Gabriel Turinici

Bio: Gabriel Turinici is an academic researcher from Paris Dauphine University. The author has contributed to research in topics: Controllability & Epidemic model. The author has an hindex of 34, co-authored 144 publications receiving 4832 citations. Previous affiliations of Gabriel Turinici include CEREMADE & ParisTech.

Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods

Abstract: We present a technique for the rapid and reliable prediction of linear-functional outputs of elliptic (and parabolic) partial differential equations with affine parameter dependence. The essential components are (i) (provably) rapidly convergent global reduced basis approximations, Galerkin projection onto a space W(sub N) spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) a posteriori error estimation, relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures, methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage, in which, given a new parameter value, we calculate the output of interest and associated error bound, depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.

Résolution d'EDP par un schéma en temps « pararéel »

Abstract: Resume On propose dans cette Note un schema permettant de profiter d'une architecture parallele pour la discretisation en temps d'une equation d'evolution aux derivees partielles. Cette methode, basee sur un schema d'Euler, combine des resolutions grossieres et des resolutions fines et independantes en temps en s'inspirant de ce qui est classique en espace. La parallelisation qui en resulte se fait dans la direction temporelle ce qui est en revanche non classique. Elle a pour principale motivation les problemes en temps reel, d'ou la terminologie proposee de «parareel ».

A priori convergence of the greedy algorithm for the parametrized reduced basis method

Abstract: The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the "reduced basis". The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we prove that three greedy algorithms converge; the last algorithm, based on the use of an a posteriori estimator, is the approach actually employed in the calculations.

A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations

Abstract: We consider “Lagrangian” reduced-basis methods for single-parameter symmetric coercive elliptic partial differential equations. We show that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced–basis approximation converges exponentially to the exact solution uniformly in parameter space. Furthermore, the convergence rate depends only weakly on the continuity-coercivity ratio of the operator: thus very low-dimensional approximations yield accurate solutions even for very wide parametric ranges. Numerical tests (reported elsewhere) corroborate the theoretical predictions.

COPASI---a COmplex PAthway SImulator

Abstract: Motivation: Simulation and modeling is becoming a standard approach to understand complex biochemical processes. Therefore, there is a big need for software tools that allow access to diverse simulation and modeling methods as well as support for the usage of these methods. Results: Here, we present COPASI, a platform-independent and user-friendly biochemical simulator that offers several unique features. We discuss numerical issues with these features; in particular, the criteria to switch between stochastic and deterministic simulation methods, hybrid deterministic--stochastic methods, and the importance of random number generator numerical resolution in stochastic simulation. Availability: The complete software is available in binary (executable) for MS Windows, OS X, Linux (Intel) and Sun Solaris (SPARC), as well as the full source code under an open source license from http://www.copasi.org. Contact: mendes@vbi.vt.edu

Nonlinear Model Reduction via Discrete Empirical Interpolation

Abstract: A dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition (POD) method for constructing reduced-order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem. The original empirical interpolation method (EIM) is a modification of POD that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM), a variant that is suitable for reducing the dimension of systems of ordinary differential equations (ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite difference discretization of time dependent PDEs and/or parametrically dependent steady state problems. However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Our contribution is a greatly simplified description of the EIM in a finite-dimensional setting that possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the one-dimensional FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures nonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.

On the Navier-Stokes equations

Abstract: A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.