Library

Description: Finding collective variables to describe some important coarse-grained information on physical systems, in particular metastable states, remains a key issue in molecular dynamics. Recently, machine learning techniques have been intensively used to complement and possibly bypass expert knowledge in order to construct collective variables. Our focus here is on neural network approaches based on autoencoders. We study some relevant mathematical properties of the loss function considered for training autoencoders, and provide physical interpretations based on conditional variances and minimum energy paths. We also consider various extensions in order to better describe physical systems, by incorporating more information on transition states at saddle points, and/or allowing for multiple decoders in order to describe several transition paths. Our results are illustrated on toy two dimensional systems and on alanine dipeptide.

Description: Mixed-Integer Linear Programming (MILP) is a ubiquitous and practical modelling paradigm that is essential for optimising a broad range of real-world systems. The backbone of all modern MILP solvers is the branch-and-cut algorithm, which is a hybrid of the branch-and-bound and cutting planes algorithms. Cutting planes (cuts) are linear inequalities that tighten the relaxation of a MILP. While a lot of research has gone into deriving valid cuts for MILPs, less emphasis has been put on determining which cuts to select. Cuts in general are generated in rounds, and a subset of the generated cuts must be added to the relaxation. The decision on which subset of cuts to add is called cut selection. This is a crucial task since adding too many cuts makes the relaxation large and slow to optimise over. Conversely, adding too few cuts results in an insufficiently tightened relaxation, and more relaxations need to be enumerated. To further emphasise the difficulty, the effectiveness of an applied cut is both dependent on the other applied cuts, and the state of the MILP solver. In this thesis, we present theoretical results on the importance and difficulty of cut selection, as well as practical results that use cut selection to improve general MILP solver performance. Improving general MILP solver performance is of great importance for practitioners and has many runoff effects. Reducing the solve time of currently solved systems can directly improve efficiency within the application area. In addition, improved performance enables larger systems to be modelled and optimised, and MILP to be used in areas where it was previously impractical due to time restrictions. Each chapter of this thesis corresponds to a publication on cut selection, where the contributions of this thesis can naturally be divided into four components. The first two components are motivated by instance-dependent performance. In practice, for each subroutine, including cut selection, MILP solvers have adjustable parameters with hard-coded default values. It is ultimately unrealistic to expect these default values to perform well for every instance. Rather, it would be ideal if the parameters were dependent on the given instance. To show this motivation is well founded, we first introduce a family of parametric MILP instances and cuts to showcase worst-case performance of cut selection for any fixed parameter value. We then introduce a graph neural network architecture and reinforcement learning framework for learning instance-dependent cut scoring parameters. In the following component, we formalise language for determining if a cut has theoretical usefulness from a polyhedral point of view in relation to other cuts. In addition, to overcome issues of infeasible projections and dual degeneracy, we introduce analytic center based distance measures. We then construct a lightweight multi-output regression model that predicts relative solver performance of an instance for a set of distance measures. The final two components are motivated by general MILP solver improvement via cut selection. Such improvement was shown to be possible, albeit difficult to achieve, by the first half of this thesis. We relate branch-and-bound and cuts through their underlying disjunctions. Using a history of previously computed Gomory mixed-integer cuts, we reduce the solve time of SCIP over the 67% of affected MIPLIB 2017 instances by 4%. In the final component, we introduce new cut scoring measures and filtering methods based on information from other MILP solving processes. The new cut selection techniques reduce the solve time of SCIP over the 97% of affected MIPLIB 2017 instances by 5%.

Description: Cardiac electrograms are an important tool to study the spread of excitation waves inside the heart, which in turn underlie muscle contraction. Electrograms can be used to analyse the dynamics of these waves, e.g. in fibrotic tissue. In computational models, these analyses can be done with greater detail than during minimally invasive in vivo procedures. Whilst homogenised models have been used to study electrogram genesis, such analyses have not yet been done in cellularly resolved models. Such high resolution may be required to develop a thorough understanding of the mechanisms behind abnormal excitation patterns leading to arrhythmias. In this study, we derived electrograms from an excitation propagation simulation in the Extracellular, Membrane, Intracellular (EMI) model, which represents these three domains explicitly in the mesh. We studied the effects of the microstructural excitation dynamics on electrogram genesis and morphology. We found that electrograms are sensitive to the myocyte alignment and connectivity, which translates into micro-fractionations in the electrograms.

Description: The Robust Perron Cluster Analysis (PCCA+) has become a popular spectral clustering algorithm for coarse-graining transition matrices of nearly decomposable Markov chains with transition states. Originally developed for reversible Markov chains, the algorithm only worked for transition matrices with real eigenvalues. In this paper, we therefore extend the theoretical framework of PCCA+ to Markov chains with a complex eigen-decomposition. We show that by replacing a complex conjugate pair of eigenvectors by their real and imaginary components, a real representation of the same subspace is obtained, which is suitable for the cluster analysis. We show that our approach leads to the same results as the generalized PCCA+ (GPCCA), which replaces the complex eigen-decomposition by a conceptually more difficult real Schur decomposition. We apply the method on non-reversible Markov chains, including circular chains, and demonstrate its efficiency compared to GPCCA. The experiments are performed in the Matlab programming language and codes are provided.

Description: Periodic timetabling is a challenging planning task in public transport. As safety requirements are crucial, track allocation is indispensable for validating the practical feasibility of a railway timetable. For busy stations with limited capacities, this requires a detailed planning of turnarounds. It is therefore desirable to integrate timetabling not only with track allocation, but also with vehicle scheduling and line planning. This is captured by the Integrated Line Planning and Turn-Sensitive Periodic Timetabling Problem with Track Choice, whose MIP formulation has been demonstrated to be effective for construction site railway rescheduling, as long as a good quality initial solution is available. In this paper, we discuss how to generate such a solution by extending the SAT formulation of the Periodic Event Scheduling Problem with track choice, track occupation, and minimum service frequency components. The SAT approach is superior to pure MIP on real-world instances of the S-Bahn Berlin network.

Description: In our combined experimental, theoretical and numerical work, we study the out of equilibrium deformations in a shrinking ring of optically trapped, interacting colloidal particles. Steerable optical tweezers are used to confine dielectric microparticles along a circle of discrete harmonic potential wells, and to reduce the ring radius at a controlled quench speed. We show that excluded-volume interactions are enough to induce particle sliding from their equilibrium positions and nonequilibrium zigzag roughening of the colloidal structure. Our work unveils the underlying mechanism of interfacial deformation in radially driven microscopic discrete rings.

