arXiv subjects

Armin Iske

Publications and source records attributed to Armin Iske.

15 recordsLinked to original sources

arXiv2026

Why summation by parts is not enough

We investigate the construction and performance of summation-by-parts (SBP) operators, which offer a powerful framework for the systematic development of structure-preserving numerical discretizations of partial differential equations. Previous approaches for the construction of SBP operators have usually relied on either local methods or sparse differentiation matrices, as commonly used in finite difference schemes. However, these methods often impose implicit requirements that are not part of the formal SBP definition. We demonstrate that adherence to the SBP definition alone does not guarantee the desired accuracy, and we identify conditions for SBP operators to achieve both accuracy and stability. Specifically, we analyze the error minimization for an augmented basis, discuss the role of sparsity, and examine the importance of nullspace consistency in the construction of SBP operators. Furthermore, we show how these design criteria can be integrated into a recently proposed optimization-based construction procedure for function space SBP (FSBP) operators on arbitrary grids. Our findings are supported by numerical experiments that illustrate the improved accuracy for the numerical solution using the proposed SBP operators.

math.NA↗

arXiv2025

Spectral equivalence of unsymmetric kernel matrices and applications

Symmetric kernel matrices are a well-researched topic in the literature of kernel based approximation. In particular stability properties in terms of lower bounds on the smallest eigenvalue of such symmetric kernel matrices are thoroughly investigated, as they play a fundamental role in theory and practice. In this work, we focus on unsymmetric kernel matrices and derive stability properties under small shifts by establishing a spectral equivalence to their unshifted, symmetric versions. This extends and generalizes results for translational invariant kernels upon Quak et al [SIAM Journal on Math. Analysis, 1993] and Sivakumar and Ward [Numerische Mathematik, 1993], however focussing instead on finitely smooth kernels. As applications, we consider convolutional kernels over domains, which are no longer translational invariant, but which are still an important class of kernels for applications. For these, we derive novel lower bounds for the smallest eigenvalue of the kernel matrices in terms of the separation distance of the data points, and thus derive stability bounds in terms of the condition number.

math.NA↗

arXiv2025

Irregular Sampling of High-Dimensional Functions in Reproducing Kernel Hilbert Spaces

We develop sampling formulas for high-dimensional functions in reproducing kernel Hilbert spaces, where we rely on irregular samples that are taken at determining sequences of data points. We place particular emphasis on sampling formulas for tensor product kernels, where we show that determining irregular samples in lower dimensions can be composed to obtain a tensor of determining irregular samples in higher dimensions. This in turn reduces the computational complexity of sampling formulas for high-dimensional functions quite significantly.

cs.LG↗

arXiv2025

On the Convergence of Irregular Sampling in Reproducing Kernel Hilbert Spaces

We analyse the convergence of sampling algorithms for functions in reproducing kernel Hilbert spaces (RKHS). To this end, we discuss approximation properties of kernel regression under minimalistic assumptions on both the kernel and the input data. We first prove error estimates in the kernel's RKHS norm. This leads us to new results concerning uniform convergence of kernel regression on compact domains. For Lipschitz continuous and H\"older continuous kernels, we prove convergence rates.

stat.ML↗

arXiv2024

Learning phase-space flows using time-discrete implicit Runge-Kutta PINNs

We present a computational framework for obtaining multidimensional phase-space solutions of systems of non-linear coupled differential equations, using high-order implicit Runge-Kutta Physics- Informed Neural Networks (IRK-PINNs) schemes. Building upon foundational work originally solving differential equations for fields depending on coordinates [J. Comput. Phys. 378, 686 (2019)], we adapt the scheme to a context where the coordinates are treated as functions. This modification enables us to efficiently solve equations of motion for a particle in an external field. Our scheme is particularly useful for explicitly time-independent and periodic fields. We apply this approach to successfully solve the equations of motion for a mass particle placed in a central force field and a charged particle in a periodic electric field.

cs.LG↗

arXiv2024

Bounds on the Generalization Error in Active Learning

We establish empirical risk minimization principles for active learning by deriving a family of upper bounds on the generalization error. Aligning with empirical observations, the bounds suggest that superior query algorithms can be obtained by combining both informativeness and representativeness query strategies, where the latter is assessed using integral probability metrics. To facilitate the use of these bounds in application, we systematically link diverse active learning scenarios, characterized by their loss functions and hypothesis classes to their corresponding upper bounds. Our results show that regularization techniques used to constraint the complexity of various hypothesis classes are sufficient conditions to ensure the validity of the bounds. The present work enables principled construction and empirical quality-evaluation of query algorithms in active learning.

stat.ML↗

arXiv2024

Spectral alignment of kernel matrices and applications

Kernel matrices are a key quantity in kernel-based approximation, and important properties such as stability and algorithmic convergence can be analyzed with their help. In this work we refine a multivariate Ingham-type theorem, which is then leveraged to obtain novel and refined stability estimates on kernel matrices. For this, we focus on the case of finitely smooth kernels, such as the family of Mat\'ern or Wendland kernels, while noting that the results also extend to norm-equivalent kernels. In particular we obtain results that relate the Rayleigh quotients of kernel matrices for kernels of different smoothness to each other. Finally we comment on conclusions for the eigenvectors of these kernel matrices.

math.NA↗

arXiv2024

On the convergence of generalized kernel-based interpolation by greedy data selection algorithms

We analyze the convergence of generalized kernel-based interpolation methods. This is done under minimalistic assumptions on both the kernel and the target function. On these grounds, we further prove convergence of popular greedy data selection algorithms for totally bounded sets of sampling functionals. Supporting numerical results concerning computerized tomography are provided for illustration.

