Certified Reduced Basis Methods for Parametrized Partial Differential Equations
- J. Hesthaven, G. Rozza, B. Stamm
- Mathematics
- 20 August 2015
This book provides a thorough introduction to the mathematical and algorithmic aspects of certified reduced basis methods for parametrized partial differential equations. Central aspects ranging from…
The reduced basis method for the electric field integral equation
- M. Fares, J. Hesthaven, Y. Maday, B. Stamm
- Computer ScienceJournal of Computational Physics
- 1 June 2011
Parameter multi‐domain ‘hp’ empirical interpolation
- J. L. Eftang, B. Stamm
- Mathematics, Computer Science
- 27 April 2012
Two parameter multi‐domain ‘ hp’ techniques for the empirical interpolation method (EIM) are introduced and it is shown that these approaches provide significant reduction in the EIM approximation space dimension and thus significantly reduce the computational cost associated with EIM approximations.
EFFICIENT GREEDY ALGORITHMS FOR HIGH-DIMENSIONAL PARAMETER SPACES WITH APPLICATIONS TO EMPIRICAL INTERPOLATION AND REDUCED BASIS METHODS ∗
- J. Hesthaven, B. Stamm, Shun Zhang
- Computer Science
- 2014
Two new algorithms to improve greedy sampling of high-dimensional parametrized functions are proposed, based on a saturation assumption of the error in the greedy algorithm and an algorithm in which the train set for the greedy approach is adaptively sparsified and enriched.
hp-Optimal discontinuous Galerkin methods for linear elliptic problems
A class of hp-DG methods that is closely related to other DG schemes, however, combines both p-optimal jump penalty as well as lifting stabilization and it is proved that the resulting error estimates are optimal with respect to both the local element sizes and polynomial degrees.
Low Order Discontinuous Galerkin Methods for Second Order Elliptic Problems
It is proved that both the symmetric and the nonsymmetric versions of the DG-method have regular system matrices without penalization of the interelement solution jumps provided boundary conditions are imposed in a certain weak manner.
Certified Reduced Basis Method for the Electric Field Integral Equation
- J. Hesthaven, B. Stamm, S. Zhang
- Computer ScienceSIAM Journal on Scientific Computing
- 28 June 2012
This paper proposes a certified RBM for the EFIE based on a mathematically rigorous a posteriori estimator and uses the $\boldsymbol{H}({\rm div})$-norm, which is computable and demonstrated to not degrade the quality of the error estimator.
Domain decomposition for implicit solvation models.
The methodology is detailed, the theoretical foundations of the approach are set up, the accuracies and convergence rates of the resulting algorithms are studied, and the full efficiency of the method and its applicability to large molecular systems of biological interest is demonstrated.
Locally Adaptive Greedy Approximations for Anisotropic Parameter Reduced Basis Spaces
This work introduces a framework where local approximation spaces (in parameter space) are used to define the reduced order approximation in order to have explicit control over the on-line cost, and adapts the local approximation Spaces to local anisotropic behavior in the parameter space.
Comparison of some Reduced Representation Approximations
- M. Bebendorf, Y. Maday, B. Stamm
- Computer Science
- 22 May 2013
The Adaptive Cross Approximation (ACA) and the Empirical Interpolation Method (EIM) together with other approaches that enter in the same category are presented.
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