arXiv subjects

Stefan Ulbrich

Publications and source records attributed to Stefan Ulbrich.

At least 19 recordsLinked to original sources

arXiv2026

Global Optimization of Flexible District Heating Networks

District heating networks are a central tool to achieve low-carbon heat supplies. In this realm, they face the challenge of dealing with increasingly heterogeneous, partially time-varying renewable sources, thermal storage, and meshed topologies. This paper examines global optimization of the operation of such district heating networks over multiple time steps, based on a stationary, yet realistical nonlinear network model. To accelerate the solution performance of a spatial branch-and-bound algorithm for one time step, the following new methodological ingredients are introduced: exclusion of cyclic flow, symmetry exploitation between supply and return networks, reduction of temperature mixing constraints, novel primal heuristics and branching rules. The proposed methods are evaluated on a set of generated and real-world benchmark network instances with cycles and several suppliers. On the generated benchmark instances, using these methods more than doubles the number of solved instances and more than halves the runtime. For the real-world benchmark instances, the resulting algorithm produces solutions with guaranteed quality in reasonable run time. For multiple time steps that are coupled by a storage, a time decomposition approach is investigated. Under assumptions that are reasonable in practice, this approach is shown to yield an optimal solution. On a small example network, this decomposition is able to compute optimal solutions in less than a second, while solving the complete time-coupled problem is not possible within one hour.

math.OC↗

arXiv2026

Efficient Pareto-Front Generation for Electric Machines using IGA and Second Order Derivatives

The multiobjective optimization of electric machines always involves a trade-off caused by various competing objectives such as performance and cost. A suitable design is usually determined by comparing variants from the Pareto front, which has been generated by a large number of simulation runs. This paper addresses the efficient generation of the Pareto front using a continuation method based on a homotopy method that exploits second-order derivative information to achieve superlinear convergence, enabling the fast generation of new Pareto-optimal points within only a few iterations. A key contribution is the derivation of formulas to compute the Hessian with respect to geometry parameters and shape, thus enabling direct modifications of the motor geometry in the context of Isogeometric Analysis. We apply our method to nonlinear 2D magnetostatic simulations of a permanent magnet synchronous motor and demonstrate its effectiveness by optimizing the cost, mean torque and torque ripple of the motor. Compared to a first-order optimization method, this approach reduces the number of iterations and function evaluations needed, making the pareto optimization fast and efficient.

math.NA↗

arXiv2025

Scaling-aware rating of Poisson-limited demand forecasts

Forecast quality should be assessed in the context of what is possible in theory and what is reasonable to expect in practice. Often, one can identify an approximate upper bound to a probabilistic forecast's sharpness, which sets a lower, not necessarily achievable, limit to error metrics. In retail forecasting, a simple, but often unconquerable sharpness limit is given by the Poisson distribution. When evaluating forecasts using traditional metrics such as Mean Absolute Error, it is hard to judge whether a certain achieved value reflects unavoidable Poisson noise or truly indicates an over-dispersed prediction model. Moreover, every evaluation metric suffers from precision scaling: The metric's value is mostly defined by the selling rate and by the resulting rate-dependent Poisson noise, and only secondarily by the forecast quality. Comparing two groups of forecasted products often yields "the slow movers are performing worse than the fast movers" or vice versa, which we call the naïve scaling trap. To distill the intrinsic quality of a forecast, we stratify predictions into buckets of approximately equal predicted values and evaluate metrics separately per bucket. By comparing the achieved value per bucket to benchmarks defined by the theoretical expectation value of the metric, we obtain an intuitive visualization of forecast quality. This representation can be summarized by a single rating that makes forecast quality comparable among different products or even industries. The thereby developed scaling-aware forecast rating is applied to forecasting models used on the M5 competition dataset as well as to real-life forecasts provided by Blue Yonder's Demand Edge for Retail solution for grocery products in Sainsbury's supermarkets in the United Kingdom. The results permit a clear interpretation and high-level understanding of model quality by non-experts.

stat.AP↗

arXiv2025

A Variational Calculus for Optimal Control of the Generalized Riemann Problem for Hyperbolic Systems of Conservation Laws

