Browsing by Author "Haasdonk, Bernard"

Now showing 1 - 13 of 13

Open Access
Analysis of target data-dependent greedy kernel algorithms : convergence rates for f-, f· P- and f/P-greedy
(2022) Wenzel, Tizian; Santin, Gabriele; Haasdonk, Bernard
Data-dependent greedy algorithms in kernel spaces are known to provide fast converging interpolants, while being extremely easy to implement and efficient to run. Despite this experimental evidence, no detailed theory has yet been presented. This situation is unsatisfactory, especially when compared to the case of the data-independent P-greedy algorithm, for which optimal convergence rates are available, despite its performances being usually inferior to the ones of target data-dependent algorithms. In this work, we fill this gap by first defining a new scale of greedy algorithms for interpolation that comprises all the existing ones in a unique analysis, where the degree of dependency of the selection criterion on the functional data is quantified by a real parameter. We then prove new convergence rates where this degree is taken into account, and we show that, possibly up to a logarithmic factor, target data-dependent selection strategies provide faster convergence. In particular, for the first time we obtain convergence rates for target data adaptive interpolation that are faster than the ones given by uniform points, without the need of any special assumption on the target function. These results are made possible by refining an earlier analysis of greedy algorithms in general Hilbert spaces. The rates are confirmed by a number of numerical examples.
Open Access
Dictionary-based online-adaptive structure-preserving model order reduction for parametric Hamiltonian systems
(2024) Herkert, Robin; Buchfink, Patrick; Haasdonk, Bernard
Classical model order reduction (MOR) for parametric problems may become computationally inefficient due to large sizes of the required projection bases, especially for problems with slowly decaying Kolmogorov n -widths. Additionally, Hamiltonian structure of dynamical systems may be available and should be preserved during the reduction. In the current presentation, we address these two aspects by proposing a corresponding dictionary-based, online-adaptive MOR approach. The method requires dictionaries for the state-variable, non-linearities, and discrete empirical interpolation (DEIM) points. During the online simulation, local basis extensions/simplifications are performed in an online-efficient way, i.e., the runtime complexity of basis modifications and online simulation of the reduced models do not depend on the full state dimension. Experiments on a linear wave equation and a non-linear Sine-Gordon example demonstrate the efficiency of the approach.
Open Access
Efficient parametric analysis of the chemical master equation through model order reduction
(2012) Waldherr, Steffen; Haasdonk, Bernard
BACKGROUND: Stochastic biochemical reaction networks are commonly modelled by the chemical master equation, and can be simulated as first order linear differential equations through a finite state projection. Due to the very high state space dimension of these equations, numerical simulations are computationally expensive. This is a particular problem for analysis tasks requiring repeated simulations for different parameter values. Such tasks are computationally expensive to the point of infeasibility with the chemical master equation.RESULTS:In this article, we apply parametric model order reduction techniques in order to construct accurate low-dimensional parametric models of the chemical master equation. These surrogate models can be used in various parametric analysis task such as identifiability analysis, parameter estimation, or sensitivity analysis. As biological examples, we consider two models for gene regulation networks, a bistable switch and a network displaying stochastic oscillations. CONCLUSIONS: The results show that the parametric model reduction yields efficient models of stochastic biochemical reaction networks, and that these models can be useful for systems biology applications involving parametric analysis problems such as parameter exploration, optimization, estimation or sensitivity analysis.
Open Access
Greedy kernel methods for approximating breakthrough curves for reactive flow from 3D porous geometry data
(2024) Herkert, Robin; Buchfink, Patrick; Wenzel, Tizian; Haasdonk, Bernard; Toktaliev, Pavel; Iliev, Oleg
We address the challenging application of 3D pore scale reactive flow under varying geometry parameters. The task is to predict time-dependent integral quantities, i.e., breakthrough curves, from the given geometries. As the 3D reactive flow simulation is highly complex and computationally expensive, we are interested in data-based surrogates that can give a rapid prediction of the target quantities of interest. This setting is an example of an application with scarce data, i.e., only having a few available data samples, while the input and output dimensions are high. In this scarce data setting, standard machine learning methods are likely to fail. Therefore, we resort to greedy kernel approximation schemes that have shown to be efficient meshless approximation techniques for multivariate functions. We demonstrate that such methods can efficiently be used in the high-dimensional input/output case under scarce data. Especially, we show that the vectorial kernel orthogonal greedy approximation (VKOGA) procedure with a data-adapted two-layer kernel yields excellent predictors for learning from 3D geometry voxel data via both morphological descriptors or principal component analysis.
