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Simultaneous Empirical Interpolation and Reduced Basis method for non-linear problems
TLDR
This talk introduces a Simultaneous EIM Reduced basis algorithm (SER) based on the use of reduced basis approximations into the EIM building step and assesses its performances for large scale problems it is designed for.Abstract:
In this paper, we focus on the reduced basis methodology in the context of non-linear non-affinely parametrized partial differential equations in which affine decomposition necessary for the reduced basis methodology are not obtained [4, 3]. To deal with this issue, it is now standard to apply the EIM methodology [8, 9] before deploying the Reduced Basis (RB) methodology. However the computational cost is generally huge as it requires many finite element solves, hence making it inefficient, to build the EIM approximation of the non-linear terms [9, 1]. We propose a simultaneous EIM Reduced basis algorithm, named SER, that provides a huge computational gain and requires as little as N + 1 finite element solves where N is the dimension of the RB approximation. The paper is organized as follows: we first review the EIM and RB methodologies applied to non-linear problems and identify the main issue, then we present SER and some variants and finally illustrates its performances in a benchmark proposed in [9].read more
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Dissertation
Reduced Basis Methods for Urban Air Quality Modeling
TL;DR: The Parameterized- Background Data-Weak (PBDW) method is extended to physically-based AQMs and shows promise for the real-time reconstruction of a pollution field in large-scale problems, providing state estimation with approximation error generally under 10% when applied to an imperfect model.
Proceedings ArticleDOI
A Non-Intrusive Reduced Basis Method for Urban Flows Simulation
TL;DR: This approach speeds up the CFD simulation while remaining non-intrusive in relation to the high fidelity model, which can allow to avoid practical problems associated to model reduction for complex air flows involved in many sophisticated methods of urban air quality modeling.
Dissertation
Réduction de modèles en thermique et mécanique non-linéaires
TL;DR: A new methodology, the Progressive RB-EIM (PREIM) which aims at reducing the cost of the phase during which the reduced model is constructed without compromising the accuracy of the final RB approximation, and a new algorithm that produces better reduced spaces while minimizing the number of measurements required for the final reduced problem.
Posted ContentDOI
PREIM for nonlinear parabolic problems
TL;DR: The purpose of PREIM (Progressive RB-EIM) is to reduce the offline costs of nonlinear parabolic reduced order models with accurate RB approximations in the online stage with a progressive enrichment of both the EIM approximation and the RB space.
Accelerating greedy algorithm for model reduction of complex systems by multi-fidelity error estimation
TL;DR: In this paper , the authors propose a solution to solve the problem of the problem: this paper ] of "uniformity" and "uncertainty" of the solution.
References
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Journal ArticleDOI
Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations
TL;DR: (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations are considered.
Journal ArticleDOI
Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods
Christophe Prud'Homme,Dimitrios V. Rovas,Karen Veroy,Luc Machiels,Yvon Maday,Anthony T. Patera,Gabriel Turinici +6 more
TL;DR: The method is ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control.
Proceedings ArticleDOI
A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations
TL;DR: In this paper, a technique for the prediction of linear-functional outputs of elliptic partial differential equations with affine parameter dependence is presented, where the essential components are (i) rapidly convergent global reduced-basis approximations -Galerkin projection onto a space WN 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 bounds for the error in the outputs of interest; and (iii) off-line/
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Certified reduced basis approximation for parametrized partial differential equations and applications
TL;DR: The reduced basis methods (built upon a high-fidelity ‘truth’ finite element approximation) for a rapid and reliable approximation of parametrized partial differential equations are reviewed, and their potential impact on applications of industrial interest is commented on.
Journal ArticleDOI
Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds
TL;DR: This work presents rigorous, sharp, and inexpensive a posteriori error bounds for reduced-basis approximations of the viscosity-parametrized Burgers equation and results confirm the performance of the error bounds.