Description: The SCIP Optimization Suite provides a collection of software packages for mathematical optimization, centered around the constraint integer programming framework SCIP. This report discusses the enhancements and extensions included in the SCIP Optimization Suite 9.0. The updates in SCIP 9.0 include improved symmetry handling, additions and improvements of nonlinear handlers and primal heuristics, a new cut generator and two new cut selection schemes, a new branching rule, a new LP interface, and several bug fixes. The SCIP Optimization Suite 9.0 also features new Rust and C++ interfaces for SCIP, new Python interface for SoPlex, along with enhancements to existing interfaces. The SCIP Optimization Suite 9.0 also includes new and improved features in the LP solver SoPlex, the presolving library PaPILO, the parallel framework UG, the decomposition framework GCG, and the SCIP extension SCIP-SDP. These additions and enhancements have resulted in an overall performance improvement of SCIP in terms of solving time, number of nodes in the branch-and-bound tree, as well as the reliability of the solver.

Description: In this article, we propose an interval constraint programming method for globally solving catalog-based categorical optimization problems. It supports catalogs of arbitrary size and properties of arbitrary dimension, and does not require any modeling effort from the user. A novel catalog-based contractor (or filtering operator) guarantees consistency between the categorical properties and the existing catalog items. This results in an intuitive and generic approach that is exact, rigorous (robust to roundoff errors) and can be easily implemented in an off-the-shelf interval-based continuous solver that interleaves branching and constraint propagation. We demonstrate the validity of the approach on a numerical problem in which a categorical variable is described by a two-dimensional property space. A Julia prototype is available as open-source software under the MIT license.

Description: Agent-based models (ABMs) provide an intuitive and powerful framework for studying social dynamics by modeling the interactions of individuals from the perspective of each individual. In addition to simulating and forecasting the dynamics of ABMs, the demand to solve optimization problems to support, for example, decision-making processes naturally arises. Most ABMs, however, are non-deterministic, high-dimensional dynamical systems, so objectives defined in terms of their behavior are computationally expensive. In particular, if the number of agents is large, evaluating the objective functions often becomes prohibitively time-consuming. We consider data-driven reduced models based on the Koopman generator to enable the efficient solution of multi-objective optimization problems involving ABMs. In a first step, we show how to obtain data-driven reduced models of non-deterministic dynamical systems (such as ABMs) that depend on potentially nonlinear control inputs. We then use them in the second step as surrogate models to solve multi-objective optimal control problems. We first illustrate our approach using the example of a voter model, where we compute optimal controls to steer the agents to a predetermined majority, and then using the example of an epidemic ABM, where we compute optimal containment strategies in a prototypical situation. We demonstrate that the surrogate models effectively approximate the Pareto-optimal points of the ABM dynamics by comparing the surrogate-based results with test points, where the objectives are evaluated using the ABM. Our results show that when objectives are defined by the dynamic behavior of ABMs, data-driven surrogate models support or even enable the solution of multi-objective optimization problems.

Description: Epidemiological models can not only be used to forecast the course of a pandemic like COVID-19, but also to propose and design non-pharmaceutical interventions such as school and work closing. In general, the design of optimal policies leads to nonlinear optimization problems that can be solved by numerical algorithms. Epidemiological models come in different complexities, ranging from systems of simple ordinary differential equations (ODEs) to complex agent-based models (ABMs). The former allow a fast and straightforward optimization, but are limited in accuracy, detail, and parameterization, while the latter can resolve spreading processes in detail, but are extremely expensive to optimize. We consider policy optimization in a prototypical situation modeled as both ODE and ABM, review numerical optimization approaches, and propose a heterogeneous multilevel approach based on combining a fine-resolution ABM and a coarse ODE model. Numerical experiments, in particular with respect to convergence speed, are given for illustrative examples.

Description: We study the solution of the rolling stock rotation problem with predictive maintenance (RSRP-PdM) by an iterative refinement approach that is based on a state-expanded event-graph. In this graph, the states are parameters of a failure distribution, and paths correspond to vehicle rotations with associated health state approximations. An optimal set of paths including maintenance can be computed by solving an integer linear program. Afterwards, the graph is refined and the procedure repeated. An associated linear program gives rise to a lower bound that can be used to determine the solution quality. Computational results for six instances derived from real-world timetables of a German railway company are presented. The results show the effectiveness of the approach and the quality of the solutions.

Description: Statistical shape models are an essential tool for various tasks in medical image analysis, including shape generation, reconstruction and classification. Shape models are learned from a population of example shapes, which are typically obtained through segmentation of volumetric medical images. In clinical practice, highly anisotropic volumetric scans with large slice distances are prevalent, e.g., to reduce radiation exposure in CT or image acquisition time in MR imaging. For existing shape modeling approaches, the resolution of the emerging model is limited to the resolution of the training shapes. Therefore, any missing information between slices prohibits existing methods from learning a high-resolution shape prior. We propose a novel shape modeling approach that can be trained on sparse, binary segmentation masks with large slice distances. This is achieved through employing continuous shape representations based on neural implicit functions. After training, our model can reconstruct shapes from various sparse inputs at high target resolutions beyond the resolution of individual training examples. We successfully reconstruct high-resolution shapes from as few as three orthogonal slices. Furthermore, our shape model allows us to embed various sparse segmentation masks into a common, low-dimensional latent space — independent of the acquisition direction, resolution, spacing, and field of view. We show that the emerging latent representation discriminates between healthy and pathological shapes, even when provided with sparse segmentation masks. Lastly, we qualitatively demonstrate that the emerging latent space is smooth and captures characteristic modes of shape variation. We evaluate our shape model on two anatomical structures: the lumbar vertebra and the distal femur, both from publicly available datasets.

Description: In multipartite Bell scenarios, we study the nonlocality robustness of the Greenberger-Horne-Zeilinger (GHZ) state. When each party performs planar measurements forming a regular polygon, we exploit the symmetry of the resulting correlation tensor to drastically accelerate the computation of (i) a Bell inequality via Frank-Wolfe algorithms and (ii) the corresponding local bound. The Bell inequalities obtained are facets of the symmetrized local polytope and they give the best-known upper bounds on the nonlocality robustness of the GHZ state for three to ten parties. Moreover, for four measurements per party, we generalize our facets and hence show, for any number of parties, an improvement on Mermin's inequality in terms of noise robustness. We also compute the detection efficiency of our inequalities and show that some give rise to the activation of nonlocality in star networks, a property that was only shown with an infinite number of measurements.