We develop a variational calculus for entropy solutions of the Generalized Riemann Problem (GRP) for strictly hyperbolic systems of conservation laws where the control is the initial state. The GRP has a discontinuous initial state with exactly one discontinuity and continuously differentiable ($C^1$) states left and right of it. The control consists of the $C^1$ parts of the initial state and the position of the discontinuity. Solutions of the problem are generally discontinuous since they contain shock curves. We assume the time horizon $T>0$ to be sufficiently small such that no shocks interact and no new shocks are generated. Moreover, we assume that no rarefaction waves occur and that the jump of the initial state is sufficiently small. Since the shock positions depend on the control, a transformation to a reference space is used to fix the shock positions. In the reference space, we prove that the solution of the GRP between the shocks is continuously differentiable from the control space to $C^0$. In physical coordinates, this implies that the shock curves in $C^1$ and the states between the shocks in the topology of $C^0$ depend continuously differentiable on the control. As a consequence, we obtain the differentiability of tracking type objective functionals.

math.OC↗

arXiv2025

Robust Topology Optimization of Electric Machines using Topological Derivatives

Designing high-performance electric machines that maintain their efficiency and reliability under uncertain material and operating conditions is crucial for industrial applications. In this paper, we present a novel framework for robust topology optimization with partial differential equation constraints to address this challenge. The robust optimization problem is formulated as a min-max optimization problem, where the inner maximization is the worst case with respect to predefined uncertainties, while the outer minimization aims to find an optimal topology that remains robust under these uncertainties using the topological derivative. The shape of the domain is represented by a level set function, which allows for arbitrary perturbation of the domain. The robust optimization problem is solved using a theorem of Clarke to compute subgradients of the worst case function. This allows the min-max problem to be solved efficiently and ensures that we find a design that performs well even in the presence of uncertainty. Finally, numerical results for a two-material permanent magnet synchronous machine demonstrate both the effectiveness of the method and the improved performance of robust designs under uncertain conditions.

math.OC↗

arXiv2025

Combined Parameter and Shape Optimization of Electric Machines with Isogeometric Analysis

In structural optimization, both parameters and shape are relevant for the model performance. Yet, conventional optimization techniques usually consider either parameters or the shape separately. This work addresses this problem by proposing a simple yet powerful approach to combine parameter and shape optimization in a framework using Isogeometric Analysis (IGA). The optimization employs sensitivity analysis by determining the gradients of an objective function with respect to parameters and control points that represent the geometry. The gradients with respect to the control points are calculated in an analytical way using the adjoint method, which enables straightforward shape optimization by altering of these control points. Given that a change in a single geometry parameter corresponds to modifications in multiple control points, the chain rule is employed to obtain the gradient with respect to the parameters in an efficient semi-analytical way. The presented method is exemplarily applied to nonlinear 2D magnetostatic simulations featuring a permanent magnet synchronous motor and compared to designs, which were optimized using parameter and shape optimization separately. It is numerically shown that the permanent magnet mass can be reduced and the torque ripple can be eliminated almost completely by simultaneously adjusting rotor parameters and shape. The approach allows for novel designs to be created with the potential to reduce the optimization time substantially.

math.OC↗

arXiv2024

Scale-bridging within a complex model hierarchy for investigation of a metal-fueled circular energy economy by use of Bayesian model calibration with model error quantification

Metal energy carriers recently gained growing interest in research as a promising storage and transport material for renewable electricity. Within the development of a metal-fueled circular energy economy, research involves a model hierarchy spanning from micro to macro scales, making the transfer of information among different levels of complexity a crucial task for the implementation of the new technology. Chemical reactor networks (CRNs) are models of reduced complexity and a promising approach to accomplish the scale-bridging task. This holds if valid information from CRNs can be obtained on a much denser set of operating conditions than available from experiments and elaborated simulation methods like Computational Fluid Dynamics (CFD). An approach for CRN calibration from recent literature, including model error quantification, is further developed to construct a CRN model of a laboratory reactor for flash ironmaking, using data from the literature. By introducing a meta model of a CRN parameter, a simple CRN model on an extended set of operating conditions has successfully been calibrated. This way, the employed coupled calibration and uncertainty quantification framework has proven promising for the task of scale-bridging in the model hierarchy under investigation.

physics.comp-ph↗

arXiv2024

On the problem of optimal fluid transport in capillaries

In this note, we revisit the problem of the pressure-driven transport of a meniscus through a narrow cylindrical capillary or pore. This generic process finds many applications in science and technology. As it is known that Direct Numerical Simulations of moving contact line problems are highly demanding in terms of computational costs, simplified models in the form of ordinary differential equations offer an interesting alternative to perform a mathematical optimization of the flow. Blake and De Coninck studied the pressure-driven transport of a meniscus and identified two major competing mechanisms. While a hydrophilic surface is favorable to enhance the spontaneous imbibition into the pore, the friction is known to be significantly reduced on a hydrophobic surface. Blake and De Coninck showed that, depending on the applied pressure difference, there exists an optimal wettability that minimizes the time required to move the meniscus over a certain distance. We revisit this problem and derive analytical solutions in the limiting cases of negligible inertia and negligible contact line friction.

