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Felix Schindler

Publications and source records attributed to Felix Schindler.

At least 19 recordsLinked to original sources
arXiv2026

Data-Driven Model Order Reduction with pyMOR

pyMOR is a free and open-source software library of model order reduction algorithms for the Python programming language. Designed with classical model-based reduction methods for large-scale parametric partial differential equation problems in mind, algorithms in pyMOR are implemented in terms of operations on abstract VectorArray, Operator and Model interfaces, allowing for a seamless integration with external solver codes implementing the full-order model. For cases where a tight integration with the full-order model code is not feasible, data-driven model order reduction algorithms, which only require simulation or measurement data of the full-order model, are an attractive alternative. In this work we discuss the data-driven methods that have been recently added to pyMOR, show practical examples of their application using pyMOR and compare their performance with classical model-based methods. We show that pyMOR serves as a unified framework for combining model-based and data-driven methods, enabling the construction of flexible and efficient hierarchical model reduction pipelines.

cs.MS↗
arXiv2024

Local training and enrichment based on a residual localization strategy

To efficiently tackle parametrized multi and/or large scale problems, we propose an adaptive localized model order reduction framework combining both local offline training and local online enrichment with localized error control. For the latter, we adapt the residual localization strategy introduced in [Buhr, Engwer, Ohlberger, Rave, SIAM J. Sci. Comput., 2017] which allows to derive a localized a posteriori error estimator that can be employed to adaptively enrich the reduced solution space locally where needed. Numerical experiments demonstrate the potential of the proposed approach.

math.NA↗
arXiv2023

Application of Deep Kernel Models for Certified and Adaptive RB-ML-ROM Surrogate Modeling

In the framework of reduced basis methods, we recently introduced a new certified hierarchical and adaptive surrogate model, which can be used for efficient approximation of input-output maps that are governed by parametrized partial differential equations. This adaptive approach combines a full order model, a reduced order model and a machine-learning model. In this contribution, we extend the approach by leveraging novel kernel models for the machine learning part, especially structured deep kernel networks as well as two layered kernel models. We demonstrate the usability of those enhanced kernel models for the RB-ML-ROM surrogate modeling chain and highlight their benefits in numerical experiments.

math.NA↗
arXiv2022

An adaptive model hierarchy for data-augmented training of kernel models for reactive flow

We consider machine-learning of time-dependent quantities of interest derived from solution trajectories of parabolic partial differential equations. For large-scale or long-time integration scenarios, where using a full order model (FOM) to generate sufficient training data is computationally prohibitive, we propose an adaptive hierarchy of intermediate Reduced Basis reduced order models (ROM) to augment the FOM training data by certified ROM training data required to fit a kernel model.

math.NA↗
arXiv2022

On the Performance of Machine Learning Methods for Breakthrough Curve Prediction

Reactive flows are important part of numerous technical and environmental processes. Often monitoring the flow and species concentrations within the domain is not possible or is expensive, in contrast, outlet concentration is straightforward to measure. In connection with reactive flows in porous media, the term breakthrough curve is used to denote the time dependency of the outlet concentration with prescribed conditions at the inlet. In this work we apply several machine learning methods to predict breakthrough curves from the given set of parameters. In our case the parameters are the Damköhler and Peclet numbers. We perform a thorough analysis for the one-dimensional case and also provide the results for the three-dimensional case.

physics.flu-dyn↗
arXiv2022

Collapse of Coherent Large Scale Flow in Strongly Turbulent Liquid Metal Convection

The large-scale flow structure and the turbulent transfer of heat and momentum are directly measured in highly turbulent liquid metal convection experiments for Rayleigh numbers varied between $4 \times 10^5$ and $\leq 5 \times 10^9$ and Prandtl numbers of $0.025~\leq~Pr~\leq ~0.033$. Our measurements are performed in two cylindrical samples of aspect ratios $Γ=$ diameter/height $= 0.5$ and 1 filled with the eutectic alloy GaInSn. The reconstruction of the three-dimensional flow pattern by 17 ultrasound Doppler velocimetry sensors detecting the velocity profiles along their beamlines in different planes reveals a clear breakdown of coherence of the large-scale circulation for $Γ= 0.5$. As a consequence, the scaling laws for heat and momentum transfer inherit a dependence on the aspect ratio. We show that this breakdown of coherence is accompanied with a reduction of the Reynolds number $Re$. The scaling exponent $β$ of the power law $Nu\propto Ra^β$ crosses \FIN{eventually} over from $β=0.221$ to 0.124 when the liquid metal flow at $Γ=0.5$ reaches $Ra\gtrsim 2\times 10^8$ and the coherent large-scale flow is completely collapsed.

physics.flu-dyn↗
arXiv2021

Jump rope vortex flow in liquid metal Rayleigh-Bénard convection in a cuboid container of aspect ratio five