Open Access
Hermite kernel surrogates for the value function of high-dimensional nonlinear optimal control problems
(2024) Ehring, Tobias; Haasdonk, Bernard
Numerical methods for the optimal feedback control of high-dimensional dynamical systems typically suffer from the curse of dimensionality. In the current presentation, we devise a mesh-free data-based approximation method for the value function of optimal control problems, which partially mitigates the dimensionality problem. The method is based on a greedy Hermite kernel interpolation scheme and incorporates context knowledge by its structure. Especially, the value function surrogate is elegantly enforced to be 0 in the target state, non-negative and constructed as a correction of a linearized model. The algorithm allows formulation in a matrix-free way which ensures efficient offline and online evaluation of the surrogate, circumventing the large-matrix problem for multivariate Hermite interpolation. Additionally, an incremental Cholesky factorization is utilized in the offline generation of the surrogate. For finite time horizons, both convergence of the surrogate to the value function and for the surrogate vs. the optimal controlled dynamical system are proven. Experiments support the effectiveness of the scheme, using among others a new academic model with an explicitly given value function. It may also be useful for the community to validate other optimal control approaches.
Open Access
Improved a posteriori error bounds for reduced port-Hamiltonian systems
(2024) Rettberg, Johannes; Wittwar, Dominik; Buchfink, Patrick; Herkert, Robin; Fehr, Jörg; Haasdonk, Bernard
Projection-based model order reduction of dynamical systems usually introduces an error between the high-fidelity model and its counterpart of lower dimension. This unknown error can be bounded by residual-based methods, which are typically known to be highly pessimistic in the sense of largely overestimating the true error. This work applies two improved error bounding techniques, namely (a) a hierarchical error bound and (b) an error bound based on an auxiliary linear problem , to the case of port-Hamiltonian systems. The approaches rely on a secondary approximation of (a) the dynamical system and (b) the error system. In this paper, these methods are adapted to port-Hamiltonian systems. The mathematical relationship between the two methods is discussed both theoretically and numerically. The effectiveness of the described methods is demonstrated using a challenging three-dimensional port-Hamiltonian model of a classical guitar with fluid–structure interaction.
Open Access
Improving determination of pigment contents in microalgae suspension with absorption spectroscopy : light scattering effect and Bouguer-Lambert-Beer law
(2023) Yeh, Yen-Cheng; Ebbing, Tobias; Frick, Konstantin; Schmid-Staiger, Ulrike; Haasdonk, Bernard; Tovar, Günter E. M.
The Bouguer-Lambert-Beer (BLB) law serves as the fundamental basis for the spectrophotometric determination of pigment content in microalgae. Although it has been observed that the applicability of the BLB law is compromised by the light scattering effect in microalgae suspensions, in-depth research concerning the relationship between the light scattering effect and the accuracy of spectrophotometric pigment determination remains scarce. We hypothesized that (1) the precision of spectrophotometric pigment content determination using the BLB law would diminish with increasing nonlinearity of absorbance, and (2) employing the modified version of the BLB (mBLB) law would yield superior performance. To assess our hypotheses, we cultivated Phaeodactylum tricornutum under varying illumination conditions and nitrogen supplies in controlled indoor experiments, resulting in suspensions with diverse pigment contents. Subsequently, P. tricornutum samples were diluted into subsamples, and spectral measurements were conducted using different combinations of biomass concentrations and path lengths. This was carried out to assess the applicability of the BLB law and the nonlinearity of absorbance. The chlorophyll a and fucoxanthin contents in the samples were analyzed via high-performance liquid chromatography (HPLC) and subsequently used in our modeling. Our findings confirm our hypotheses, showing that the modified BLB law outperforms the original BLB law in terms of the normalized root mean square error (NRMSE): 6.3% for chlorophyll a and 5.8% for fucoxanthin, compared to 8.5% and 7.9%, respectively.
Open Access
A new method to design energy-conserving surrogate models for the coupled, nonlinear responses of intervertebral discs
(2024) Hammer, Maria; Wenzel, Tizian; Santin, Gabriele; Meszaros-Beller, Laura; Little, Judith Paige; Haasdonk, Bernard; Schmitt, Syn
The aim of this study was to design physics-preserving and precise surrogate models of the nonlinear elastic behaviour of an intervertebral disc (IVD). Based on artificial force-displacement data sets from detailed finite element (FE) disc models, we used greedy kernel and polynomial approximations of second, third and fourth order to train surrogate models for the scalar force-torque-potential. Doing so, the resulting models of the elastic IVD responses ensured the conservation of mechanical energy through their structure. At the same time, they were capable of predicting disc forces in a physiological range of motion and for the coupling of all six degrees of freedom of an intervertebral joint. The performance of all surrogate models for a subject-specific L4|5 disc geometry was evaluated both on training and test data obtained from uncoupled (one-dimensional), weakly coupled (two-dimensional), and random movement trajectories in the entire six-dimensional (6d) physiological displacement range, as well as on synthetic kinematic data. We observed highest precisions for the kernel surrogate followed by the fourth-order polynomial model. Both clearly outperformed the second-order polynomial model which is equivalent to the commonly used stiffness matrix in neuro-musculoskeletal simulations. Hence, the proposed model architectures have the potential to improve the accuracy and, therewith, validity of load predictions in neuro-musculoskeletal spine models.
