arXiv subjects

Martin Stoll

Publications and source records attributed to Martin Stoll.

At least 37 records · Page 2Linked to original sources

arXiv2025

Preconditioned Additive Gaussian Processes with Fourier Acceleration

Gaussian processes (GPs) are crucial in machine learning for quantifying uncertainty in predictions. However, their associated covariance matrices, defined by kernel functions, are typically dense and large-scale, posing significant computational challenges. This paper introduces a matrix-free method that utilizes the Non-equispaced Fast Fourier Transform (NFFT) to achieve nearly linear complexity in the multiplication of kernel matrices and their derivatives with vectors for a predetermined accuracy level. To address high-dimensional problems, we propose an additive kernel approach. Each sub-kernel in this approach captures lower-order feature interactions, allowing for the efficient application of the NFFT method and potentially increasing accuracy across various real-world datasets. Additionally, we implement a preconditioning strategy that accelerates hyperparameter tuning, further improving the efficiency and effectiveness of GPs.

cs.LG↗

arXiv2025

Preconditioning for a Cahn-Hilliard-Navier-Stokes model for morphology formation in organic solar cells

We present a model for the morphology evolution of printed organic solar cells which occurs during the drying of a mixture of polymer, the non-fullerene acceptor and the solvent. Our model uses a phase field approach coupled to a Navier-Stokes equation describing the macroscopic movement of the fluid. Additionally, we incorporate the evaporation process of the solvent using an Allen-Cahn equation. The model is discretized using a finite-element approach with a semi-implicit discretization in time. The resulting (non)linear systems are coupled and of large dimensionality. We present a preconditioned iterative scheme to solve them robustly with respect to changes in the discretization parameters. We illustrate that the preconditioned solver shows parameter-robust iteration numbers and that the model qualitatively captures the behavior of the film morphology during drying.

math.NA↗

arXiv2024

Gradient flow-based modularity maximization for community detection in multiplex networks

We propose two methods for the unsupervised detection of communities in undirected multiplex networks. These networks consist of multiple layers that record different relationships between the same entities or incorporate data from different sources. Both methods are formulated as gradient flows of suitable energy functionals: the first (MPBTV) builds on the minimization of a balanced total variation functional, which we show to be equivalent to multiplex modularity maximization, while the second (DGFM3) directly maximizes multiplex modularity. The resulting non-linear matrix-valued ordinary differential equations (ODEs) are solved efficiently by a graph Merriman--Bence--Osher (MBO) scheme. Key to the efficiency is the approximate integration of the discrete linear differential operators by truncated eigendecompositions in the matrix exponential function. Numerical experiments on several real-world multiplex networks show that our methods are competitive with the state of the art with respect to various metrics. Their major benefit is a significant reduction of computational complexity leading to runtimes that are orders of magnitude faster for large multiplex networks.

math.NA↗

arXiv2024

The Integration of Prediction and Planning in Deep Learning Automated Driving Systems: A Review

Automated driving has the potential to revolutionize personal, public, and freight mobility. Beside accurately perceiving the environment, automated vehicles must plan a safe, comfortable, and efficient motion trajectory. To promote safety and progress, many works rely on modules that predict the future motion of surrounding traffic. Modular automated driving systems commonly handle prediction and planning as sequential, separate tasks. While this accounts for the influence of surrounding traffic on the ego vehicle, it fails to anticipate the reactions of traffic participants to the ego vehicle's behavior. Recent methods increasingly integrate prediction and planning in a joint or interdependent step to model bidirectional interactions. To date, a comprehensive overview of different integration principles is lacking. We systematically review state-of-the-art deep learning-based planning systems, and focus on how they integrate prediction. Different facets of the integration ranging from system architecture to high-level behavioral aspects are considered and related to each other. Moreover, we discuss the implications, strengths, and limitations of different integration principles. By pointing out research gaps, describing relevant future challenges, and highlighting trends in the research field, we identify promising directions for future research.

cs.RO↗

arXiv2024

Can Vehicle Motion Planning Generalize to Realistic Long-tail Scenarios?