We study the topology and the temporal dynamics of turbulent Rayleigh-Benard convection in a liquid metal with a Prandtl number of 0.03 located inside a box with a square base area and an aspect ratio of five. Experiments and numerical simulations are focused on the moderate Rayleigh number range, where a new cellular flow regime has been reported by a previous study (Akashi et al., Phys. Rev. Fluids, vol.4, 2019, 033501). This flow structure shows symmetries with respect to the vertical planes crossing at the center of the container. The dynamic behaviour is dominated by strong three-dimensional oscillations with a period length that corresponds to the turnover time. Our analysis reveals that the flow structure in the aspect ratio five box corresponds in key features to the jump rope vortex structure, which has recently been discovered in an aspect ratio two cylinder (Vogt et al., Proc. Natl Acad. Sci. USA, vol.115, 2018, 12674-12679). While in the cylinder a single jump rope vortex occurs, the coexistence of four recirculating swirls is detected in this study. %Their cycling movement is restrained by the limited height of the fluid layer in the aspect ratio five box. Their approach to the lid or the bottom of the convection box causes a temporal deceleration of both the horizontal velocity at the respective boundary and the vertical velocity in the bulk, which in turn is reflected in Nusselt number oscillations. The cellular flow regime shows remarkable similarities to properties commonly attributed to turbulent superstructures

physics.flu-dyn↗
arXiv2020

An adaptive projected Newton non-conforming dual approach for trust-region reduced basis approximation of PDE-constrained parameter optimization

In this contribution we device and analyze improved variants of the non-conforming dual approach for trust-region reduced basis (TR-RB) approximation of PDE-constrained parameter optimization that has recently been introduced in [Keil et al.. A non-conforming dual approach for adaptive Trust-Region Reduced Basis approximation of PDE-constrained optimization. arXiv:2006.09297, 2020]. The proposed methods use model order reduction techniques for parametrized PDEs to significantly reduce the computational demand of parameter optimization with PDE constraints in the context of large-scale or multi-scale applications. The adaptive TR approach allows to localize the reduction with respect to the parameter space along the path of optimization without wasting unnecessary resources in an offline phase. The improved variants employ projected Newton methods to solve the local optimization problems within each TR step to benefit from high convergence rates. This implies new strategies in constructing the RB spaces, together with an estimate for the approximation of the hessian. Moreover, we present a new proof of convergence of the TR-RB method based on infinite-dimensional arguments, not restricted to the particular case of an RB approximation and provide an a posteriori error estimate for the approximation of the optimal parameter. Numerical experiments demonstrate the efficiency of the proposed methods.

math.NA↗
arXiv2020

Free-fall velocities and heat transport enhancement in liquid metal magneto-convection

In geo- and astrophysics, low Prandtl number convective flows often interact with magnetic fields. Although a static magnetic field acts as a stabilizing force on such flow fields, we find that self-organized convective flow structures reach an optimal state where the heat transport significantly increases and convective velocities reach the theoretical free-fall limit, i.e. the maximum possible velocity a fluid parcel can achieve when its potential buoyant energy is fully converted into kinetic energy. Our measurements show that the application of a static magnetic field leads to an anisotropic, highly ordered flow structure and a decrease of the turbulent fluctuations. When the magnetic field strength is increased beyond the optimum, Hartmann braking becomes dominant and leads to a reduction of the heat and momentum transport. The results are relevant for the understanding of magneto-hydrodynamic convective flows in planetary cores and stellar interiors in regions with strong toroidal magnetic fields oriented perpendicular to temperature gradients.

physics.flu-dyn↗
arXiv2020

A non-conforming dual approach for adaptive Trust-Region Reduced Basis approximation of PDE-constrained optimization

In this contribution we propose and rigorously analyze new variants of adaptive Trust-Region methods for parameter optimization with PDE constraints and bilateral parameter constraints. The approach employs successively enriched Reduced Basis surrogate models that are constructed during the outer optimization loop and used as model function for the Trust-Region method. Each Trust-Region sub-problem is solved with the projected BFGS method. Moreover, we propose a non-conforming dual (NCD) approach to improve the standard RB approximation of the optimality system. Rigorous improved a posteriori error bounds are derived and used to prove convergence of the resulting NCD-corrected adaptive Trust-Region Reduced Basis algorithm. Numerical experiments demonstrate that this approach enables to reduce the computational demand for large scale or multi-scale PDE constrained optimization problems significantly.

math.NA↗
arXiv2020

Generating a tide-like flow in a cylindrical vessel by electromagnetic forcing

We show and compare numerical and experimental results on the electromagnetic generation of a tide-like flow structure in a cylindrical vessel which is filled with the eutectic liquid metal alloy GaInSn. Fields of various strengths and frequencies are applied to drive liquid metal flows. The impact of the field variations on amplitude and structure of the flows is investigated. The results represent the basis for a future Rayleigh-Bénard experiment, in which a modulated tide-like flow perturbation is expected to synchronize the typical sloshing mode of the large-scale circulation. A similar entrainment mechanism for the helicity in the Sun may be responsible for the synchronization of the solar dynamo with the alignment cycle of the tidally dominant planets Venus, Earth and Jupiter.

