is smoothly varying with respect to the regularization parameter.