The 10 most recently published documents
We study the existence, stability and optimal control of classical solutions of networked strictly hyperbolic systems of balance laws. Such networks arise, for instance, in the modeling of the gas dynamics in a network of gas pipelines, traffic networks, or networks of water channels.
It is assumed that characteristic speeds are nonzero and do not change sign. Node conditions are stated in a general way by requiring that the boundary traces of all states adjacent to a vertex satisfy an algebraic equation. With a suitable assumption, this equation can be solved for outgoing characteristic variables as a function of the incoming ones.
We prove the unique existence of a classical solution for arbitrarily large initial and boundary values on a possibly small time horizon. We derive continuity and differentiability properties of the solution operator of the quasilinear problem w.r.t. the control term located in the node conditions. Then we analyze an optimal control problem for the networked system, where the control is of boundary type. We prove the existence of optimal controls and the differentiability of the objective functional w.r.t. the controls.
Branch-and-cut for mixed-integer robust chance-constrained optimization with discrete distributions
(2025)
We study robust chance-constrained problems with mixed-integer design variables and ambiguity sets consisting of discrete probability distributions. Allowing general non-convex constraint functions, we develop a branch-and-cut framework using scenario-based cutting planes to generate lower bounds. The cutting planes are obtained by exploiting the classical big-M reformulation of the chance-constrained problem in the case of discrete distributions. Furthermore, we include the calculation of initial feasible solutions based on a bundle method applied to an approximation of the original problem into the branch-and-cut procedure. We conclude with a detailed discussion about the practical performance of the branch-and-cut framework with and without initial feasible solutions. In our experiments we focus on gas transport problems under uncertainty and provide a comparison of our method with solving the classical reformulation directly for various real-world sized instances.
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.
We propose a novel online learning framework for robust Bayesian optimization of uncertain black-box functions. While Bayesian optimization is well-suited for data-efficient optimization of expensive objectives, its standard form can be sensitive to hidden or varying parameters. To address this issue, we consider a min–max robust counterpart of the optimization problem and develop a practically efficient solution algorithm, BROVER (Bayesian Robust Optimization via Exploration with Regret minimization). Our method combines Gaussian process regression with a decomposition approach: the minimax structure is split into a non-convex online learner based on the Follow-the-Perturbed-Leader algorithm together with a subsequent minimization step in the decision variables. We prove that the theoretical regret bound converges under mild assumptions, ensuring asymptotic convergence to robust solutions. Numerical experiments on synthetic data validate the regret guarantees and demonstrate fast convergence to the robust optimum. Furthermore, we apply our method to the robust optimization of organic solar cell performance, where hidden process parameters and experimental variability naturally induce uncertainty. Our results on real-world datae show that BROVER identifies solutions with strong robustness properties within relatively few iterations, thereby offering a modern and practical approach for data-driven black-box optimization under uncertainty.
We prove an existence result for the steady state flow of gas mixtures
on networks. The basis of the model are the physical principles of the isothermal
Euler equation, coupling conditions for the flow and pressure, and the mixing of
incoming flow at nodes. The state equation is based on a convex combination of
the ideal gas equations of state for natural gas and hydrogen. We analyze mathematical
properties of the model allowing us to prove the existence of solutions in
particular for tree-shaped networks and networks with exactly one cycle. Numerical
examples illustrate the results and explore the applicability of our approach
to different network topologies.
In this paper, topological derivatives are defined and employed for gas transport networks
governed by nonlinear hyperbolic systems of PDEs. The concept of topological derivatives of a shape functional is introduced for optimum design and control of gas networks. First, the dynamic model for the network is considered. The cost for the control problem includes the deviations of the pressure at the inflow and outflow nodes. For dynamic control problems of gas networks when the turnpike property occurs, the synthesis of control and optimum design of the network can be simplified. That is, the design of the network can be performed for optimal control of the steady-state network model. The cost of design is defined by the optimal control cost for the steady-state network model. The topological derivative of the design cost, given by the optimal control cost with respect to the nucleation of a small cycle, is
determined. Tree-structured networks can be decomposed into single network junctions. The topological derivative of the design cost is systematically evaluated at each junction of the decomposed network. This allows for the identification of internal nodes with negative topological derivatives, where replacing the node with a small cycle leads to an improved design cost. As the set of network junctions is finite, the iterative procedure is convergent. This design procedure is applied to representative examples and it can be generalized to arbitrary network graphs. A key feature of such modeling approach is the availability of exact steady-state solutions, enabling a fully analytical topological analysis of the design cost without numerical approximations.