physics.flu-dyn↗
arXiv2020

Combined measurement of velocity and temperature in liquid metal convection

Combined measurements of velocity components and temperature in a turbulent Rayleigh-Bénard convection flow at a low Prandtl number of $\mathit{Pr}= 0.029$ and Rayleigh numbers between $10^6 \le \mathit{Ra} \le 6 \times 10^7$ are conducted in a series of experiments with durations of more than a thousand free-fall time units. Multiple crossing ultrasound beam lines and an array of thermocouples at mid-height allow for a detailed analysis and characterization of the complex three-dimensional dynamics of the single large-scale circulation (LSC) roll in the cylindrical convection cell of unit aspect ratio which is filled with the liquid metal alloy GaInSn. We measure the internal temporal correlations of the complex large-scale flow and distinguish between short-term oscillations associated with a sloshing motion in the mid-plane as well as varying orientation angles of the velocity close to the top/bottom plates and the slow azimuthal drift of the mean orientation of the roll as a whole that proceeds on an up to a hundred times slower time scale. The coherent LSC drives a vigorous turbulence in the whole cell that is quantified by direct Reynolds number measurements at different locations in the cell. The velocity increment statistics in the bulk of the cell displays characteristic properties of intermittent small-scale fluid turbulence. We also show that the impact of the symmetry-breaking large-scale flow persists to small-scale velocity fluctuations thus preventing the establishment of fully isotropic turbulence in the cell centre. Reynolds number amplitudes depend sensitively on beam line position in the cell such that different definitions have to be compared. The global momentum and heat transfer scalings with Rayleigh number are found to agree with those of direct numerical simulations and other laboratory experiments.

physics.flu-dyn↗
arXiv2020

Flow regimes of Rayleigh-Bénard convection in a vertical magnetic field

The effects of a vertical static magnetic field on the flow structure and global transport properties of momentum and heat in liquid metal Rayleigh-Bénard convection are investigated. Experiments are conducted in a cylindrical convection cell of unity aspect ratio, filled with the alloy GaInSn at a low Prandtl number of $\mathit{Pr}=0.029$. Changes of the large-scale velocity structure with increasing magnetic field strength are probed systematically using multiple ultrasound Doppler velocimetry sensors and thermocouples for a parameter range that is spanned by Rayleigh numbers of $10^6 \le \mathit{Ra} \le 6\times 10^7$ and Hartmann numbers of $\mathit{Ha} \le 1000$. Our simultaneous multi-probe temperature and velocity measurements demonstrate how the large-scale circulation is affected by an increasing magnetic field strength (or Hartmann number). Lorentz forces induced in the liquid metal first suppress the oscillations of the large-scale circulation at low $\mathit{Ha}$, then transform the one-roll structure into a cellular large-scale pattern consisting of multiple up- and downwellings for intermediate $\mathit{Ha}$, before finally expelling any fluid motion out of the bulk at the highest accessible $\mathit{Ha}$ leaving only a near-wall convective flow that persists even below Chandrasekhar's linear instability threshold. Our study thus proves experimentally the existence of wall modes in confined magnetoconvection. The magnitude of the transferred heat remains nearly unaffected by the steady decrease of the fluid momentum over a large range of Hartmann numbers. We extend the experimental global transport analysis to momentum transfer and include the dependence of the Reynolds number on the Hartmann number.

physics.flu-dyn↗
arXiv2019

Localized model reduction for parameterized problems

In this contribution we present a survey of concepts in localized model order reduction methods for parameterized partial differential equations. The key concept of localized model order reduction is to construct local reduced spaces that have only support on part of the domain and compute a global approximation by a suitable coupling of the local spaces. In detail, we show how optimal local approximation spaces can be constructed and approximated by random sampling. An overview of possible conforming and non-conforming couplings of the local spaces is provided and corresponding localized a posteriori error estimates are derived. We introduce concepts of local basis enrichment, which includes a discussion of adaptivity. Implementational aspects of localized model reduction methods are addressed. Finally, we illustrate the presented concepts for multiscale, linear elasticity and fluid-flow problems, providing several numerical experiments. This work has been accepted as a chapter in P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Sileira. Handbook on Model Order Reduction. Walter De Gruyter GmbH, Berlin, 2019+.

math.NA↗
arXiv2019

A locally conservative reduced flux reconstruction for elliptic problems

In the context of model order reduction of parametric elliptic problems, we present a methodology to reconstruct a conforming flux from a given reduced solution, that is locally conservative with respect to the underlying finite element grid. All components of the procedure depend separably on the parameter and allow for further use in offline/online decomposed computations, for instance in the context of a posterior error estimation or flow problems.

math.NA↗
arXiv2016

True Error Control for the Localized Reduced Basis Method for Parabolic Problems

We present an abstract framework for a posteriori error estimation for approximations of scalar parabolic evolution equations, based on elliptic reconstruction techniques [10, 9, 3, 5]. In addition to its original application (to derive error estimates on the discretization error), we extend the scope of this framework to derive offline/online decomposable a posteriori estimates on the model reduction error in the context of Reduced Basis (RB) methods. In addition, we present offline/online decomposable a posteriori error estimates on the full approximation error (including discretization as well as model reduction error) in the context of the localized RB method [14]. Hence, this work generalizes the localized RB method with true error certification to parabolic problems. Numerical experiments are given to demonstrate the applicability of the approach.

math.NA↗