Real-world autonomous driving systems must make safe decisions in the face of rare and diverse traffic scenarios. Current state-of-the-art planners are mostly evaluated on real-world datasets like nuScenes (open-loop) or nuPlan (closed-loop). In particular, nuPlan seems to be an expressive evaluation method since it is based on real-world data and closed-loop, yet it mostly covers basic driving scenarios. This makes it difficult to judge a planner's capabilities to generalize to rarely-seen situations. Therefore, we propose a novel closed-loop benchmark interPlan containing several edge cases and challenging driving scenarios. We assess existing state-of-the-art planners on our benchmark and show that neither rule-based nor learning-based planners can safely navigate the interPlan scenarios. A recently evolving direction is the usage of foundation models like large language models (LLM) to handle generalization. We evaluate an LLM-only planner and introduce a novel hybrid planner that combines an LLM-based behavior planner with a rule-based motion planner that achieves state-of-the-art performance on our benchmark.

cs.RO↗

arXiv2024

A nonlinear spectral core-periphery detection method for multiplex networks

Core-periphery detection aims to separate the nodes of a complex network into two subsets: a core that is densely connected to the entire network and a periphery that is densely connected to the core but sparsely connected internally. The definition of core-periphery structure in multiplex networks that record different types of interactions between the same set of nodes on different layers is nontrivial since a node may belong to the core in some layers and to the periphery in others. We propose a nonlinear spectral method for multiplex networks that simultaneously optimises a node and a layer coreness vector by maximising a suitable nonconvex homogeneous objective function by a provably convergent alternating fixed point iteration. We derive a quantitative measure for the quality of a given multiplex core-periphery structure that allows the determination of the optimal core size. Numerical experiments on synthetic and real-world networks illustrate that our approach is robust against noisy layers and significantly outperforms baseline methods while improving the latter with our novel optimised layer coreness weights. As the runtime of our method depends linearly on the number of edges of the network it is scalable to large-scale multiplex networks.

math.NA↗

arXiv2024

IETI-based Low-Rank method for PDE-constrained optimization

Isogeometric Analysis (IgA) is a versatile method for the discretization of partial differential equations on complex domains, which arise in various applications of science and engineering. Some complex geometries can be better described as a computational domain by a multi-patch approach, where each patch is determined by a tensor product Non-Uniform Rational Basis Splines (NURBS) parameterization. This allows on the one hand to consider the problem of the complex assembly of mass or stiffness matrices (or tensors) over the whole geometry locally on the individual smaller patches, and on the other hand it is possible to perform local mesh refinements independently on each patch, allowing efficient local refinement in regions of high activity where higher accuracy is required, while coarser meshes can be used elsewhere. Furthermore, the information about differing material models or properties that are to apply in a subdomain of the geometry can be included in the patch in which this subdomain is located. For this it must be ensured that the approximate solution is continuous over the entire computational domain and therefore at the interfaces of two (or more) patches. The most promising approach for this problem, which transfers the idea of Finite Element Tearing and Interconnecting (FETI) methods into the isogeometric setup, was the IsogEometric Tearing and Interconnecting (IETI) method, where by introducing a constraints matrix and associated Lagrange multipliers and formulating it into a dual problem, depending only on the Lagrange multipliers, continuity at the interfaces was ensured in solving the resulting system. In this paper we illustrate that low-rank methods based on the tensor-train format can be generalised for a multi-patch IgA setup, which follows the IETI idea.

math.NA↗

arXiv2024

Fast Evaluation of Additive Kernels: Feature Arrangement, Fourier Methods, and Kernel Derivatives

One of the main computational bottlenecks when working with kernel based learning is dealing with the large and typically dense kernel matrix. Techniques dealing with fast approximations of the matrix vector product for these kernel matrices typically deteriorate in their performance if the feature vectors reside in higher-dimensional feature spaces. We here present a technique based on the non-equispaced fast Fourier transform (NFFT) with rigorous error analysis. We show that this approach is also well suited to allow the approximation of the matrix that arises when the kernel is differentiated with respect to the kernel hyperparameters; a problem often found in the training phase of methods such as Gaussian processes. We also provide an error analysis for this case. We illustrate the performance of the additive kernel scheme with fast matrix vector products on a number of data sets. Our code is available at https://github.com/wagnertheresa/NFFTAddKer

cs.LG↗

arXiv2024

Stay on Track: A Frenet Wrapper to Overcome Off-road Trajectories in Vehicle Motion Prediction

Predicting the future motion of observed vehicles is a crucial enabler for safe autonomous driving. The field of motion prediction has seen large progress recently with state-of-the-Art (sotA) models achieving impressive results on large-scale public benchmarks. However, recent work revealed that learning-based methods are prone to predict off-road trajectories in challenging scenarios. These can be created by perturbing existing scenarios with additional turns in front of the target vehicle while the motion history is left unchanged. We argue that this indicates that SotA models do not consider the map information sufficiently and demonstrate how this can be solved by representing model inputs and outputs in a Frenet frame defined by lane centreline sequences. To this end, we present a general wrapper that leverages a Frenet representation of the scene, and that can be applied to SotA models without changing their architecture. We demonstrate the effectiveness of this approach in a comprehensive benchmark using two SotA motion prediction models. Our experiments show that this reduces the off-road rate on challenging scenarios by more than 90% without sacrificing average performance.