physics.flu-dyn↗

arXiv2024

Spline-Based Rotor and Stator Optimization of a Permanent Magnet Synchronous Motor

This work features the optimization of a Permanent Magnet Synchronous Motor using 2D nonlinear simulations in an Isogeometric Analysis framework. The rotor and stator designs are optimized for both geometric parameters and surface shapes via modifications of control points. The scaling laws for magnetism are employed to allow for axial and radial scaling, enabling a thorough optimization of all critical machine parameters for multiple operating points. The process is carried out in a gradient-based fashion with the objectives of lowering motor material cost, torque ripple and losses. It is shown that the optimization can be efficiently conducted for many optimization variables and all objective values can be reduced.

math.OC↗

arXiv2022

Reactive Neural Path Planning with Dynamic Obstacle Avoidance in a Condensed Configuration Space

We present a biologically inspired approach for path planning with dynamic obstacle avoidance. Path planning is performed in a condensed configuration space of a robot generated by self-organizing neural networks (SONN). The robot itself and static as well as dynamic obstacles are mapped from the Cartesian task space into the configuration space by precomputed kinematics. The condensed space represents a cognitive map of the environment, which is inspired by place cells and the concept of cognitive maps in mammalian brains. The generation of training data as well as the evaluation were performed on a real industrial robot accompanied by simulations. To evaluate the reactive collision-free online planning within a changing environment, a demonstrator was realized. Then, a comparative study regarding sample-based planners was carried out. So we could show that the robot is able to operate in dynamically changing environments and re-plan its motion trajectories within impressing 0.02 seconds, which proofs the real-time capability of our concept.

cs.RO↗

arXiv2022

Comparing SONN Types for Efficient Robot Motion Planning in the Configuration Space

Motion planning in the configuration space (C-space) induces benefits, such as smooth trajectories. It becomes more complex as the degrees of freedom (DOF) increase. This is due to the direct relation between the dimensionality of the search space and the DOF. Self-organizing neural networks (SONN) and their famous candidate, the Self-Organizing Map, have been proven to be useful tools for C-space reduction while preserving its underlying topology, as presented in [29]. In this work, we extend our previous study with additional models and adapt the approach from human motion data towards robots' kinematics. The evaluation includes the best performant models from [29] and three additional SONN architectures, representing the consequent continuation of this previous work. Generated Trajectories, planned with the different SONN models, were successfully tested in a robot simulation.

cs.RO↗

arXiv2021

A simplified Newton method to generate snapshots for POD models of semilinear optimal control problems

In PDE-constrained optimization, proper orthogonal decomposition (POD) provides a surrogate model of a (potentially expensive) PDE discretization, on which optimization iterations are executed. Because POD models usually provide good approximation quality only locally, they have to be updated during optimization. Updating the POD model is usually expensive, however, and therefore often impossible in a model-predictive control (MPC) context. Thus, reduced models of mediocre quality might be accepted. We take the view of a simplified Newton method for solving semilinear evolution equations to derive an algorithm that can serve as an offline phase to produce a POD model. Approaches that build the POD model with impulse response snapshots can be regarded as the first Newton step in this context. In particular, POD models that are based on impulse response snapshots are extended by adding a second simplified Newton step. This procedure improves the approximation quality of the POD model significantly by introducing a moderate amount of extra computational costs during optimization or the MPC loop. We illustrate our findings with an example satisfying our assumptions.

math.OC↗

arXiv2021

On the Characterization of Generalized Derivatives for the Solution Operator of the Bilateral Obstacle Problem

We consider optimal control problems for a wide class of bilateral obstacle problems where the control appears in a possibly nonlinear source term. The non-differentiability of the solution operator poses the main challenge for the application of efficient optimization methods and the characterization of Bouligand generalized derivatives of the solution operator is essential for their theoretical foundation and numerical realization. In this paper, we derive specific elements of the Bouligand generalized differential if the control operator satisfies natural monotonicity properties. We construct monotone sequences of controls where the solution operator is Gâteaux differentiable and characterize the corresponding limit element of the Bouligand generalized differential as being the solution operator of a Dirichlet problem on a quasi-open domain. In contrast to a similar recent result for the unilateral obstacle problem [RU19], we have to deal with an opposite monotonic behavior of the active and strictly active sets corresponding to the upper and lower obstacle. Moreover, the residual is no longer a nonnegative functional on H^{-1} and its representation as the difference of two nonnegative Radon measures requires special care. This necessitates new proof techniques that yield two elements of the Bouligand generalized differential. Also for the unilateral case we obtain an additional element to that derived in [RU19].

