arXiv subjects

Lukas Pflug

Publications and source records attributed to Lukas Pflug.

At least 19 recordsLinked to original sources

arXiv2026

A general framework for nonlocal traffic flow models on networks

We present a general computational framework for macroscopic nonlocal traffic flow models on networks with multiple commodities. The model combines scalar conservation laws on edges with nonlocal velocity functions and couples them via buffer-based junction dynamics. Routing is defined in general as a prescription for distributing drivers across outgoing road segments. As an example, we implement dynamic k-shortest-path routing, where travel times along roads and waiting times at intersections are used to compute shortest paths to the commodities' destinations at each time step, and drivers are distributed accordingly. In another example, we optimize routing over a considered time horizon to minimize the total travel time. This framework naturally creates a feedback loop between traffic evolution and route choice. Numerical examples, ranging from small test cases to large grid-like networks, demonstrate the robustness of the approach and allow for a comparison of different routing strategies.

math.OC↗

arXiv2026

The obstacle problem for scalar conservation laws with nonlocal dynamics

In this article, we present a method to find a solution to a one-dimensional nonlocal conservation law that respects a space-dependent mapping, referred to as the obstacle. This is achieved by generalizing existing results for the local conservation law: We consider a relaxation of the velocity, that explicitly depends on the obstacle. We prove existence of solutions to the relaxed problem and show that, as the relaxation mapping converges to a Heaviside-type function, the corresponding solutions converge to a weak solution of a discontinuous nonlocal conservation law. Moreover, we can characterize the limiting flux in several cases.\\ The paper concludes with a numerical study that illustrates the aforementioned convergence.

math.AP↗

arXiv2026

Nonlocal Approximation Principle for Entropy Solutions of Scalar Conservation Laws

We establish a general nonlocal approximation principle for the entropy solutions of scalar conservation laws on $\mathbb{R}$. More precisely, we show that the entropy solution to a nonnegative initial datum can be obtained as a weak-star limit of a corresponding scalar nonlocal conservation law. The flux function of the nonlocal conservation law depends on suitable spatial averages of the density. The proof is based on a reformulation on the Hamilton--Jacobi level: working with the primitives, we identify the limit via the stability properties of viscosity solutions; we then recover the entropy solution using the classical relation between Hamilton--Jacobi equations and scalar conservation laws. We further show that the approximation extends, after a suitable shift, to sign-changing initial data, and we prove a quantitative convergence estimate for convex fluxes in terms of the first moments of the nonlocal kernels. This result makes it possible to define entropy solutions for general fluxes using their nonlocal approximations, which satisfy the requirement for a finite speed of mass propagation, a key feature of hyperbolic conservation laws.

math.AP↗

arXiv2026

On The Convergence of the Discretized Linear Static State-Based Peridynamic Equations

In this paper, the convergence of the solutions for a discretized linear state-based static peridynamic system to the corresponding continuous solution is analytically proven. To obtain an implementable model, we further apply one-point-quadrature to the terms in the discrete equations. The resulting system coincides with the commonly used meshfree discretization using a regular lattice, including the possibility of using partial area algorithms to improve the numerical behavior. We again prove convergence, this time for fixed choices of a weighting function commonly used in literature and stronger assumptions on the input data. We note however, that these assumptions are not significantly restrictive for practical purposes. In particular, they still allow discontinuities in the material parameters and external body forces.

math.NA↗

arXiv2025

Nonlocal conservation laws with p-norm, the singular limit problem and applications to traffic flow