cs.RO↗

arXiv2024

A splitting-based KPIK method for eddy current optimal control problems in an all-at-once approach

In this paper, we focus on efficient methods to solve discretized linear systems obtained from eddy current optimal control problems in an all-at-once approach. We construct a new low-rank matrix equation method based on a special splitting of the coefficient matrix and the Krylov-plus-inverted-Krylov (KPIK) algorithm. Firstly, we rewrite the resulting discretized linear system in a matrix-equation form. Then using the KPIK algorithm, we can obtain the low-rank approximation solution. The new method is named the splitting-based Krylov-plus-inverted-Krylov (SKPIK) method. The SKPIK method can not only solve the large and sparse discretized systems fast but also overcomes the storage problem. Theoretical results about the existence of the low-rank solutions are given. Numerical experiments are used to illustrate the performance of the new low-rank matrix equation method by compared with some existing classical efficient methods.

math.NA↗

arXiv2024

Scaling Planning for Automated Driving using Simplistic Synthetic Data

We challenge the perceived consensus that the application of deep learning to solve the automated driving planning task necessarily requires huge amounts of real-world data or highly realistic simulation. Focusing on a roundabout scenario, we show that this requirement can be relaxed in favour of targeted, simplistic simulated data. A benefit is that such data can be easily generated for critical scenarios that are typically underrepresented in realistic datasets. By applying vanilla behavioural cloning almost exclusively to lightweight simulated data, we achieve reliable and comfortable driving in a real-world test vehicle. We leverage an incremental development approach that includes regular in-vehicle testing to identify sim-to-real gaps, targeted data augmentation, and training scenario variations. In addition to a detailed description of the methodology, we share our lessons learned, touching upon scenario generation, simulation features, and evaluation metrics.

cs.RO↗

arXiv2023

Efficient training of Gaussian processes with tensor product structure

To determine the optimal set of hyperparameters of a Gaussian process based on a large number of training data, both a linear system and a trace estimation problem must be solved. In this paper, we focus on establishing numerical methods for the case where the covariance matrix is given as the sum of possibly multiple Kronecker products, i.e., can be identified as a tensor. As such, we will represent this operator and the training data in the tensor train format. Based on the AMEn method and Krylov subspace methods, we derive an efficient scheme for computing the matrix functions required for evaluating the gradient and the objective function in hyperparameter optimization.

math.OC↗

arXiv2023

A Preconditioned Interior Point Method for Support Vector Machines Using an ANOVA-Decomposition and NFFT-Based Matrix-Vector Products

In this paper we consider the numerical solution to the soft-margin support vector machine optimization problem. This problem is typically solved using the SMO algorithm, given the high computational complexity of traditional optimization algorithms when dealing with large-scale kernel matrices. In this work, we propose employing an NFFT-accelerated matrix-vector product using an ANOVA decomposition for the feature space that is used within an interior point method for the overall optimization problem. As this method requires the solution of a linear system of saddle point form we suggest a preconditioning approach that is based on low-rank approximations of the kernel matrix together with a Krylov subspace solver. We compare the accuracy of the ANOVA-based kernel with the default LIBSVM implementation. We investigate the performance of the different preconditioners as well as the accuracy of the ANOVA kernel on several large-scale datasets.

math.NA↗

arXiv2023

Adaptive rational Krylov methods for exponential Runge--Kutta integrators

We consider the solution of large stiff systems of ordinary differential equations with explicit exponential Runge--Kutta integrators. These problems arise from semi-discretized semi-linear parabolic partial differential equations on continuous domains or on inherently discrete graph domains. A series of results reduces the requirement of computing linear combinations of $φ$-functions in exponential integrators to the approximation of the action of a smaller number of matrix exponentials on certain vectors. State-of-the-art computational methods use polynomial Krylov subspaces of adaptive size for this task. They have the drawback that the required number of Krylov subspace iterations to obtain a desired tolerance increase drastically with the spectral radius of the discrete linear differential operator, e.g., the problem size. We present an approach that leverages rational Krylov subspace methods promising superior approximation qualities. We prove a novel a-posteriori error estimate of rational Krylov approximations to the action of the matrix exponential on vectors for single time points, which allows for an adaptive approach similar to existing polynomial Krylov techniques. We discuss pole selection and the efficient solution of the arising sequences of shifted linear systems by direct and preconditioned iterative solvers. Numerical experiments show that our method outperforms the state of the art for sufficiently large spectral radii of the discrete linear differential operators. The key to this are approximately constant numbers of rational Krylov iterations, which enable a near-linear scaling of the runtime with respect to the problem size.

