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

Tensor methods are among the most prominent tools for the numerical solution of high-dimensional problems where functions of multiple variables have to be approximated. These methods exploit the tensor structure of function spaces and apply to many problems in computational science which are formulated in tensor spaces, such as problems arising in stochastic calculus, uncertainty quantification or parametric analyses. Here, we present complexity reduction methods based on low-rank approximation methods. We analyze the problem of best approximation in subsets of low-rank tensors and discuss its connection with the problem of optimal model reduction in low-dimensional reduced spaces. We present different algorithms for computing approximations of a function in low-rank formats. In particular, we present constructive algorithms which are based either on a greedy construction of an approximation (with successive corrections in subsets of low-rank tensors) or on the greedy construction of tensor subspaces (for subspace-based low-rank formats). These algorithms can be applied for tensor compression, tensor completion or for the numerical solution of equations in low-rank tensor formats. A special emphasis is given to the solution of stochastic or parameter-dependent models. Different approaches are presented for the approximation of vector-valued or multivariate functions (identified with tensors), based on samples of the functions (black-box approaches) or on the models equations which are satisfied by the functions.

Low-rank methods for high-dimensional approximation and model order reduction

This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities — which are mainly related either to nonlinear convection terms and/or some geometric variability — that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and — in the unsteady case — long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

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Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

A local convergence theorem for calculating canonical low-rank tensor approximations (PARAFAC, CANDECOMP) by the alternating least squares algorithm is established. The main assumption is that the Hessian matrix of the problem is positive definite modulo the scaling indeterminacy. A discussion, whether this is realistic, and numerical illustrations are included. Also regularization is addressed.

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Local Convergence of the Alternating Least Squares Algorithm for Canonical Tensor Approximation

We propose stable variational formulations for certain linear, unsymmetric operators with first order transport equations in bounded domains serving as the primary focus of this paper. The central objective is to develop for such classes adaptive solution concepts with provable error reduction. To adaptively resolve anisotropic solution features such as propagating singularities, the presently proposed variational formulations allow, in particular, the employment of trial spaces spanned by directional representation systems. Since such systems, typically given as frames, are known to be stable only in $L_2$, special emphasis is placed on $L_2$-stable formulations. The proposed stability concept is based on perturbations of certain “ideal” test spaces in Petrov--Galerkin formulations; see also [L. F. Demkowicz and J. Gopalakrishnan, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 1558--1572], [L. Demkowicz and J. Gopalakrishnan, Numer. Methods Partial Differential Equations, 27 (2011), pp. 70--105], [J...

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Adaptive Petrov-Galerkin methods for first order transport equations !

In this paper we propose and analyze stable variational formulations for convection diffu- sion problems starting from concepts introduced by Sangalli. We derive efficient and reliable ap osteri- ori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli's work, reveals partly unexpected phe- nomena related to the specific nature of the norms induced by the variational formulation. Several remedies, based on other specifications, are explored and illustrated by numerical experiments.

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Adaptivity and variational stabilization for convection-diffusion equations∗

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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Gerrit Welper

Papers

Adaptive Petrov-Galerkin methods for first order transport equations !

Adaptivity and variational stabilization for convection-diffusion equations∗

Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated Problems

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

Adaptive Petrov-Galerkin methods for first order transport equations