Description: We propose two graph neural network layers for graphs with features in a Riemannian manifold. First, based on a manifold-valued graph diffusion equation, we construct a diffusion layer that can be applied to an arbitrary number of nodes and graph connectivity patterns. Second, we model a tangent multilayer perceptron by transferring ideas from the vector neuron framework to our general setting. Both layers are equivariant with respect to node permutations and isometries of the feature manifold. These properties have been shown to lead to a beneficial inductive bias in many deep learning tasks. Numerical examples on synthetic data as well as on triangle meshes of the right hippocampus to classify Alzheimer's disease demonstrate the very good performance of our layers.

Description: Knee osteoarthritis (KOA) is a degenerative disease that leads to pain and loss of function. It is estimated to affect over 500 million humans world-wide and is one of the most common reasons for disability. KOA is usually diagnosed by radiologists or clinical experts by anamnesis, physical examination, and by assessing medical image data. The latter is typically acquired using X-Ray or magnetic resonance imaging. Since manual image reading is subjective, tedious and time-consuming, automated methods are required for a fast and objective decision support and for a better understanding of the pathogenesis of KOA. This thesis sets a foundation towards automated computation of image-based KOA biomarkers for holistic assessment of the knee. This involves the assessment of multiple knee bones and soft tissues. An assessment of particular structures requires localization of these tissues. In order to automate a faithful localization of anatomical structures, deep learning-based methods are investigated and utilized. Additionally, convolutional neural networks (CNNs) are used for classification of medical image data, i.e., for a direct determination of the disease status and to detect anatomical structures and landmarks. The automatically computed anatomical volumes, locations, and other measurements are finally compared to values acquired by clinical experts and evaluated for clustering of KOA groups, classification of KOA severity, prediction of KOA progression, and prediction of total knee replacement. In various experiments it is shown that CNN-based methods are suitable for accurate medical image segmentation, object detection, landmark detection, and direct classification of disease stages from the image data. Computed features related to the menisci are found to be most expressive in terms of clustering of KOA groups and predicting of future disease states, thus allowing diagnosis of current KOA conditions and prediction of future conditions. The conclusion of this thesis is that machine learning-based, fully automated processing of medical image data shows potential for diagnosis and prediction of KOA grades. Future studies could investigate additional features in order to achieve an assessment of the whole knee or validate the findings of this work in clinical studies.

Description: The dominant eigenfunctions of the Koopman operator characterize the metastabilities and slow-timescale dynamics of stochastic diffusion processes. In the context of molecular dynamics and Markov state modeling, they allow for a description of the location and frequencies of rare transitions, which are hard to obtain by direct simulation alone. In this article, we reformulate the eigenproblem in terms of the ISOKANN framework, an iterative algorithm that learns the eigenfunctions by alternating between short burst simulations and a mixture of machine learning and classical numerics, which naturally leads to a proof of convergence. We furthermore show how the intermediate iterates can be used to reduce the sampling variance by importance sampling and optimal control (enhanced sampling), as well as to select locations for further training (adaptive sampling). We demonstrate the usage of our proposed method in experiments, increasing the approximation accuracy by several orders of magnitude.

Description: We present EPR-Net, a novel and effective deep learning approach that tackles a crucial challenge in biophysics: constructing potential landscapes for high-dimensional non-equilibrium steady-state (NESS) systems. EPR-Net leverages a nice mathematical fact that the desired negative potential gradient is simply the orthogonal projection of the driving force of the underlying dynamics in a weighted inner-product space. Remarkably, our loss function has an intimate connection with the steady entropy production rate (EPR), enabling simultaneous landscape construction and EPR estimation. We introduce an enhanced learning strategy for systems with small noise, and extend our framework to include dimensionality reduction and state-dependent diffusion coefficient case in a unified fashion. Comparative evaluations on benchmark problems demonstrate the superior accuracy, effectiveness, and robustness of EPR-Net compared to existing methods. We apply our approach to challenging biophysical problems, such as an 8D limit cycle and a 52D multi-stability problem, which provide accurate solutions and interesting insights on constructed landscapes. With its versatility and power, EPR-Net offers a promising solution for diverse landscape construction problems in biophysics.

Description: Grazing-incidence X-ray diffraction (GIXRD) is a scattering technique which allows one to characterize the structure of fluid interfaces down to the molecular scale, including the measurement of the surface tension and of the interface roughness. However, the corresponding standard data analysis at non-zero wave numbers has been criticized as to be inconclusive because the scattering intensity is polluted by the unavoidable scattering from the bulk. Here we overcome this ambiguity by proposing a physically consistent model of the bulk contribution which is based on a minimal set of assumptions of experimental relevance. To this end, we derive an explicit integral expression for the background scattering, which can be determined numerically from the static structure factors of the coexisting bulk phases as independent input. Concerning the interpretation of GIXRD data inferred from computer simulations, we account also for the finite sizes of the bulk phases, which are unavoidable in simulations. The corresponding leading-order correction beyond the dominant contribution to the scattered intensity is revealed by asymptotic analysis, which is characterized by the competition between the linear system size and the X-ray penetration depth in the case of simulations. Specifically, we have calculated the expected GIXRD intensity for scattering at the planar liquid--vapor interface of Lennard-Jones fluids with truncated pair interactions via extensive, high-precision simulations. The reported data cover interfacial and bulk properties of fluid states along the whole liquid--vapor coexistence line. A sensitivity analysis demonstrates the robustness of our findings concerning the detailed definition of the mean interface position. We conclude that previous claims of an enhanced surface tension at mesoscopic scales are amenable to unambiguous tests via scattering experiments.

Description: Estimating the rate of rare conformational changes in molecular systems is one of the goals of molecular dynamics simulations. In the past few decades, a lot of progress has been done in data-based approaches toward this problem. In contrast, model-based methods, such as the Square Root Approximation (SqRA), directly derive these quantities from the potential energy functions. In this article, we demonstrate how the SqRA formalism naturally blends with the tensor structure obtained by coupling multiple systems, resulting in the tensor-based Square Root Approximation (tSqRA). It enables efficient treatment of high-dimensional systems using the SqRA and provides an algebraic expression of the impact of coupling energies between molecular subsystems. Based on the tSqRA, we also develop the projected rate estimation, a hybrid data-model-based algorithm that efficiently estimates the slowest rates for coupled systems. In addition, we investigate the possibility of integrating low-rank approximations within this framework to maximize the potential of the tSqRA.

