A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations

By Martin A. Grepl, Anthony T. Patera. 2005

418 citations recorded by the source corpus.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c82e7782b4b902f51ba9982c1db47abb84ecb504

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Citations in the corpus

Physically based modeling in catchment hydrology at 50: Survey and outlook (2015, methodology): “Recent research in this field has been spurred by advances in proper orthogonal decomposition and reduced basis finite element methods [Kunisch and Volkwein, 2001; Vermeulen et al., 2004; Grepl and Patera, 2005; Bui-Thanh et al., 2008; Quarteroni et al., 2011; Manzoni, 2014].”
Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Parabolic PDEs: Application to Real‐Time Bayesian Parameter Estimation (2010, methodology): “norm of the residual Grepl and Patera (2005) .”
A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems (2015, positive): “It was further developed in reduced basis models for elliptic PDEs in [218], the steady incompressible Navier–Stokes equations in [217], and parabolic PDEs in [111, 112].”
REDUCED BASIS METHOD FOR FINITE VOLUME APPROXIMATIONS OF PARAMETRIZED LINEAR EVOLUTION EQUATIONS (2008, methodology): “The general procedure of incremental basis-construction in the offline-phase [14,15] can be divided into the following steps: (1) compute a small initial reduced basis ΦN0 of N0 basis-vectors; (2) set ΦN := ΦN0 and N := N0; (3) determine all offline-simulation quantities for the current RB-basis ΦN ”
Nonlinear Model Reduction via Discrete Empirical Interpolation (2010, methodology): “Basis(RB) framework [39], whose basis functions would be the snapshots selected by an adaptive greedy selection process.”
Certified Real-Time Solution of Parametrized Partial Differential Equations (2005, background): “Finally, a third question, again related to generality: What class of PDEs can be treated? In addition to the elliptic equations discussed in this paper, parabolic equations can also be addressed satisfactorily from both the approximation and error estimation points of view [24, 34, 35 ]:† much of ”
Stabilized reduced basis method for parametrized advection-diffusion PDEs (2014, methodology): “Discretization and RB formulation To discretize the time-dependent problem (44) we follow the approach used in [10, 22, 25] which consists in using finite differences in time and FE in space discretization [27].”
Surrogate and reduced-order modeling: a comparison of approaches for large-scale statistical inverse problems [Chapter 7] (2010, methodology): “…the challenge of sampling a high-dimensional parmeter space to build a reduced basis, the greedy sampling method was introduced in Grepl (2005); Grepl and Patera (2005); Veroy and Patera (2005); Veroy et al. (2003) to adaptively choose samples by finding the location at which the estimate of…”
Reduced basis methods for partial differential equations with stochastic influences (2013, methodology): “However, it is straightforward to apply the introduced methodology also for instationary problems, as considered for example in [27, 38, 40].”
Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations (2007, methodology): “In particular, in Part IV we extend to parabolic PDEs the primal-dual approximations, greedy sampling strategies, a posteriori error estimation concepts, and Offline-Online computational strategies developed in Part II for elliptic PDEs — with particular focus on the accommodation of an “evolution” ”
Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations (2007, background): “The exploration of the “parameter + time” framework in the important context of affine linear (stable) parabolic PDEs — such as the heat equation and the passive scalar convection-diffusion equation (also the Black-Scholes equation of derivative theory [118]) — is carried out in [52, 55 ,60,133]; ma”
Model Order Reduction in Fluid Dynamics: Challenges and Perspectives (2014, methodology): “In this case, snapshots are typically chosen iteratively by measuring the error of the current ROM at different trial points of the parameter space, then computing snapshots at the parameter point where the maximum error (estimate) is obtained and adding them to the ROM, like in the greedy-RB algori”
Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space (2008, methodology): “It was also combined with a posteriori error estimators for parameterized parabolic PDEs and applied to several optimal control and inverse problems [15, 14].”
Certified reduced basis approximation for parametrized partial differential equations and applications (2011, methodology): “It is important to note that the POD-greedy method readily accommodates a repeat μ∗ in successive greedy cycles – new information will always be available and old information rejected; in contrast, a pure greedy approach in both t and μ [31], though often generating good spaces, can “stall.”
Application of POD and DEIM on dimension reduction of non-linear miscible viscous fingering in porous media (2011, background): “It is possible to consider varying multiple parameters at the same time with a larger range for each of them as done in, for example, [36,37].”