Interpolation of the inverse of parameter dependent operator for preconditioning parametric and stochastic equations

When solving partial differential equations with parametrized (or random) coefficents, one usually needs to solve high dimensional problems for a large number of realizations of the coefficients, which is computationaly expensive. Model reduction techniques such as the Reduced Basis Method [4] or the Proper Generalized Decomposition [2] are now comonly used for the construction of approximation of the solution of such problems. The idea is to build a subspace on which a projection of the solution can be computed with a low computational cost. From a practical point of view, that subspace is constructed so that it minimizes some norm of the residual associated with the equation. In practice, we observe that a bad condition number of the operator leads to a poor approximation : preconditioning is necessary to achieve efficient model reduction.There exist in the literature different definitions for the preconditioner. A widely used preconditioner is the inverse of the operator at a given parameter value [4], or the inverse of the mean operator in the context of uncertainty quantification. In [1], the authors propose an analytical interpolation of the inverse of the operator, and show the benefits of using a parameter dependent preconditioner.We propose here different interpolation methods of the inverse of the operator for the construction of the preconditioner, and compare them. In particular, we show that the interpolation based on the projection of the identity matrix [3] with respect to the Frobenius norm seems to be the most appropriated strategy. In addition, we introduce a greedy algorithm for the construction of the preconditioner: the corresponding set of interpolation points results in a better preconditioner. Finally, numerical examples show that the quality of the model reduction (Reduced Basis or PGD) is significantly better when using the proposed preconditioner

A Tensor-Based Algorithm for the Optimal Model Reduction of High Dimensional Problems

Abstract. We propose a method for the approximation of the solution of high-dimensional problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of an ideal minimal residual method with a residual norm corresponding to the error in a solution norm of interest. We introduce and analyze an algorithm for the approximation of the best approximation in a given low-rank tensor subset. A weak greedy algorithm based on this ideal minimal residual formulation is introduced and its convergence is proven under some conditions. The robustness of the method is illustrated on numerical examples in uncertainty propagation

A multi-material CCALE-MOF approach in cylindrical geometry

International audienceIn this paper we present recent developments concerning a Cell-Centered Arbitrary Lagrangian Eulerian (CCALE) strategy using the Moment Of Fluid (MOF) interface reconstruction for the numerical simulation of multi-material compressible fluid flows on unstructured grids in cylindrical geometries. Especially, our attention is focused here on the following points. First, we propose a new formulation of the scheme used during the Lagrangian phase in the particular case of axisymmetric geometries. Then, the MOF method is considered for multi-interface reconstruction in cylindrical geometry. Subsequently, a method devoted to the rezoning of polar meshes is detailed. Finally, a generalization of the hybrid remapping to cylindrical geometries is presented. These explorations are validated by mean of several test cases using unstructured grid that clearly illustrate the robustness and accuracy of the new method