Description: The microphysical structure of the lunar regolith provides information on the geologic history of the Moon. We used remote sensing measurements of thermal emission and a thermophysical model to determine the microphysical properties of the lunar regolith. We expand upon previous investigations by developing a microphysical thermal model, which more directly simulates regolith properties, such as grain size and volume filling factor. The modeled temperatures are matched with surface temperatures measured by the Diviner Lunar Radiometer Experiment on board the Lunar Reconnaissance Orbiter. The maria and highlands are investigated separately and characterized in the model by a difference in albedo and grain density. We find similar regolith temperatures for both terrains, which can be well described by similar volume filling factor profiles and mean grain sizes obtained from returned Apollo samples. We also investigate a significantly lower thermal conductivity for highlands, which formally also gives a very good solution, but in a parameter range that is well outside the Apollo data. We then study the latitudinal dependence of regolith properties up to ±80° latitude. When assuming constant regolith properties, we find that a variation of the solar incidence-dependent albedo can reduce the initially observed latitudinal gradient between model and Diviner measurements significantly. A better match between measurements and model can be achieved by a variation in intrinsic regolith properties with a decrease in bulk density with increasing latitude. We find that a variation in grain size alone cannot explain the Diviner measurements at higher latitudes.

Description: The decarbonization of the European energy system demands a rapid and comprehensive transformation while securing energy supplies at all times. Still, natural gas plays a crucial role in this process. Recent unexpected events forced drastic changes in gas routes throughout Europe. Therefore, operational-level analysis of the gas transport networks and technical capacities to cope with these transitions using unconventional scenarios has become essential. Unfortunately, data limitations often hinder such analyses. To overcome this challenge, we propose a mathematical model-based scenario generator that enables operational analysis of the European gas network using open data. Our approach focuses on the consistent analysis of specific partitions of the gas transport network, whose network topology data is readily available. We generate reproducible and consistent node-based gas in/out-flow scenarios for these defined network partitions to enable feasibility analysis and data quality assessment. Our proposed method is demonstrated through several applications that address the feasibility analysis and data quality assessment of the German gas transport network. By using open data and a mathematical modeling approach, our method allows for a more comprehensive understanding of the gas transport network's behavior and assists in decision-making during the transition to decarbonization.

Description: In this work, we study the geodesics of the space of certain geometrically and physically motivated subspaces of the space of immersed curves endowed with a first order Sobolev metric. This includes elastic curves and also an extension of some results on planar concentric circles to surfaces. The work focuses on intrinsic and constructive approaches.

Description: For decades, de Casteljau's algorithm has been used as a fundamental building block in curve and surface design and has found a wide range of applications in fields such as scientific computing, and discrete geometry to name but a few. With increasing interest in nonlinear data science, its constructive approach has been shown to provide a principled way to generalize parametric smooth curves to manifolds. These curves have found remarkable new applications in the analysis of parameter-dependent, geometric data. This article provides a survey of the recent theoretical developments in this exciting area as well as its applications in fields such as geometric morphometrics and longitudinal data analysis in medicine, archaeology, and meteorology.

Description: The rolling stock rotation problem with predictive maintenance (RSRP-PdM) involves the assignment of trips to a fleet of vehicles with integrated maintenance scheduling based on the predicted failure probability of the vehicles. These probabilities are determined by the health states of the vehicles, which are considered to be random variables distributed by a parameterized family of probability distribution functions. During the operation of the trips, the corresponding parameters get updated. In this article, we present a dual solution approach for RSRP-PdM and generalize a linear programming based lower bound for this problem to families of probability distribution functions with more than one parameter. For this purpose, we define a rounding function that allows for a consistent underestimation of the parameters and model the problem by a state-expanded event-graph in which the possible states are restricted to a discrete set. This induces a flow problem that is solved by an integer linear program. We show that the iterative refinement of the underlying discretization leads to solutions that converge from below to an optimal solution of the original instance. Thus, the linear relaxation of the considered integer linear program results in a lower bound for RSRP-PdM. Finally, we report on the results of computational experiments conducted on a library of test instances.

Description: Protein–protein interactions (PPIs) offer great opportunities to expand the druggable proteome and therapeutically tackle various diseases, but remain challenging targets for drug discovery. Here, we provide a comprehensive pipeline that combines experimental and computational tools to identify and validate PPI targets and perform early-stage drug discovery. We have developed a machine learning approach that prioritizes interactions by analyzing quantitative data from binary PPI assays or AlphaFold-Multimer predictions. Using the quantitative assay LuTHy together with our machine learning algorithm, we identified high-confidence interactions among SARS-CoV-2 proteins for which we predicted three-dimensional structures using AlphaFold-Multimer. We employed VirtualFlow to target the contact interface of the NSP10-NSP16 SARS-CoV-2 methyltransferase complex by ultra-large virtual drug screening. Thereby, we identified a compound that binds to NSP10 and inhibits its interaction with NSP16, while also disrupting the methyltransferase activity of the complex, and SARS-CoV-2 replication. Overall, this pipeline will help to prioritize PPI targets to accelerate the discovery of early-stage drug candidates targeting protein complexes and pathways.

Description: Task-adapted image reconstruction methods using end-to-end trainable neural networks (NNs) have been proposed to optimize reconstruction for subsequent processing tasks, such as segmentation. However, their training typically requires considerable hardware resources and thus, only relatively simple building blocks, e.g. U-Nets, are typically used, which, albeit powerful, do not integrate model-specific knowledge. In this work, we extend an end-to-end trainable task-adapted image reconstruction method for a clinically realistic reconstruction and segmentation problem of bone and cartilage in 3D knee MRI by incorporating statistical shape models (SSMs). The SSMs model the prior information and help to regularize the segmentation maps as a final post-processing step. We compare the proposed method to a state-of-the-art (SOTA) simultaneous multitask learning approach for image reconstruction and segmentation (MTL) and to a complex SSMs-informed segmentation pipeline (SIS). Our experiments show that the combination of joint end-to-end training and SSMs to further regularize the segmentation maps obtained by MTL highly improves the results, especially in terms of mean and maximal surface errors. In particular, we achieve the segmentation quality of SIS and, at the same time, a substantial model reduction that yields a five-fold decimation in model parameters and a computational speedup of an order of magnitude. Remarkably, even for undersampling factors of up to R=8, the obtained segmentation maps are of comparable quality to those obtained by SIS from ground-truth images.