Uncertainty plays a crucial role in modeling and optimization of complex systems across various applications. In this paper, uncertain gas transport through pipeline networks is considered and a novel strategy to measure the robustness of deterministically computed compressor and valve controls, the probabilistic robustness, is presented.
\noindent Initially, an optimal control for a deterministic gas network problem is computed such that the total control cost is minimized with respect to box constraints for the pressure. Subsequently, the model is perturbed by uncertain gas demands. The probability, that the uncertain gas pressures - based on the a priori deterministic optimal control - satisfy the box constraints, is evaluated. Moreover, buffer zones are introduced in order to tighten the pressure bounds in the deterministic scenario. Optimal controls for the deterministic scenario with buffer zones are also applied to the uncertain scenario, allowing to analyze the impact of the buffer zones on the probabilistic robustness of the optimal controls.
\noindent For the computation of the probability, we apply a kernel density estimator based on samples of the uncertain pressure at chosen locations. In order to reduce the computational effort of generating the samples, we combine the kernel density estimator approach with a stochastic collocation method which approximates the pressure at the chosen locations in the stochastic space. Finally, we discuss generalizations of the probabilistic robustness check and we present numerical results for a gas network taken from the public gas library.
Constructing ambiguity sets in distributionally robust optimization is difficult and currently receives increased attention. In this paper, we focus on mixture models with finitely many reference distributions. We present two different solution concepts for robust joint chance-constrained optimization problems with these ambiguity sets and non-convex constraint functions. Both concepts rely on solving an approximation problem that is based on well-known smoothing and penalization techniques. On the one side, we consider a classical bundle method together with an approach for finding good starting points. On the other side, we integrate the Continuous Stochastic Gradient method, a variant of the stochastic gradient descent that is able to exploit regularity in the data. On the example of gas networks we compare the two algorithmic concepts for different topologies and two types of mixture ambiguity sets with Gaussian reference distributions and polyhedral and ϕ-divergence based feasible sets for the mixing coefficients. The results show that both solution approaches are well-suited to solve this difficult problem class. Based on the numerical results we provide some general advices for choosing the more efficient algorithm depending on the main challenges of the considered optimization problem. We give an outlook for the applicability of the method in a wider context.
We consider an incremental variant of the rooted prize-collecting Steiner-tree problem with a growing budget constraint. While no incremental solution exists that simultaneously approximates the optimum for all budgets, we show that a bicriterial (α,μ)-approximation is possible, i.e., a solution that with budget B+α for all B∈R≥0 is a multiplicative μ-approximation compared to the optimum solution with budget B. For the case that the underlying graph is a tree, we present a polynomial-time density-greedy algorithm that computes a (χ,1)-approximation, where χ denotes the eccentricity of the root vertex in the underlying graph, and show that this is best possible. An adaptation of the density-greedy algorithm for general graphs is (γ,2)-competitive where γ is the maximal length of a vertex-disjoint path starting in the root. While this algorithm does not run in polynomial time, it can be adapted to a (γ,3)-competitive algorithm that runs in polynomial time. We further devise a capacity-scaling algorithm that guarantees a (3χ,8)-approximation and, more generally, a ((4ℓ−1)χ,(2^(ℓ+2))/(2^ℓ−1))-approximation for every fixed ℓ∈N.
A posteriori error control for a finite volume scheme for a cross-diffusion model of ion transport
(2025)
We derive a reliable a posteriori error estimate for a cell-centered finite volume scheme approximating a cross-diffusion system modeling ion transport through nanopores. To this end we derive an abstract stability framework that is independent of the numerical scheme and introduce a suitable (conforming) reconstruction of the numerical solution. The stability framework relies on some simplifying assumption that coincide with those made in weak uniqueness results for this system. This is the first a posteriori error estimate for a cross-diffusion system. Along the way, we derive a pointwise a posteriori error estimate for a finite volume scheme approximating the diffusion equation. We conduct numerical experiments showing that the error estimator scales with the same order as the true error.