math.NA↗

arXiv2023

From Prediction to Planning With Goal Conditioned Lane Graph Traversals

The field of motion prediction for automated driving has seen tremendous progress recently, bearing ever-more mighty neural network architectures. Leveraging these powerful models bears great potential for the closely related planning task. In this letter we propose a novel goal-conditioning method and show its potential to transform a state-of-the-art prediction model into a goal-directed planner. Our key insight is that conditioning prediction on a navigation goal at the behaviour level outperforms other widely adopted methods, with the additional benefit of increased model interpretability. We train our model on a large open-source dataset and show promising performance in a comprehensive benchmark.

cs.RO↗

arXiv2023

Uncertainty Propagation of Initial Conditions in Thermal Models

The operation of machine tools often demands a highly accurate knowledge of the tool center point's (TCP) position. The displacement of the TCP over time can be inferred from thermal models, which comprise a set of geometrically coupled heat equations. Each of these equations represents the temperature in part of the machine, and they are often formulated on complicated geometries. The accuracy of the TCP prediction depends highly on the accuracy of the model parameters, such as heat exchange parameters, and the initial temperature. Thus it is of utmost interest to determine the influence of these parameters on the TCP displacement prediction. In turn, the accuracy of the parameter estimate is essentially determined by the measurement accuracy and the sensor placement. Determining the accuracy of a given sensor configuration is a key prerequisite of optimal sensor placement. We develop here a thermal model for a particular machine tool. On top of this model we propose two numerical algorithms to evaluate any given thermal sensor configuration with respect to its accuracy. We compute the posterior variances from the posterior covariance matrix with respect to an uncertain initial temperature field. The full matrix is dense and potentially very large, depending on the model size. Thus, we apply a low-rank method to approximate relevant entries, i.e. the variances on its diagonal. We first present a straightforward way to compute this approximation which requires computation of the model sensitivities with with respect to the initial values. Additionally, we present a low-rank tensor method which exploits the underlying system structure. We compare the efficiency of both algorithms with respect to runtime and memory requirements and discuss their respective advantages with regard to optimal sensor placement problems.

math.NA↗

arXiv2023

A weighted subspace exponential kernel for support tensor machines

High-dimensional data in the form of tensors are challenging for kernel classification methods. To both reduce the computational complexity and extract informative features, kernels based on low-rank tensor decompositions have been proposed. However, what decisive features of the tensors are exploited by these kernels is often unclear. In this paper we propose a novel kernel that is based on the Tucker decomposition. For this kernel the Tucker factors are computed based on re-weighting of the Tucker matrices with tuneable powers of singular values from the HOSVD decomposition. This provides a mechanism to balance the contribution of the Tucker core and factors of the data. We benchmark support tensor machines with this new kernel on several datasets. First we generate synthetic data where two classes differ in either Tucker factors or core, and compare our novel and previously existing kernels. We show robustness of the new kernel with respect to both classification scenarios. We further test the new method on real-world datasets. The proposed kernel has demonstrated a higher test accuracy than the state-of-the-art tensor train multi-way multi-level kernel, and a significantly lower computational time.

stat.ML↗

arXiv2022

Gibbs-Helmholtz Graph Neural Network: capturing the temperature dependency of activity coefficients at infinite dilution

The accurate prediction of physicochemical properties of chemical compounds in mixtures (such as the activity coefficient at infinite dilution $γ_{ij}^\infty$) is essential for developing novel and more sustainable chemical processes. In this work, we analyze the performance of previously-proposed GNN-based models for the prediction of $γ_{ij}^\infty$, and compare them with several mechanistic models in a series of 9 isothermal studies. Moreover, we develop the Gibbs-Helmholtz Graph Neural Network (GH-GNN) model for predicting $\ln γ_{ij}^\infty$ of molecular systems at different temperatures. Our method combines the simplicity of a Gibbs-Helmholtz-derived expression with a series of graph neural networks that incorporate explicit molecular and intermolecular descriptors for capturing dispersion and hydrogen bonding effects. We have trained this model using experimentally determined $\ln γ_{ij}^\infty$ data of 40,219 binary-systems involving 1032 solutes and 866 solvents, overall showing superior performance compared to the popular UNIFAC-Dortmund model. We analyze the performance of GH-GNN for continuous and discrete inter/extrapolation and give indications for the model's applicability domain and expected accuracy. In general, GH-GNN is able to produce accurate predictions for extrapolated binary-systems if at least 25 systems with the same combination of solute-solvent chemical classes are contained in the training set and a similarity indicator above 0.35 is also present. This model and its applicability domain recommendations have been made open-source at https://github.com/edgarsmdn/GH-GNN.

physics.chem-ph↗