Description: We present a heuristic solution approach for the rolling stock rotation problem with predictive maintenance (RSRP-PdM). The task of this problem is to assign a sequence of trips to each of the vehicles and to schedule their maintenance such that all trips can be operated. Here, the health states of the vehicles are considered to be random variables distributed by a family of probability distribution functions, and the maintenance services should be scheduled based on the failure probability of the vehicles. The proposed algorithm first generates a solution by solving an integer linear program and then heuristically improves this solution by applying a local search procedure. For this purpose, the trips assigned to the vehicles are split up and recombined, whereby additional deadhead trips can be inserted between the partial assignments. Subsequently, the maintenance is scheduled by solving a shortest path problem in a state-expanded version of a space-time graph restricted to the trips of the individual vehicles. The solution approach is tested and evaluated on a set of test instances based on real-world timetables.

Description: The stability of shear layers in fluid flows is a crucial factor in forming vortices and jets and plays a fundamental role in the development of turbulence. Such shear layer instabilities are ubiquitous in natural phenomena, such as atmospheric and oceanic flows, contributing to the formation of weather systems and predicting tsunamis. This study specifically focuses on the stability of a shear layer sandwiched between two semi-infinite layers within a two-dimensional flow. The velocity profile of the shear layer is assumed to be linearly dependent on the vertical coordinate, while the velocity of the other layers remains uniform with differing strengths. The effect of viscosity and surface tension is ignored to simplify the analysis. The shallow water equations are used to analyze the interface stability of the shear layer, and the resulting dispersion relation between wave frequency and other wave characteristics is obtained. This relation incorporates Whittaker functions and their first derivatives and is used to derive appropriate limits corresponding to various physical conditions. Our study thus contributes to a deeper understanding of the stability of shear layers and their implications for natural phenomena.

Description: This paper proposes an almost feasible Sequential Linear Programming (afSLP) algorithm. In the first part, the practical limitations of previously proposed Feasible Sequential Linear Programming (FSLP) methods are discussed along with illustrative examples. Then, we present a generalization of FSLP based on a tolerance-tube method that addresses the shortcomings of FSLP. The proposed algorithm afSLP consists of two phases. Phase I starts from random infeasible points and iterates towards a relaxation of the feasible set. Once the tolerance-tube around the feasible set is reached, phase II is started and all future iterates are kept within the tolerance-tube. The novel method includes enhancements to the originally proposed tolerance-tube method that are necessary for global convergence. afSLP is shown to outperform FSLP and the state-of-the-art solver IPOPT on a SCARA robot optimization problem.

Description: The multi-grid reaction-diffusion master equation (mgRDME) provides a generalization of stochastic compartment-based reaction-diffusion modelling described by the standard reaction-diffusion master equation (RDME). By enabling different resolutions on lattices for biochemical species with different diffusion constants, the mgRDME approach improves both accuracy and efficiency of compartment-based reaction-diffusion simulations. The mgRDME framework is examined through its application to morphogen gradient formation in stochastic reaction-diffusion scenarios, using both an analytically tractable first-order reaction network and a model with a second-order reaction. The results obtained by the mgRDME modelling are compared with the standard RDME model and with the (more detailed) particle-based Brownian dynamics simulations. The dependence of error and numerical cost on the compartment sizes is defined and investigated through a multi-objective optimization problem.

Description: This paper is concerned with collective variables, or reaction coordinates, that map a discrete-in-time Markov process X_n in R^d to a (much) smaller dimension k≪d. We define the effective dynamics under a given collective variable map ξ as the best Markovian representation of X_n under ξ. The novelty of the paper is that it gives strict criteria for selecting optimal collective variables via the properties of the effective dynamics. In particular, we show that the transition density of the effective dynamics of the optimal collective variable solves a relative entropy minimization problem from certain family of densities to the transition density of X_n. We also show that many transfer operator-based data-driven numerical approaches essentially learn quantities of the effective dynamics. Furthermore, we obtain various error estimates for the effective dynamics in approximating dominant timescales / eigenvalues and transition rates of the original process X_n and how optimal collective variables minimize these errors. Our results contribute to the development of theoretical tools for the understanding of complex dynamical systems, e.g. molecular kinetics, on large timescales. These results shed light on the relations among existing data-driven numerical approaches for identifying good collective variables, and they also motivate the development of new methods.

Description: The Koopman operator has entered and transformed many research areas over the last years. Although the underlying concept–representing highly nonlinear dynamical systems by infinite-dimensional linear operators–has been known for a long time, the availability of large data sets and efficient machine learning algorithms for estimating the Koopman operator from data make this framework extremely powerful and popular. Koopman operator theory allows us to gain insights into the characteristic global properties of a system without requiring detailed mathematical models. We will show how these methods can also be used to analyze complex networks and highlight relationships between Koopman operators and graph Laplacians.

Description: Compressible flows are prevalent in natural and technological processes, particularly in the energy transition to renewable energy systems. Consequently, extensive research has focused on understanding the stability of tangential--velocity discontinuity in compressible media. Despite recent advancements that address industrial challenges more realistically, many studies have ignored viscous stress tensors' impact, leading to inaccuracies in predicting interface stability. This omission becomes critical, especially in high Reynolds or low Mach number flows, where viscous forces dissipate kinetic energy across interfaces, affect total energy dissipation, and dampen flow instabilities. Our work is thus motivated to analyze the viscosity force effect by including the viscous stress tensor terms in the motion equations. Our results show that by considering the effect of viscous forces, the tangential-velocity discontinuity interface is constantly destabilized for the entire range of the Mach number.

Description: Modeling-Simulation-Optimization workflows play a fundamental role in applied mathematics. The Mathematical Research Data Initiative, MaRDI, responded to this by developing a FAIR and machine-interpretable template for a comprehensive documentation of such workflows. MaRDMO, a Plugin for the Research Data Management Organiser, enables scientists from diverse fields to document and publish their workflows on the MaRDI Portal seamlessly using the MaRDI template. Central to these workflows are mathematical models. MaRDI addresses them with the MathModDB ontology, offering a structured formal model description. Here, we showcase the interaction between MaRDMO and the MathModDB Knowledge Graph through an algebraic modeling workflow from the Digital Humanities. This demonstration underscores the versatility of both services beyond their original numerical domain.

Description: We have investigated how Langevin dynamics is affected by the friction coefficient using the novel algorithm ISOKANN, which combines the transfer operator approach with modern machine learning techniques. ISOKANN describes the dynamics in terms of an invariant subspace projection of the Koopman operator defined in the entire state space, avoiding approximations due to dimensionality reduction and discretization. Our results are consistent with the Kramers turnover and show that in the low and moderate friction regimes, metastable macro-states and transition rates are defined in phase space, not only in position space.