Open Access
A novel model extended from the Bouguer-Lambert-Beer law can describe the non-linear absorbance of potassium dichromate solutions and microalgae suspensions
(2023) Yeh, Yen-Cheng; Haasdonk, Bernard; Schmid-Staiger, Ulrike; Stier, Matthias; Tovar, Günter E. M.
The Bouguer-Lambert-Beer law is widely used as the fundamental equation for quantification in absorption spectroscopy. However, deviations from the Bouguer-Lambert-Beer law have also been observed, such as chemical deviation and light scattering effect. While it has been proven and shown that the Bouguer-Lambert-Beer law is valid only under very restricted limitations, there are only a few alternatives of analytical models to this law. Based on the observation in the experiments, we propose a novel model to solve the problem of chemical deviation and light scattering effect. To test the proposed model, a systematic verification was conducted using potassium dichromate solutions and two types of microalgae suspensions with varying concentrations and path lengths. Our proposed model demonstrated excellent performance, with a correlation coefficient (R2) exceeding 0.995 for all tested materials, significantly surpassing the Bouguer-Lambert-Beer law, which had an R2 as low as 0.94. Our results confirm that the absorbance of pure pigment solutions follows the Bouguer-Lambert-Beer law, while the microalgae suspensions do not due to the light scattering effect. We also show that this scattering effect leads to huge deviations for the commonly used linear scaling of the spectra, and we provide a better solution based on the proposed model. This work provides a powerful tool for chemical analysis and especially for the quantification of microorganisms, such as the concentration of biomass or intracellular biomolecules. Not only the high accuracy but also the simplicity of the model makes it a practical alternative to the existing Bouguer-Lambert-Beer law.
Open Access
Port-Hamiltonian fluid-structure interaction modelling and structure-preserving model order reduction of a classical guitar
(2023) Rettberg, Johannes; Wittwar, Dominik; Buchfink, Patrick; Brauchler, Alexander; Ziegler, Pascal; Fehr, Jörg; Haasdonk, Bernard
Open Access
Rigorous and effective a-posteriori error bounds for nonlinear problems : application to RB methods
(2020) Schmidt, Andreas; Wittwar, Dominik; Haasdonk, Bernard
Quantifying the error that is induced by numerical approximation techniques is an important task in many fields of applied mathematics. Two characteristic properties of error bounds that are desirable are reliability and efficiency. In this article, we present an error estimation procedure for general nonlinear problems and, in particular, for parameter-dependent problems. With the presented auxiliary linear problem (ALP)-based error bounds and corresponding theoretical results, we can prove large improvements in the accuracy of the error predictions compared with existing error bounds. The application of the procedure in parametric model order reduction setting provides a particularly interesting setup, which is why we focus on the application in the reduced basis framework. Several numerical examples illustrate the performance and accuracy of the proposed method.
Open Access
Symplectic model order reduction with non-orthonormal bases
(2019) Buchfink, Patrick; Bhatt, Ashish; Haasdonk, Bernard
Parametric high-fidelity simulations are of interest for a wide range of applications. But the restriction of computational resources renders such models to be inapplicable in a real-time context or in multi-query scenarios. Model order reduction (MOR) is used to tackle this issue. Recently, MOR is extended to preserve specific structures of the model throughout the reduction, e.g. structure-preserving MOR for Hamiltonian systems. This is referred to as symplectic MOR. It is based on the classical projection-based MOR and uses a symplectic reduced order basis (ROB). Such a ROB can be derived in a data-driven manner with the Proper Symplectic Decomposition (PSD) in the form of a minimization problem. Due to the strong nonlinearity of the minimization problem, it is unclear how to efficiently find a global optimum. In our paper, we show that current solution procedures almost exclusively yield suboptimal solutions by restricting to orthonormal ROBs. As new methodological contribution, we propose a new method which eliminates this restriction by generating non-orthonormal ROBs. In the numerical experiments, we examine the different techniques for a classical linear elasticity problem and observe that the non-orthonormal technique proposed in this paper shows superior results with respect to the error introduced by the reduction.
Open Access
Well-scaled, a-posteriori error estimation for model order reduction of large second-order mechanical systems
(2019) Grunert, Dennis; Fehr, Jörg; Haasdonk, Bernard
Model Order Reduction is used to vastly speed up simulations but it also introduces an error to the simulation results, which needs to be controlled. The performance of the general to use, a-posteriori error estimator of Ruiner et al. for second-order systems is analyzed and a bottleneck is found in the offline stage making it unusable for larger models. We use the spectral theorem, power series expansions, monotonicity properties, and self-tailored algorithms to speed up the offline stage largely by one polynomial order both in terms of computation time as well as storage complexity. All properties are proven rigorously. This eliminates the aforementioned bottleneck. Hence, the error estimator of Ruiner et al. can finally be used for large, linear, second-order mechanical systems reduced by any model reduction method based on Petrov-Galerkin reduction. The examples show speedups of up to 28.000 and the ability to compute much larger systems with a fixed amount of memory.