Reduction of complexity and variational data assimilation

Conferencia plenaria por invitaciónReduced basis methods belong to a class of approaches of \emph{model reduction} for the approximation of the solution of mathematical models involved in many fields of research or decision making and in data assimilation. These approaches allow to tackle, in --- close to --- real time, problems requiring, a priori, a large number of computations by formalizing two steps : one known as "offline stage” that is a preparation step and is quite costly and an ``online stage'' that is used on demand and is very cheap. The strategy uses the fact that the solutions we are interested in belong to a family, a manifold, parametrized by input coefficients, shapes or stochastic data, that has a small complexity. The complexity is measured in terms of a quantity like the ``Kolmogorov width'' that, when it is small, formalizes the fact that some small dimensional vectorial spaces allow to provide a good approximation of the elements on the manifold. We shall make a review of the fundamental background and state some results proving that such a dimension is small for a large class of problems of interest, then use this fact to propose approximation strategies in various cases depending on the knowledge we have of the solution we want to approximate : either explicit through values at points, or through outputs evaluated from the solution, or implicit through the Partial Differential Equation it satisfies. We shall also present a strategy available when a mixed of the above informations is available allowing to propose new efficient approaches in data assimilation and data mining. The theory on the numerical analysis (a priori and a posteriori) of these approaches will also be presented together with results on numerical simulations. Work done in close collaboration with A. T. Patera (MIT, Cambridge) and has benefited from the collaboration with A. Buffa (IAN, Pavia), R. Chakir (IFSTAR, Paris), Y. Chen (U. of Massachusetts, Dartmouth), Y. Hesthaven (EPFL, Lausanne), E. Lovgren (Simula, Oslo), O. Mula (UPMC, Paris), NC Nguyen (MIT, Cambridge), J. Pen (MIT, Cambridge), C. Prud'homme (U. Strasbourg), J. Rodriguez (U. Santiago de Compostella), E. M. Ronquist (U. Trondheim), B. Stamm (UPMC, Paris), G. Turinici (Dauphine, Paris), M. Yano (MIT, Cambridge).Universidad de Málaga. Campus de Excelencia Internacional Andalucia Tech. Conferencias del plan propio de investigación UM

Some spectral approximation of one-dimensional fourth-order problems

Some spectral type collocation method well suited for the approximation of fourth-order systems are proposed. The model problem is the biharmonic equation, in one and two dimensions when the boundary conditions are periodic in one direction. It is proved that the standard Gauss-Lobatto nodes are not the best choice for the collocation points. Then, a new set of nodes related to some generalized Gauss type quadrature formulas is proposed. Also provided is a complete analysis of these formulas including some new issues about the asymptotic behavior of the weights and we apply these results to the analysis of the collocation method

Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models

We provide a priori error estimates for the spectral and pseudospectral Fourier (also called planewave) discretizations of the periodic Thomas-Fermi-von Weizs\"{a}cker (TFW) model and for the spectral discretization of the Kohn-Sham model, within the local density approximation (LDA). These models allow to compute approximations of the ground state energy and density of molecular systems in the condensed phase. The TFW model is stricly convex with respect to the electronic density, and allows for a comprehensive analysis. This is not the case for the Kohn-Sham LDA model, for which the uniqueness of the ground state electronic density is not guaranteed. Under a coercivity assumption on the second order optimality condition, we prove that for large enough energy cut-offs, the discretized Kohn-Sham LDA problem has a minimizer in the vicinity of any Kohn-Sham ground state, and that this minimizer is unique up to unitary transform. We then derive optimal a priori error estimates for the spectral discretization method.Comment: 50 page

Robin Schwarz algorithm for the NICEM Method: the Pq finite element case

In Gander et al. [2004] we proposed a new non-conforming domain decomposition paradigm, the New Interface Cement Equilibrated Mortar (NICEM) method, based on Schwarz type methods that allows for the use of Robin interface conditions on non-conforming grids. The error analysis was done for P1 finite elements, in 2D and 3D. In this paper, we provide new numerical analysis results that allow to extend this error analysis in 2D for piecewise polynomials of higher order and also prove the convergence of the iterative algorithm in all these cases.Comment: arXiv admin note: substantial text overlap with arXiv:0705.028

Using Spectral Method as an Approximation for Solving Hyperbolic PDEs

We demonstrate an application of the spectral method as a numerical approximation for solving Hyperbolic PDEs. In this method a finite basis is used for approximating the solutions. In particular, we demonstrate a set of such solutions for cases which would be otherwise almost impossible to solve by the more routine methods such as the Finite Difference Method. Eigenvalue problems are included in the class of PDEs that are solvable by this method. Although any complete orthonormal basis can be used, we discuss two particularly interesting bases: the Fourier basis and the quantum oscillator eigenfunction basis. We compare and discuss the relative advantages of each of these two bases.Comment: 19 pages, 14 figures. to appear in Computer Physics Communicatio

Parareal in time intermediate targets methods for optimal control problem

In this paper, we present a method that enables solving in parallel the Euler-Lagrange system associated with the optimal control of a parabolic equation. Our approach is based on an iterative update of a sequence of intermediate targets that gives rise to independent sub-problems that can be solved in parallel. This method can be coupled with the parareal in time algorithm. Numerical experiments show the efficiency of our method.Comment: 14 page

Optimal error analysis of spectral methods with emphasis on non-constant coefficients and deformed geometries

The numerical analysis of spectral methods when non-constant coefficients appear in the equation, either due to the original statement of the equations or to take into account the deformed geometry, is presented. Particular attention is devoted to the optimality of the discretization even for low values of the discretization parameter. The effect of some overintegration is also addressed, in order to possibly improve the accuracy of the discretization

Nonconforming mortar element methods: Application to spectral discretizations

Spectral element methods are p-type weighted residual techniques for partial differential equations that combine the generality of finite element methods with the accuracy of spectral methods. Presented here is a new nonconforming discretization which greatly improves the flexibility of the spectral element approach as regards automatic mesh generation and non-propagating local mesh refinement. The method is based on the introduction of an auxiliary mortar trace space, and constitutes a new approach to discretization-driven domain decomposition characterized by a clean decoupling of the local, structure-preserving residual evaluations and the transmission of boundary and continuity conditions. The flexibility of the mortar method is illustrated by several nonconforming adaptive Navier-Stokes calculations in complex geometry