math.NA↗

arXiv2024

Inducing Riesz and orthonormal bases in $L^2$ via composition operators

Let $C_h$ be a composition operator mapping $L^2(\Omega_1)$ into $L^2(\Omega_2)$ for some open sets $\Omega_1, \Omega_2 \subseteq \mathbb{R}^n$. We characterize the mappings $h$ that transform Riesz bases of $L^2(\Omega_1)$ into Riesz bases of $L^2(\Omega_2)$. Restricting our analysis to differentiable mappings, we demonstrate that mappings $h$ that preserve Riesz bases have Jacobian determinants that are bounded away from zero and infinity. We discuss implications of these results for approximation theory, highlighting the potential of using bijective neural networks to construct Riesz bases with favorable approximation properties.

math.FA↗

arXiv2024

Finetuning greedy kernel models by exchange algorithms

Kernel based approximation offers versatile tools for high-dimensional approximation, which can especially be leveraged for surrogate modeling. For this purpose, both "knot insertion" and "knot removal" approaches aim at choosing a suitable subset of the data, in order to obtain a sparse but nevertheless accurate kernel model. In the present work, focussing on kernel based interpolation, we aim at combining these two approaches to further improve the accuracy of kernel models, without increasing the computational complexity of the final kernel model. For this, we introduce a class of kernel exchange algorithms (KEA). The resulting KEA algorithm can be used for finetuning greedy kernel surrogate models, allowing for an reduction of the error up to 86.4% (17.2% on average) in our experiments.

cs.LG↗

arXiv2023

Product kernels are efficient and flexible tools for high-dimensional scattered data interpolation

This work concerns the construction and characterization of product kernels for multivariate approximation from a finite set of discrete samples. To this end, we consider composing different component kernels, each acting on a low-dimensional Euclidean space. Due to Aronszajn (1950), the product of positive semi-definite kernel functions is again positive semi-definite, where, moreover, the corresponding native space is a particular instance of a tensor product, referred to as Hilbert tensor product. We first analyze the general problem of multivariate interpolation by product kernels. Then, we further investigate the tensor product structure, in particular for grid-like samples. We use this case to show that the product of positive definite kernel functions is again positive definite. Moreover, we develop an efficient computation scheme for the well-known Newton basis. Supporting numerical examples show the good performance of product kernels, especially for their flexibility.

math.NA↗

arXiv2023

Computing excited states of molecules using normalizing flows

Calculations of highly excited and delocalized molecular vibrational states are computationally challenging tasks, which strongly depends on the choice of coordinates for describing vibrational motions. We introduce a new method that leverages normalizing flows -- parametrized invertible functions -- to learn optimal vibrational coordinates that satisfy the variational principle. This approach produces coordinates tailored to the vibrational problem at hand, significantly increasing the accuracy and enhancing basis-set convergence of the calculated energy spectrum. The efficiency of the method is demonstrated in calculations of the 100 lowest excited vibrational states of H$_2$S, H$_2$CO, and HCN/HNC. The method effectively captures the essential vibrational behavior of molecules by enhancing the separability of the Hamiltonian and hence allows for an effective assignment of approximate quantum numbers. We demonstrate that the optimized coordinates are transferable across different levels of basis-set truncation, enabling a cost-efficient protocol for computing vibrational spectra of high-dimensional systems.

physics.chem-ph↗

arXiv2022

Augmenting Basis Sets by Normalizing Flows

Approximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential and integral equations. Commonly used concepts of approximation methods are well-posed and convergent, by provable approximation orders. On the down side, however, these methods often suffer from the curse of dimensionality, which limits their approximation behavior, especially in situations of highly oscillatory target functions. Nonlinear approximation methods, such as neural networks, were shown to be very efficient in approximating high-dimensional functions. We investigate nonlinear approximation methods that are constructed by composing standard basis sets with normalizing flows. Such models yield richer approximation spaces while maintaining the density properties of the initial basis set, as we show. Simulations to approximate eigenfunctions of a perturbed quantum harmonic oscillator indicate convergence with respect to the size of the basis set.

math.NA↗

arXiv2022

Dimensionality Reduction and Wasserstein Stability for Kernel Regression

In a high-dimensional regression framework, we study consequences of the naive two-step procedure where first the dimension of the input variables is reduced and second, the reduced input variables are used to predict the output variable with kernel regression. In order to analyze the resulting regression errors, a novel stability result for kernel regression with respect to the Wasserstein distance is derived. This allows us to bound errors that occur when perturbed input data is used to fit the regression function. We apply the general stability result to principal component analysis (PCA). Exploiting known estimates from the literature on both principal component analysis and kernel regression, we deduce convergence rates for the two-step procedure. The latter turns out to be particularly useful in a semi-supervised setting.

stat.ML↗

arXiv2021

Active learning of potential-energy surfaces of weakly-bound complexes with regression-tree ensembles

Several pool-based active learning algorithms (AL) were employed to model potential energy surfaces (PESs) with a minimum number of electronic structure calculations. Theoretical and empirical results suggest that superior strategies can be obtained by sampling molecular structures corresponding to large uncertainties in their predictions while at the same time not deviating much from the true distribution of the data. To model PESs in an AL framework we propose to use a regression version of stochastic query by forest, a hybrid method that samples points corresponding to large uncertainties while avoiding collecting too many points from sparse regions of space. The algorithm is implemented with decision trees that come with relatively small computational costs. We empirically show that this algorithm requires around half the data to converge to the same accuracy in comparison to the uncertainty-based query-by-committee algorithm. Moreover, the algorithm is fully automatic and does not require any prior knowledge of the PES. Simulations on a 6D PES of \pyrrolew show that $\mathord{<}15\,000$ configurations are enough to build a PES with a generalization error of 16~\invcm, whereas the final model with around 50\,000 configurations has a generalization error of 11~\invcm.

physics.chem-ph↗