In this contribution, we study scalar nonlocal conservation laws with the $p$-norm. Here, 'nonlocal' means that the velocity of the conservation law depends on an integral term in space. Typically, the nonlocal term consists of integrating the solution in $L^{1}$, whereas here we will study the case when the solution is integrated in the $L^{p}$-norm. We consider even the case of the $L^{p}$ metric when $p\in (0,1)$ and establish, for an initial datum which is uniformly bounded away from zero, the existence and uniqueness of weak solutions. We then demonstrate that there are also solutions to the initial datum being zero under more restrictive assumptions. Furthermore, we investigate the singular limit, i.e., what happens when the nonlocal kernel converges to a Dirac distribution. Indeed, for the one-sided exponential kernel, we recover the (entropy) solution of the corresponding local conservation law for all $p\in(0,\infty)$ with further restrictions for $p\in(0,1)$. This generalizes the celebrated singular limit result for nonlocal conservation laws for $p=1$ significantly and showcases the robustness of the approximation of local conservation laws by nonlocal ones. We investigate also the monotonicity of the solution when assuming that the initial datum is monotone. Finally, we prove the convergence of solutions for $p\rightarrow 0$ on a small time horizon, resulting in a different kind of nonlocal conservation law. Numerical studies showcasing the effect of $p$ on the singular limit convergence and more conclude the contribution.

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arXiv2025

Optimal Control Problems with Nonlocal Conservation Laws: Existence of Optimizers and Singular Limits in Approximations of Local Conservation Laws

This contribution considers optimal control problems subject to nonlocal conservation laws -- those in which the velocity depends nonlocally (i.e., via a convolution) on the solution -- and the so-called singular limit. First, the existence of minimizers is demonstrated for a broad class of optimal control problems, involving optimization over the initial datum, velocity, and nonlocal kernel for classical tracking-type $L^2$ cost functionals. Then, it is proven that the obtained minimizers converge to minimizers of the corresponding local optimal control problem when the kernel function of the convolution is of exponential type and approaches a Dirac distribution. Finally, some numerical results are presented.

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arXiv2024

A stochastic method of moving asymptotes for topology optimization under uncertainty

Topology optimization under uncertainty or reliability-based topology optimization is usually numerically very expensive. This is mainly due to the fact that an accurate evaluation of the probabilistic model requires the system to be simulated for a large number of varying parameters. Traditional gradient-based optimization schemes thus face the difficulty that reasonable accuracy and numerical efficiency often seem mutually exclusive. In this work, we propose a stochastic optimization technique to tackle this problem. To be precise, we combine the well-known method of moving asymptotes (MMA) with a stochastic sample-based integration strategy. By adaptively recombining gradient information from previous steps, we obtain a noisy gradient estimator that is asymptotically correct, i.e., the approximation error vanishes over the course of iterations. As a consequence, the resulting stochastic method of moving asymptotes (sMMA) allows us to solve chance constraint topology optimization problems for a fraction of the cost compared to traditional approaches from literature. To demonstrate the efficiency of sMMA, we analyze structural optimization problems in two and three dimensions.

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arXiv2024

T-matrix representation of optical scattering response: Suggestion for a data format

The transition matrix, frequently abbreviated as T-matrix, contains the complete information in a linear approximation of how a spatially localized object scatters an incident field. The T-matrix is used to study the scattering response of an isolated object and describes the optical response of complex photonic materials made from ensembles of individual objects. T-matrices of certain common structures, potentially, have been repeatedly calculated all over the world again and again. This is not necessary and constitutes a major challenge for various reasons. First, the resources spent on their computation represent an unsustainable financial and ecological burden. Second, with the onset of machine learning, data is the gold of our era, and it should be freely available to everybody to address novel scientific challenges. Finally, the possibility of reproducing simulations could tremendously improve if the considered T-matrices could be shared. To address these challenges, we found it important to agree on a common data format for T-matrices and to enable their collection from different sources and distribution. This document aims to develop the specifications for storing T-matrices and associated metadata. The specifications should allow maximum freedom to accommodate as many use cases as possible without introducing any ambiguity in the stored data. The common format will assist in setting up a public database of T-matrices.

physics.optics↗

arXiv2024

The obstacle problem for linear scalar conservation laws with constant velocity

In this contribution, we present a novel approach for solving the obstacle problem for (linear) conservation laws. Usually, given a conservation law with an initial datum, the solution is uniquely determined. How to incorporate obstacles, i.e., inequality constraints on the solution so that the resulting solution is still "physically reasonable" and obeys the obstacle, is unclear. The proposed approach involves scaling down the velocity of the conservation law when the solution approaches the obstacle. We demonstrate that this leads to a reasonable solution and show that, when scaling down is performed in a discontinuous fashion, we still obtain a suitable velocity - and the solution satisfying a discontinuous conservation law. We illustrate the developed solution concept using numerical approximations.