Description: The imperative to decarbonize energy systems has intensified the need for efficient transformations within the heating sector, with a particular focus on district heating networks. This study addresses this challenge by proposing a comprehensive optimization approach evaluated on the district heating network of the Märkisches Viertel of Berlin. Our objective is to simultaneously optimize heat production with three targets: minimizing costs, minimizing CO2-emissions, and maximizing heat generation from Combined Heat and Power (CHP) plants for enhanced efficiency. To tackle this optimization problem, we employed a Mixed-Integer Linear Program (MILP) that encompasses the conversion of various fuels into heat and power, integration with relevant markets, and considerations for technical constraints on power plant operation. These constraints include startup and minimum downtime, activation costs, and storage limits. The ultimate goal is to delineate the Pareto front, representing the optimal trade-offs between the three targets. We evaluate variants of the 𝜖-constraint algorithm for their effectiveness in coordinating these objectives, with a simultaneous focus on the quality of the estimated Pareto front and computational efficiency. One algorithm explores solutions on an evenly spaced grid in the objective space, while another dynamically adjusts the grid based on identified solutions. Initial findings highlight the strengths and limitations of each algorithm, providing guidance on algorithm selection depending on desired outcomes and computational constraints. Our study emphasizes that the optimal choice of algorithm hinges on the density and distribution of solutions in the feasible space. Whether solutions are clustered or evenly distributed significantly influences algorithm performance. These insights contribute to a nuanced understanding of algorithm selection for multi-objective multi-energy system optimization, offering valuable guidance for future research and practical applications for planning sustainable district heating networks.

Description: At present, data management plans (DMPs) are still often perceived as mere documents for funding agencies providing clarity on how research data will be handled during a funded project, but are not usually actively involved in the processes. However, they contain a great deal of information that can be shared automatically to facilitate active research data management (RDM) by providing metadata to research infrastructures and supporting communication between all involved stakeholders. This position paper brings together a number of ideas developed and collected during interdisciplinary workshops of the Data Management Planning Working Group (infra-dmp), which is part of the section Common Infrastructures of the National Research Data Infrastructure (NFDI) in Germany. We present our vision of a possible future role of DMPs, templates, and tools in the upcoming NFDI service architecture.

Description: Markov processes serve as foundational models in many scientific disciplines, such as molecular dynamics, and their simulation forms a common basis for analysis. While simulations produce useful trajectories, obtaining macroscopic information directly from microstate data presents significant challenges. This paper addresses this gap by introducing the concept of membership functions being the macrostates themselves. We derive equations for the holding times of these macrostates and demonstrate their consistency with the classical definition. Furthermore, we discuss the application of the ISOKANN method for learning these quantities from simulation data. In addition, we present a novel method for extracting transition paths based on the ISOKANN results and demonstrate its efficacy by applying it to simulations of the 𝜇-opioid receptor. With this approach we provide a new perspective on analyzing the macroscopic behaviour of Markov systems.

Description: The computation of Voronoi Diagrams, or their dual Delauney triangulations is difficult in high dimensions. In a recent publication Polianskii and Pokorny propose an iterative randomized algorithm facilitating the approximation of Voronoi tesselations in high dimensions. In this paper, we provide an improved vertex search method that is not only exact but even faster than the bisection method that was previously recommended. Building on this we also provide a depth-first graph-traversal algorithm which allows us to compute the entire Voronoi diagram. This enables us to compare the outcomes with those of classical algorithms like qHull, which we either match or marginally beat in terms of computation time. We furthermore show how the raycasting algorithm naturally lends to a Monte Carlo approximation for the volume and boundary integrals of the Voronoi cells, both of which are of importance for finite Volume methods. We compare the Monte-Carlo methods to the exact polygonal integration, as well as a hybrid approximation scheme.

Description: Existing planning approaches for onshore wind farm siting and grid integration often do not meet minimum cost solutions or social and environmental considerations. In this paper, we develop an exact approach for the integrated layout and cable routing problem of onshore wind farm planning using the Quota Steiner tree problem. Applying a novel transformation on a known directed cut formulation, reduction techniques, and heuristics, we design an exact solver that makes large problem instances solvable and outperforms generic MIP solvers. In selected regions of Germany, the trade-offs between minimizing costs and landscape impact of onshore wind farm siting are investigated. Although our case studies show large trade-offs between the objective criteria of cost and landscape impact, small burdens on one criterion can significantly improve the other criteria. In addition, we demonstrate that contrary to many approaches for exclusive turbine siting, grid integration must be simultaneously optimized to avoid excessive costs or landscape impacts in the course of a wind farm project. Our novel problem formulation and the developed solver can assist planners in decision-making and help optimize wind farms in large regions in the future.

Description: This paper explores memory mechanisms in complex socio-technical systems, using a mobility demand model as an example case. We simplified a large-scale agent-based mobility model into a Markov process and discover that the mobility decision process is non-Markovian. This is due to its dependence on the system’s history, including social structure and local infrastructure, which evolve based on prior mobility decisions. To make the process Markovian, we extend the state space by incorporating two history-dependent components. Although our model is a very much reduced version of the original one, it remains too complex for the application of usual analytic methods. Instead, we employ simulations to examine the functionalities of the two history-dependent components. We think that the structure of the analyzed stochastic process is exemplary for many socio-technical, -economic, -ecological systems. Additionally, it exhibits analogies with the framework of extended evolution, which has previously been used to study cultural evolution.

Description: It has been shown that any 9 by 9 Sudoku puzzle must contain at least 17 clues to have a unique solution. This paper investigates the more specific question: given a particular completed Sudoku grid, what is the minimum number of clues in any puzzle whose unique solution is the given grid? We call this problem the Minimum Sudoku Clue Problem (MSCP). We formulate MSCP as a binary bilevel linear program, present a class of globally valid inequalities, and provide a computational study on 50 MSCP instances of 9 by 9 Sudoku grids. Using a general bilevel solver, we solve 95% of instances to optimality, and show that the solution process benefits from the addition of a moderate amount of inequalities. Finally, we extend the proposed model to other combinatorial problems in which uniqueness of the solution is of interest.

