Certified Real-Time Solution of Parametrized Partial Differential Equations

By Nguyen Ngoc Cuong, Karen Veroy, Anthony T. Patera. 2005

195 citations recorded by the source corpus.

Subfields: Numerical Analysis, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4bc4772827a0b64eaa91d81e58ff92d57ebe3cc6

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In Numerical Analysis

In Statistical and Nonlinear Physics

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Citations in the corpus

Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy‐based mesh sampling and weighting for computational efficiency (2014, background): “Examples include the frequency-domain analysis of linear, time-invariant dynamical systems [1], the linear static analysis of problems where the underlying mathematical operators are affine in their parameter space [2], and problems with at most quadratic nonlinearities [3, 4].”
Numerical solution of parametrized Navier–Stokes equations by reduced basis methods (2007, background): “global approximation functions) to study natural convection problem, parametrized by physical quantities (such as the Grashof number [15] or the viscosity [35]) in divergence-free spaces.”
Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation (2009, unknown): “There are examples of rigorous reduced basis a posteriori error bounds for the steady Burgers’ [37] and incompressible Navier–Stokes [28,36] equations; the new contribution of the current paper is treatment of the unsteady — parabolic — case.”
A general multipurpose interpolation procedure: the magic points (2008, unknown): “Unfortunately, the presence of strong nonlinearity in g does not allow an efficient offline-online procedure outlined in [23, 18].”
REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS (2011, background): “aEarlier work has established reduced basis approximations and associated rigorous a posteriori error bounds for nonsingular solution branches of the steady Burgers [55] and steady incompressible Navier-Stokes and Boussinesq equations [9, 14,45,54]; we focus here on the unsteady case.”
Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations (2007, background): “For the Navier-Stokes equations (and for nonlinear equations more generally) we can not appeal to any simple monotonicity arguments; our focus is thus on the computational (quantitative) realization of the general Brezzi-Rappaz-Raviart (“BRR”) a posteriori theory [28, 34] — and development of associ”
Certified reduced basis approximation for parametrized partial differential equations and applications (2011, background): “Much current effort in the last ten years in the RB framework has thus been devoted to the development of (i) a posteriori error estimation procedures – and in particular rigorous error bounds for outputs of interest – and (ii) effective sampling strategies, in particular for higher dimensional para”
Model Order Reduction in Fluid Dynamics: Challenges and Perspectives (2014, background): “We do not provide any detail about the evaluation of these quantities; the interested reader can refer, for instance, to [76, 87, 94, 124].”
Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations (2007, methodology): “Much current effort is thus devoted to development of (i ) a posteriori error estimation procedures and in particular rigorous error bounds for outputs of interest [121], and (ii ) effective sampling strategies in particular for higher (than one) dimensional parameter domains [33,32, 97 ,138,153].”
Asymptotic normality and efficiency of two Sobol index estimators (2013, background): “Well-known metamodels include Kriging [24], polynomial chaos expansion [30] and reduced bases [12, 19], to name a few.”
Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations (2007, methodology): “This particular result is special to the compliant output case; more general output and error estimates (for noncompliant, nonsymmetric, and noncoercive problems) can be found in [25, 30, 34].”