math.AP↗

arXiv2024

A nonlocal approach to graded surface modeling in topology optimization

Additively manufactured structures often exhibit a correlation between their mechanical properties, such as stiffness, strength, and porosity, and their wall thickness. This correlation stems from the interplay between the manufacturing process and the properties of the filler material. In this study, we investigate the thickness-dependent effect on structural stiffness and propose a nonlocal integral model that introduces surface grading of Young's modulus to capture this phenomenon. We incorporate this model into topology optimization for designing structures with optimized compliance subject to a volume constraint. Notably, elastically degraded surfaces penalize excessively thin features, effectively eliminating them from the optimized design. We showcase the efficacy of our proposed framework by optimizing the design of a two-dimensional cantilever beam and a bridge.

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arXiv2023

A note on nonlocal approximations of sign-unrestricted solutions of conservation laws

We study the singular limit problem for nonlocal conservation laws in which the sign of the initial datum is unrestricted and the velocity of the conservation law depends on a nonlocal approximation of the absolute value of the density. We demonstrate that the nonlocal solutions converge to the local entropy solution when the nonlocal kernel tends to a Dirac distribution, and thus obtain an approximation result for local unsigned conservation laws, generalizing the current results on the so-called sign-restricted singular limit problem. The considered model class covers special cases like a generalized Burgers' equation and scalar versions of the Keyfitz--Kranzer system.

math.AP↗

arXiv2023

On the singular limit problem for nonlocal conservation laws: A general approximation result for kernels with fixed support

We prove the convergence of solutions of nonlocal conservation laws to their local entropic counterpart for a fundamentally extended class of nonlocal kernels when these kernels approach a Dirac distribution. The nonlocal kernels are assumed to have fixed support and do not have to be monotonic. With sharp estimates of the nonlocal kernels and a surrogate nonlocal quantity, we prove compactness in $C(L^{1}_{\text{loc}})$ which allow passing to the limit in the weak formulation. A careful analysis of the entropy condition of local conservation laws together with the named estimators for the considered kernels enable it to prove the entropy admissibility of the nonlocal equation in the limit, completing the convergence proof.

math.AP↗

arXiv2023

Topology Optimization of Broadband Acoustic Transition Section: A Comparison between Deterministic and Stochastic Approaches

This paper focuses on the topology optimization of a broadband acoustic transition section that connects two cylindrical waveguides with different radii. The primary objective is to design a transition section such that it maximizes the transmission of a planar acoustic wave while ensuring the planarity of the transmitted wave. Helmholtz equation is used to model linear wave propagation in the device.We utilize the finite element method to solve the state equation on a structured mesh of square elements. Subsequently, a material distribution topology optimization problem is formulated to optimize the distribution of sound-hard material in the transition section. We employ two different gradient-based approaches to solve the optimization problem: namely, a deterministic approach using the method of moving asymptotes (MMA), and a stochastic approach utilizing both stochastic gradient (SG) and continuous stochastic gradient (CSG) methods. A comparative analysis is provided among these methodologies concerning the design feasibility and the transmission performance of the optimized designs, and the computational efficiency. The outcomes highlight the effectiveness of stochastic techniques in achieving enhanced broadband acoustic performance with reduced computational demands and improved design practicality. The insights from this investigation demonstrate the potential of stochastic approaches in acoustic applications, especially when broadband acoustic performance is desired.

math.OC↗

arXiv2023

Quality Control in Particle Precipitation via Robust Optimization

In this work, we propose a robust optimization approach to mitigate the impact of uncertainties in particle precipitation. Our model incorporates partial differential equations, more particular nonlinear and nonlocal population balance equations to describe particle synthesis. The goal of the optimization problem is to design products with desired size distributions. Recognizing the impact of uncertainties, we extend the model to hedge against them. We emphasize the importance of robust protection to ensure the production of high-quality particles. To solve the resulting robust problem, we enhance a novel adaptive bundle framework for nonlinear robust optimization that integrates the exact method of moments approach for solving the population balance equations. Computational experiments performed with the integrated algorithm focus on uncertainties in the total mass of the system as it greatly influence the quality of the resulting product. Using realistic parameter values for quantum dot synthesis, we demonstrate the efficiency of our integrated algorithm. Furthermore, we find that the unprotected process fails to achieve the desired particle characteristics, even for small uncertainties, which highlights the necessity of the robust process. The latter consistently outperforms the unprotected process in quality of the obtained product, in particular in perturbed scenarios.