math.OC↗

arXiv2021

Differentiability results and sensitivity calculation for optimal control of incompressible two-phase Navier-Stokes equations with surface tension

We analyze optimal control problems for two-phase Navier-Stokes equations with surface tension. Based on $L_p$-maximal regularity of the underlying linear problem and recent well-posedness results of the problem for sufficiently small data we show the differentiability of the solution with respect to initial and distributed controls for appropriate spaces resulting form the $L_p$-maximal regularity setting. We consider first a formulation where the interface is transformed to a hyperplane. Then we deduce differentiability results for the solution in the physical coordinates. Finally, we state an equivalent Volume-of-Fluid type formulation and use the obtained differentiability results to derive rigorosly the corresponding sensitivity equations of the Volume-of-Fluid type formulation. For objective functionals involving the velocity field or the discontinuous pressure or phase indciator field we derive differentiability results with respect to controls and state formulas for the derivative. The results of the paper form an analytical foundation for stating optimality conditions, justifying the application of derivative based optimization methods and for studying the convergence of discrete sensitivity schemes based on Volume-of-Fluid discretizations for optimal control of two-phase Navier-Stokes equations.

math.AP↗

arXiv2020

Identification of Model Uncertainty via Optimal Design of Experiments Applied to a Mechanical Press

In engineering applications almost all processes are described with the help of models. Especially forming machines heavily rely on mathematical models for control and condition monitoring. Inaccuracies during the modeling, manufacturing and assembly of these machines induce model uncertainty which impairs the controller's performance. In this paper we propose an approach to identify model uncertainty using parameter identification, optimal design of experiments and hypothesis testing. The experimental setup is characterized by optimal sensor positions such that specific model parameters can be determined with minimal variance. This allows for the computation of confidence regions in which the real parameters or the parameter estimates from different test sets have to lie. We claim that inconsistencies in the estimated parameter values, considering their approximated confidence ellipsoids as well, cannot be explained by data uncertainty but are indicators of model uncertainty. The proposed method is demonstrated using a component of the 3D Servo Press, a multi-technology forming machine that combines spindles with eccentric servo drives.

stat.ML↗

arXiv2019

A Certified Model Reduction Approach for Robust Parameter Optimization with PDE Constraints

We investigate an optimization problem governed by an elliptic partial differential equation with uncertain parameters. We introduce a robust optimization framework that accounts for uncertain model parameters. The resulting non-linear optimization problem has a bi-level structure due to the min-max formulation. To approximate the worst-case in the optimization problem we propose linear and quadratic approximations. However, this approach still turns out to be very expensive, therefore we propose an adaptive model order reduction technique which avoids long offline stages and provides a certified reduced order surrogate model for the parametrized PDE which is then utilized in the numerical optimization. Numerical results are presented to validate the presented approach.

math.OC↗

arXiv2019

Semi-automatically optimized calibration of internal combustion engines

Modern combustion engines incorporate a number of actuators and sensors that can be used to control and optimize the performance and emissions. We describe a semi-automatic method to simultaneously measure and calibrate the actuator settings and the resulting behavior of the engine. The method includes an adaptive process for refining the measurements, a data cleaning step, and an optimization procedure. The optimization works in a discretized space and incorporates the conditions to describe the dependence between the actuators and the engine behavior as well as emission bounds. We demonstrate our method on practical examples.

math.OC↗

arXiv2018

Robust Optimization Approaches for the Design of an Electric Machine

In this paper two formulations for the robust optimization of the size of the permanent magnet in a synchronous machine are discussed. The optimization is constrained by a partial differential equation to describe the electromagnetic behavior of the machine. The need for a robust optimization procedure originates from the fact that optimization parameters have deviations. The first approach, i.e., \textcolor{red}{worst-case} optimization, makes use of local sensitivities. The second approach takes into account expectation values and standard deviations. The latter are associated with global sensitivities. The geometry parametrization is elegantly handled thanks to the introduction of an affine decomposition. Since the stochastic quantities are determined by tools from uncertainty quantification (UQ) and thus require a lot of finite element evaluations, model order reduction is used in order to increase the efficiency of the procedure. It is shown that both approaches are equivalent if a linearization is carried out. \textcolor{blue}{This finding is supported by the application on an electric machine. The optimization algorithms used are sequential quadratic programming, particle swarm optimization and genetic algorithm}. While both formulations reduce the size of the magnets, the UQ based optimization approach is less pessimistic with respect to deviations and yields smaller magnets.

math.OC↗