Description: For industries like the cement industry, switching to a carbon-neutral production process is impossible. They must rely on carbon capture, utilization and storage (CCUS) technologies to reduce their production processes’ inevitable carbon dioxide (CO2) emissions. For transporting continuously large amounts of CO2, utilizing a pipeline network is the most effective solution; however, building such a network is expensive. Therefore minimizing the cost of the pipelines to be built is extremely important to make the operation financially feasible. In this context, we investigate the problem of finding optimal pipeline diameters from a discrete set of diameters for a tree-shaped network transporting captured CO2 from multiple sources to a single sink. The general problem of optimizing arc capacities in potential-based fluid networks is already a challenging mixed-integer nonlinear optimization problem. The problem becomes even more complex when adding the highly sensitive nonlinear behavior of CO2 regarding temperature and pressure changes. We propose an iterative algorithm splitting the problem into two parts: a) the pipe-sizing problem under a fixed supply scenario and temperature distribution and b) the thermophysical modeling, including mixing effects, the Joule-Thomson effect, and heat exchange with the surrounding environment. We demonstrate the effectiveness of our approach by applying our algorithm to a real-world network planning problem for a CO2 network in Western Germany. Further, we show the robustness of the algorithm by solving a large artificially created set of network instances.

Description: Collective variables (CVs) are low-dimensional projections of high-dimensional system states. They are used to gain insights into complex emergent dynamical behaviors of processes on networks. The relation between CVs and network measures is not well understood and its derivation typically requires detailed knowledge of both the dynamical system and the network topology. In this Letter, we present a data-driven method for algorithmically learning and understanding CVs for binary-state spreading processes on networks of arbitrary topology. We demonstrate our method using four example networks: the stochastic block model, a ring-shaped graph, a random regular graph, and a scale-free network generated by the Albert-Barabási model. Our results deliver evidence for the existence of low-dimensional CVs even in cases that are not yet understood theoretically.

Description: Computing an optimal cycle in a given homology class, also referred to as the homology localization problem, is known to be an NP-hard problem in general. Furthermore, there is currently no known optimality criterion that localizes classes geometrically and admits a stability property under the setting of persistent homology. We present a geometric optimization of the cycles that is computable in polynomial time and is stable in an approximate sense. Tailoring our search criterion to different settings, we obtain various optimization problems like optimal homologous cycle, minimum homology basis, and minimum persistent homology basis. In practice, the (trivial) exact algorithm is computationally expensive despite having a worst case polynomial runtime. Therefore, we design approximation algorithms for the above problems and study their performance experimentally. These algorithms have reasonable runtimes for moderate sized datasets and the cycles computed by these algorithms are consistently of high quality as demonstrated via experiments on multiple datasets.

Description: We study a complex planning and scheduling problem arising from the build-up process of air cargo pallets and containers, collectively referred to as unit load devices (ULD), in which ULDs must be assigned to workstations for loading. Since air freight usually becomes available gradually along the planning horizon, ULD build-ups must be scheduled neither too early to avoid underutilizing ULD capacity, nor too late to avoid resource conflicts with other flights. Whenever possible, ULDs should be built up in batches, thereby giving ground handlers more freedom to rearrange cargo and utilize the ULD's capacity efficiently. The resulting scheduling problem has an intricate cost function and produces large time-expanded models, especially for longer planning horizons. We propose a logic-based Benders decomposition approach that assigns batches to time intervals and workstations in the master problem, while the actual schedule is decided in a subproblem. By choosing appropriate intervals, the subproblem becomes a feasibility problem that decomposes over the workstations. Additionally, the similarity of many batches is exploited by a strengthening procedure for no-good cuts. We benchmark our approach against a time-expanded MIP formulation from the literature on a publicly available data set. It solves 15% more instances to optimality and decreases run times by more than 50% in the geometric mean. This improvement is especially pronounced for longer planning horizons of up to one week, where the Benders approach solves over 50% instances more than the baseline

Description: The Jacobi set of a bivariate scalar field is the set of points where the gradients of the two constituent scalar fields align with each other. It captures the regions of topological changes in the bivariate field. The Jacobi set is a bivariate analog of critical points, and may correspond to features of interest. In the specific case of time-varying fields and when one of the scalar fields is time, the Jacobi set corresponds to temporal tracks of critical points, and serves as a feature-tracking graph. The Jacobi set of a bivariate field or a time-varying scalar field is complex, resulting in cluttered visualizations that are difficult to analyze. This paper addresses the problem of Jacobi set simplification. Specifically, we use the time-varying scalar field scenario to introduce a method that computes a reduced Jacobi set. The method is based on a stability measure called robustness that was originally developed for vector fields and helps capture the structural stability of critical points. We also present a mathematical analysis for the method, and describe an implementation for 2D time-varying scalar fields. Applications to both synthetic and real-world datasets demonstrate the effectiveness of the method for tracking features.

Description: This article addresses reaction networks in which spatial and stochastic effects are of crucial importance. For such systems, particle-based models allow us to describe all microscopic details with high accuracy. However, they suffer from computational inefficiency if particle numbers and density get too large. Alternative coarse-grained-resolution models reduce computational effort tremendously, e.g., by replacing the particle distribution by a continuous concentration field governed by reaction-diffusion PDEs. We demonstrate how models on the different resolution levels can be combined into hybrid models that seamlessly combine the best of both worlds, describing molecular species with large copy numbers by macroscopic equations with spatial resolution while keeping the stochastic-spatial particle-based resolution level for the species with low copy numbers. To this end, we introduce a simple particle-based model for the binding dynamics of ions and vesicles at the heart of the neurotransmission process. Within this framework, we derive a novel hybrid model and present results from numerical experiments which demonstrate that the hybrid model allows for an accurate approximation of the full particle-based model in realistic scenarios.

Description: The Bay of Bengal (BoB) has maintained its salinity distribution over the years despite a continuous flow of fresh water entering it through rivers on the northern coast, which is capable of diluting the salinity. This can be attributed to the cyclic flow of high salinity water (〉= 35 psu), coming from Arabian sea and entering BoB from the south, which moves northward and mixes with this fresh water. The movement of this high salinity water has been studied and analyzed in previous work (Singh et al., 2022). This paper extends the computational methods and analysis of salinity movement. Specifically, we introduce an advection based feature definition that represents the movement of high salinity water, and describe algorithms to track their evolution over time. This method allows us to trace the movement of high salinity water caused due to ocean currents. The method is validated via comparison with established observations on the flow of high salinity water in the BoB, including its entry from the Arabian Sea and its movement near Sri Lanka. Further, the visual analysis and tracking framework enables us to compare it with previous work and analyze the contribution of advection to salinity transport.