math.OC↗

arXiv2023

Exact Method of Moments for multi-dimensional population balance equations

The unique properties of anisotropic and composite particles are increasingly being leveraged in modern particulate products. However, tailored synthesis of particles characterized by multi-dimensional dispersed properties remains in its infancy and few mathematical models for their synthesis exist. Here, we present a novel, accurate and highly efficient numerical approach to solve a multi-dimensional population balance equation, based on the idea of the exact method of moments for nucleation and growth \cite{pflug2020emom}. The transformation of the multi-dimensional population balance equation into a set of one-dimensional integro-differential equations allows us to exploit accurate and extremely efficient numerical schemes that markedly outperform classical methods (such as finite volume type methods) which is outlined by convergence tests. Our approach not only provides information about complete particle size distribution over time, but also offers insights into particle structure. The presented scheme and its performance is exmplified based on coprecipitation of nanoparticles. For this process, a generic growth law is derived and parameter studies as well as convergence series are performed.

math.AP↗

arXiv2023

Ole\u{\i}nik-type estimates for nonlocal conservation laws and applications to the nonlocal-to-local limit

We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term $W:=\mathbb{1}_{(-\infty,0]}(\cdot)\exp(\cdot) \ast \rho$ satisfy an Ole\u{\i}nik-type entropy condition. More precisely, under different sets of assumptions on the velocity function $V$, we prove that $W$ satisfies a one-sided Lipschitz condition and that $V'(W) W \partial_x W$ satisfies a one-sided bound, respectively. As a byproduct, we deduce that, as the exponential kernel is rescaled to converge to a Dirac delta distribution, the weak solution of the nonlocal problem converges to the unique entropy-admissible solution of the corresponding local conservation law, under the only assumption that the initial datum is essentially bounded and not necessarily of bounded variation.

math.AP↗

arXiv2023

On optimization of heterogeneous materials for enhanced resistance to bulk fracture

We propose a novel approach to optimize the design of heterogeneous materials, with the goal of enhancing their effective fracture toughness under mode-I loading. The method employs a Gaussian processes-based Bayesian optimization framework to determine the optimal shapes and locations of stiff elliptical inclusions within a periodic microstructure in two dimensions. To model crack propagation, the phase-field fracture method with an efficient interior-point monolithic solver and adaptive mesh refinement, is used. To account for the high sensitivity of fracture properties to initial crack location with respect to heterogeneities, we consider multiple cases of initial crack and optimize the material for the worst-case scenario. We also impose a minimum clearance constraint between the inclusions to ensure design feasibility. Numerical experiments demonstrate that the method significantly improves the fracture toughness of the material compared to the homogeneous case.

math.OC↗

arXiv2023

The Continuous Stochastic Gradient Method: Part II -- Application and Numerics

In this contribution, we present a numerical analysis of the continuous stochastic gradient (CSG) method, including applications from topology optimization and convergence rates. In contrast to standard stochastic gradient optimization schemes, CSG does not discard old gradient samples from previous iterations. Instead, design dependent integration weights are calculated to form a linear combination as an approximation to the true gradient at the current design. As the approximation error vanishes in the course of the iterations, CSG represents a hybrid approach, starting off like a purely stochastic method and behaving like a full gradient scheme in the limit. In this work, the efficiency of CSG is demonstrated for practically relevant applications from topology optimization. These settings are characterized by both, a large number of optimization variables \textit{and} an objective function, whose evaluation requires the numerical computation of multiple integrals concatenated in a nonlinear fashion. Such problems could not be solved by any existing optimization method before. Lastly, with regards to convergence rates, first estimates are provided and confirmed with the help of numerical experiments.

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