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Proper Orthogonal Decomposition (POD) is a widely used technique for the construction of low-dimensional approximation spaces from highdimensional input data. For large-scale applications and an increasing number of input data vectors, however, computing the POD often becomes prohibitively expensive. This work presents a general, easy-to-implement approach to compute an approximate POD based on arbitrary tree hierarchies of worker nodes, where each worker computes a POD of only a small number of input vectors. The tree hierarchy can be freely adapted to optimally suit the available computational resources. In particular, this hierarchical approximate POD (HAPOD) allows for both simple parallelization with low communication overhead, as well as incremental POD computation under constrained memory capacities. Rigorous error estimates ensure the reliability of our approach, and extensive numerical examples underline its performance.

Abstract. Reduced basis methods are projection-based model order reduction techniques for reducing the computational complexity of solving parametrized partial differential equation problems. In this work we discuss the design of pyMOR, a freely available software library of model order reduction algorithms, in particular reduced basis methods, implemented with the Python programming language. As its main design feature, all reduction algorithms in pyMOR are implemented generically via operations on well-defined vector array, operator and discretization interface classes. This allows for an easy integration with existing open-source high-performance partial differential equation solvers without adding any model reduction specific code to these solvers. Besides an in-depth discussion of pyMOR's design philosophy and architecture, we present several benchmark results and numerical examples showing the feasibility of our approach. Key words. model order reduction, reduced basis method, empirical interpolation, scientific computing, software, Python AMS subject classifications. 35-04, 35J20, 35L03, 65-04, 65N30, 65Y05, 68N01.1. Introduction. Over the past years, model order reduction methods have become an important part of many numerical simulation workflows for handling large-scale application problems. Reduced basis (RB) methods are a popular family of such reduction techniques, applicable to parametrized high-dimensional models described by partial differential equations (PDEs). The main ingredient of RB methods is a Galerkin projection of the differential equation onto a problem-adapted reduced subspace generated from solution snapshots of a high-dimensional approximation of the problem for certain wellchosen sampling parameters. While the high-dimensional approximation using standard discretization techniques (such as finite element methods) often yields discrete function spaces with millions of degrees of freedoms, the reduced spaces generated by RB methods typically are of order 100 or smaller, while still retaining the same approximation quality for the problem at hand as the high-dimensional space. In practice, model order reduction by RB approximation can lead to speedups of up to several orders of magnitude. By now, a large body of literature has emerged which theoretically proves and practically demonstrates the applicability of the RB approach to a large variety of application problems (see, e.g., the recent monographs [18,30], the tutorial [15], and the references therein).

Engineers manually optimizing a structure using Finite Element based simulation software often employ an iterative approach where in each iteration they change the structure slightly and resimulate. Standard Finite Element based simulation software is usually not well suited for this workflow, as it restarts in each iteration, even for tiny changes. In settings with complex local microstructure, where a fine mesh is required to capture the geometric detail, localized model reduction can improve this workflow. To this end, we introduce ArbiLoMod, a method which allows fast recomputation after arbitrary local modifications. It employs a domain decomposition and a localized form of the Reduced Basis Method for model order reduction. It assumes that the reduced basis on many of the unchanged domains can be reused after a localized change. The reduced model is adapted when necessary, steered by a localized error indicator. The global error introduced by the model order reduction is controlled by a robust and efficient localized a posteriori error estimator, certifying the quality of the result. We demonstrate ArbiLoMod for a coercive, parameterized example with changing structure.

Parametric model order reduction using reduced basis methods can be an effective tool for obtaining quickly solvable reduced order models 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.