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 model construction, error estimation and computational efficiency to empirical interpolation methods are discussed in detail for coercive problems. More advanced aspects associated with time-dependent problems, non-compliant and non-coercive problems and applications with geometric variation are also discussed as examples.

Certified Reduced Basis Methods for Parametrized Partial Differential Equations

In this part we are concerned with a class of model reduction techniques for parametric partial differential equations, the so-called Reduced Basis (RB) methods. These allow to obtain low-dimensional parametric models for various complex applications, enabling accurate and rapid numerical simulations. Important aspects are basis generation and certification of the simulation results by suitable a posteriori error control. The main terminology, ideas and assumptions will be explained for the case of linear stationary elliptic, as well as parabolic or hyperbolic instationary problems. Reproducable experiments will illustrate the theoretical findings. We close with a discussion of further recent developments.

Chapter 2: Reduced Basis Methods for Parametrized PDEs—A Tutorial Introduction for Stationary and Instationary Problems

Parametric model order reduction using reduced basis methods can be an effective tool for obtaining quickly solvable reduced order model  of parametrized partial differential equation problems. With speedups that can reach several orders of magnitude, reduced basis methods enable high fidelity real-time simulations of complex systems and dramatically reduce the computational costs in many-query applications. In this contribution we analyze the methodology, mainly focussing on the theoretical aspects of the approach. In particular we discuss what is known about the convergence properties of these methods: when they succeed and when they are bound to fail. Moreover, we highlight some recent approaches employing nonlinear  approximation techniques which aim to overcome the current limitations of reduced basis methods.

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Reduced Basis Methods: Success, Limitations and Future Challenges

Hypergraphs allow one to encode higher-order relationships in data and are thus a very flexible modeling tool. Current learning methods are either based on approximations of the hypergraphs via graphs or on tensor methods which are only applicable under special conditions. In this paper, we present a new learning framework on hypergraphs which fully uses the hypergraph structure. The key element is a family of regularization functionals based on the total variation on hypergraphs.

The Total Variation on Hypergraphs - Learning on Hypergraphs Revisited

We present reduced basis approximations and associated rigorous a posteriori error bounds for parametrized saddle point problems. First, we develop new a posteriori error estimates that, unlike earlier approaches, provide upper bounds for the errors in the approximations of the primal variable and the Lagrange multiplier separately. The proposed method is an application of Brezzi's theory for saddle point problems to the reduced basis context and exhibits significant advantages over existing methods. Second, based on an analysis of Brezzi's theory, we compare several options for the reduced basis approximation space from the perspective of approximation stability and computational cost. Finally, we introduce a new adaptive sampling procedure for saddle point problems constructing approximation spaces that are stable and, compared to earlier approaches, computationally much more efficient. The method is applied to a Stokes flow problem in a two-dimensional channel with a parametrized rectangular obstacle. ...

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Certified Reduced Basis Methods for Parametrized Saddle Point Problems

The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal performance when compared with the Kolmogorov $n$-widths of the solution sets. The central ingredient is the construction of computationally feasible "tight" surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated by numerical experiments for convection-diffusion and pure transport equations. In particular, the latter example sheds some light on the smoothness of the dependence of the solutions on the parameters.

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Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated Problems

The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal perfor- mance when compared with the Kolmogorov n-widths of the solution sets. The central ingredient is the construction of computationally feasible "tight" surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated by numerical experiments for convection-diffusion and pure transport equations. In particular, the latter example sheds some light on the smoothness of the dependence of the solutions on the parameters.

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Christian Plesken

Papers

Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated Problems

Double greedy algorithms: reduced basis methods for transport dominated problems ∗