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.

Moment models are a class of specialized approximate models for kinetic transport equations. These models transform the kinetic equation to a system of equations for weighted velocity averages of the solution, called moments, thereby removing the velocity dependency. The properties of the resulting models depend on the chosen weight functions for the moments and on the approach used to close the equations. Closing the system by specifying a linear ansatz function results in linear models that are comparatively easy to solve but may show non-physical behaviour. Minimum-entropy moment closures, on the other hand, conserve many of the fundamental physical properties but require the solution of a non-linear optimization problem for every cell in the space-time grid. In addition, these models are only well-defined if the moments can be kept within a particular subset of the real coordinate space, the so-called realizable set, which is particularly challenging for higher-order numerical solvers.Schließlich beschäftigen wir uns mit der Möglichkeit einer zusätzlichen Modellreduktion für parameterabhängige Momentenmodelle. Hierzu nutzen wir die Reduzierte-Basis-Methode und berechnen mit der Hauptkomponentenanalyse ("Proper Orthogonal Decomposition", POD) ein reduziertes Modell. Wir stellen die hierarchische approximative POD (HAPOD) vor, ein universelles und einfach zu implementierendes Verfahren um eine approximative POD zu berechnen. Die HAPOD nutzt eine fast beliebig an die verfügbaren Ressourcen anpassbare Baumstruktur, so dass jeder Rechenknoten nur die POD einer relativ kleinen Teilmenge der Eingangsvektoren berechnen muss. Wir führen eine ausführliche theoretische Analyse der HAPOD durch und zeigen die Anwendbarkeit auf lineare Momentenmodelle sowie eine gegenüber der POD deutlich verbesserte Effizienz.

We derive a second-order realizability-preserving scheme for moment models for linear kinetic equations. We apply this scheme to the first-order continuous (HFM n ) and discontinuous (PMM n ) models in slab and three-dimensional geometry derived in [54] as well as the classical full-moment M N models. We provide extensive numerical analysis as well as our code to show that the new class of models can compete or even outperform the full-moment models in reasonable test cases.

The cross Gramian matrix encodes the input-output coherence of linear control systems and is used in projection-based model reduction. The empirical cross Gramian is a data-driven variant of the cross Gramian which also extends to nonlinear systems. A drawback of the empirical cross Gramian for large-scale systems is its full order and dense structure; yet, it may be computed column-wise. Using the hierarchical approximate proper orthogonal decomposition (HAPOD), this partial computability can be exploited to obtain a truncated projection for model order reduction without assembling a full cross Gramian. Model ReductionThe simulation of large-scale input-output systems is a computationally challenging task, even though, typically, a lowdimensional input is mapped to a low-dimensional output. Yet, this mapping involves the solution of a, possibly highdimensional, differential equation system. Model order reduction (MOR) accelerates such simulations by computing reduced order models that approximate the full order model's output [3]. In this work we summarize the computation of the low-rank empirical-cross-Gramian-based model reduction technique [4]. Generally, this presented MOR method is applicable to nonlinear systems, but for the sake of exposition, we focus on the reduction of linear time-invariant systems with M := dim(u(t)) inputs, N := dim(x(t)) and Q := dim(y(t)) outputs:The reduced order model has a lower dimension in the differential equation system dim(x r (t)) dim(x(t)) =: N and exhibits a small output error y − y r 1. Cross GramianCross-Gramian-based model reduction is a balancing approach closely related to balanced truncation. Balancing refers to the transformation of the underlying system into a coordinate system in which the system-theoretic attributes controllability and observability are equalized. For square systems, M = Q, a cross Gramian matrix, which jointly encodes these properties, can be computed and a truncated singular value decomposition (tSVD) of this cross Gramian results in an approximate balancing Galerkin projection U 1 that enables the projection-based model reduction of the original system: Empirical Cross GramianThe cross Gramian can be computed in a data-driven manner by the empirical cross Gramian W X , which is equal to W X for linear systems up to numerical error [5]:for a system with M inputs and outputs, state trajectories x m (t) with a perturbed m-th input component, output trajectories y j (t) with a perturbed j-th initial state component and respective temporal averagesx m ,ȳ j . Note that the computation does not require the linear system components A, B, C.

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