Description: The landscape of applications and subroutines relying on shortest path computations continues to grow steadily. This growth is driven by the undeniable success of shortest path algorithms in theory and practice. It also introduces new challenges as the models and assessing the optimality of paths become more complicated. Hence, multiple recent publications in the field adapt existing labeling methods in an ad hoc fashion to their specific problem variant without considering the underlying general structure: they always deal with multi-criteria scenarios, and those criteria define different partial orders on the paths. In this paper, we introduce the partial order shortest path problem (POSP), a generalization of the multi-objective shortest path problem (MOSP) and in turn also of the classical shortest path problem. POSP captures the particular structure of many shortest path applications as special cases. In this generality, we study optimality conditions or the lack of them, depending on the objective functions’ properties. Our final contribution is a big lookup table summarizing our findings and providing the reader with an easy way to choose among the most recent multi-criteria shortest path algorithms depending on their problems’ weight structure. Examples range from time-dependent shortest path and bottleneck path problems to the electric vehicle shortest path problem with recharging and complex financial weight functions studied in the public transportation community. Our results hold for general digraphs and, therefore, surpass previous generalizations that were limited to acyclic graphs.

Description: Studying neural mechanisms in complementary model organisms from different ecological niches in the same animal class can leverage the comparative brain analysis at the cellular level. To advance such a direction, we developed a unified brain atlas platform and specialized tools that allowed us to quantitatively compare neural structures in two teleost larvae, medaka (Oryzias latipes) and zebrafish (Danio rerio). Leveraging this quantitative approach we found that most brain regions are similar but some subpopulations are unique in each species. Specifically, we confirmed the existence of a clear dorsal pallial region in the telencephalon in medaka lacking in zebrafish. Further, our approach allows for extraction of differentially expressed genes in both species, and for quantitative comparison of neural activity at cellular resolution. The web-based and interactive nature of this atlas platform will facilitate the teleost community’s research and its easy extensibility will encourage contributions to its continuous expansion.

Description: This work explores a synchronization-like phenomenon induced by common noise for continuous-time Markov jump processes given by chemical reaction networks. Based on Gillespie’s stochastic simulation algorithm, a corresponding random dynamical system is formulated in a two-step procedure, at first for the states of the embedded discrete-time Markov chain and then for the augmented Markov chain including random jump times. We uncover a time-shifted synchronization in the sense that—after some initial waiting time—one trajectory exactly replicates another one with a certain time delay. Whether or not such a synchronization behavior occurs depends on the combination of the initial states. We prove this partial time-shifted synchronization for the special setting of a birth-death process by analyzing the corresponding two-point motion of the embedded Markov chain and determine the structure of the associated random attractor. In this context, we also provide general results on existence and form of random attractors for discrete-time, discrete-space random dynamical systems.

Description: Derivative-based iterative methods for nonlinearly constrained non-convex optimization usually share common algorithmic components, such as strategies for computing a descent direction and mechanisms that promote global convergence. Based on this observation, we introduce an abstract framework based on four common ingredients that describes most derivative-based iterative methods and unifies their workflows. We then present Uno, a modular C++ solver that implements our abstract framework and allows the automatic generation of various strategy combinations with no programming effort from the user. Uno is meant to (1) organize mathematical optimization strategies into a coherent hierarchy; (2) offer a wide range of efficient and robust methods that can be compared for a given instance; (3) enable researchers to experiment with novel optimization strategies; and (4) reduce the cost of development and maintenance of multiple optimization solvers. Uno's software design allows user to compose new customized solvers for emerging optimization areas such as robust optimization or optimization problems with complementarity constraints, while building on reliable nonlinear optimization techniques. We demonstrate that Uno is highly competitive against state-of-the-art solvers filterSQP, IPOPT, SNOPT, MINOS, LANCELOT, LOQO, and CONOPT on a subset of 429 small problems from the CUTEst collection. Uno is available as open-source software under the MIT license at https://github.com/cvanaret/Uno .

Description: Nirmatrelvir/Ritonavir is an oral treatment for mild to moderate COVID-19 cases with a high risk for a severe course of the disease. For this paper, a comprehensive literature review was performed, leading to a summary of currently available data on Nirmatrelvir/Ritonavir’s ability to reduce the risk of progressing to a severe disease state. Herein, the focus lies on publications that include comparisons between patients receiving Nirmatrelvir/Ritonavir and a control group. The findings can be summarized as follows: Data from the time when the Delta-variant was dominant show that Nirmatrelvir/Ritonavir reduced the risk of hospitalization or death by 88.9% for unvaccinated, non-hospitalized high-risk individuals. Data from the time when the Omicron variant was dominant found decreased relative risk reductions for various vaccination statuses: between 26% and 65% for hospitalization. The presented papers that differentiate between unvaccinated and vaccinated individuals agree that unvaccinated patients benefit more from treatment with Nirmatrelvir/Ritonavir. However, when it comes to the dependency of potential on age and comorbidities, further studies are necessary. From the available data, one can conclude that Nirmatrelvir/Ritonavir cannot substitute vaccinations; however, its low manufacturing cost and easy administration make it a valuable tool in fighting COVID-19, especially for countries with low vaccination rates.

Description: The investigation of energy transition paths toward a sustainable and decarbonized future under uncertainty is a critical aspect of contemporary energy planning and policy development. There are numerous methods for analysing uncertainties and sensitivities and many studies on sustainable transformation paths, but there is a lack of combined application to relevant use-cases. In this study, we investigate the sensitivity of energy transition paths to uncertainties in operational and investment costs of power plants in the metropolitan area of Berlin and its rural surroundings. By employing the linear programming energy system model oemof-B3, we extensively focus on the system's energy technologies, such as wind turbines, photovoltaics, hydro and combustion plants, and energy storages. Greenhouse gas reduction and electrification rates per commodity are realized by selected constraints. Our research aims to discern how investments in energy production capacities are influenced by uncertainties of other energy technologies' investment and operational costs in the system. We apply a quantitative approach to investigate such interdependencies of cost variations and their impact on long-term energy planning. Thus, the analysis sheds light on the robustness of energy transition paths in the face of these uncertainties. The region Berlin-Brandenburg serves as a case study and thus reflects on the present space conflicts to meet energy demands in urban and suburban areas and their rural surroundings. An electricity-intensive scenario is selected that assumes a 100 % reduction in greenhouse gas emissions by 2050. With the results of the case study, we show how our approach enables rural and metropolitan decision-makers to collaborate in achieving sustainable energy. Decision-making in long-term energy planning can be made more robust and flexible by acknowledging the identified sensitivities and enable such regions better to navigate challenges and uncertainties associated with sustainable energy planning.