Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics
Not a member yet
7600 research outputs found
Sort by
On a Differential Generalized Nash Equilibrium Problem with Mean Field Interaction
No full textWe consider a class of N-player linear quadratic differential generalized Nash equilibrium problems (GNEPs) with bound constraints on the individual control and state variables. In addition, we assume the individual players’ optimal control problems are coupled through their dynamics and objectives via a time-dependent mean field interaction term. This assumption allows us to model the realistic setting that strategic players in large games cannot observe the individual states of their competitors. We observe that the GNEPs require a constraint qualification, which necessitates sufficient robustness of the individuals, in order to prove the existence of an open-loop pure strategy Nash equilibrium and to derive optimality conditions. In order to gain qualitative insight into the N-player game, we assume that players are identical and pass to the limit in N to derive a type of first-order constrained mean field game (MFG). We prove that the mean field interaction terms converge to an absolutely continuous curve of probability measures on the set of possible state trajectories. Using variational convergence methods, we show that the optimal control problems converge to a representative agent problem. Under additional regularity assumptions, we provide an explicit form for the mean field term as the solution of a continuity equation and demonstrate the link back to the N-player GNEP
Data‐Driven Methods for Quantitative Imaging
No full textIn the field of quantitative imaging, the image information at a pixel or voxel in an underlying domain entails crucial information about the imaged matter. This is particularly important in medical imaging applications, such as quantitative magnetic resonance imaging (qMRI), where quantitative maps of biophysical parameters can characterize the imaged tissue and thus lead to more accurate diagnoses. Such quantitative values can also be useful in subsequent, automatized classification tasks in order to discriminate normal from abnormal tissue, for instance. The accurate reconstruction of these quantitative maps is typically achieved by solving two coupled inverse problems which involve a (forward) measurement operator, typically ill-posed, and a physical process that links the wanted quantitative parameters to the reconstructed qualitative image, given some underlying measurement data. In this review, by considering qMRI as a prototypical application, we provide a mathematically-oriented overview on how data-driven approaches can be employed in these inverse problems eventually improving the reconstruction of the associated quantitative maps
Trotter-Kato Product Formulæ
No full textThe book captures a fascinating snapshot of the current state of results about the operator-norm convergent Trotter-Kato Product Formulæ on Hilbert and Banach spaces. It also includes results on the operator-norm convergent product formulæ for solution operators of the non-autonomous Cauchy problems as well as similar results on the unitary and Zeno product formulæ.After the Sophus Lie product formula for matrices was established in 1875, it was generalised to Hilbert and Banach spaces for convergence in the strong operator topology by H. Trotter (1959) and then in an extended form by T. Kato (1978). In 1993 Dzh. L. Rogava discovered that convergence of the Trotter product formula takes place in the operator-norm topology. The latter is the main subject of this book, which is dedicated essentially to the operator-norm convergent Trotter-Kato Product Formulæ on Hilbert and Banach spaces, but also to related results on the time-dependent, unitary and Zeno product formulæ.The book yields a detailed up-to-date introduction into the subject that will appeal to any reader with a basic knowledge of functional analysis and operator theory. It also provides references to the rich literature and historical remarks
Traveling fronts in a reaction–diffusion equation with a memory term
No full textBased on a recent work on traveling waves in spatially nonlocal reaction–diffusion equations, we investigate the existence of traveling fronts in reaction–diffusion equations with a memory term. We will explain how such memory terms can arise from reduction of reaction–diffusion systems if the diffusion constants of the other species can be neglected. In particular, we show that two-scale homogenization of spatially periodic systems can induce spatially homogeneous systems with temporal memory. The existence of fronts is proved using comparison principles as well as a reformulation trick involving an auxiliary speed that allows us to transform memory terms into spatially nonlocal terms. Deriving explicit bounds and monotonicity properties of the wave speed of the arising traveling front, we are able to establish the existence of true traveling fronts for the original problem with memory. Our results are supplemented by numerical simulations
Curvature effects in pattern formation: Well-posedness and optimal control of a sixth-order Cahn--Hilliard equation
No full textThis work investigates the well-posedness and optimal control of a sixth-order Cahn--Hilliard equation, a higher-order variant of the celebrated and well-established Cahn–Hilliard equation. The equation is endowed with a source term, where the control variable enters as a distributed mass regulator. The inclusion of additional spatial derivatives in the sixth-order formulation enables the model to capture curvature effects, leading to a more accurate depiction of isothermal phase separation dynamics in complex materials systems. We provide a well-posedness result for the aforementioned system when the corresponding nonlinearity of double-well shape is regular and then analyze a corresponding optimal control problem. For the latter, existence of optimal controls is established, and the first-order necessary optimality conditions are characterized via a suitable variational inequality. These results aim at contributing to improving the understanding of the mathematical properties and control aspects of the sixth-order Cahn–Hilliard equation, offering potential applications in the design and optimization of materials with tailored microstructures and properties
On a Cahn–Hilliard system with source term and thermal memory
No full textA nonisothermal phase field system of Cahn--Hilliard type is introduced and analyzed mathematically. The system constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn--Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter is lost. We provide several mathematical results under general assumptions on the source term and the double-well nonlinearity governing the evolution: existence and continuous dependence results are shown for weak and strong solutions to the corresponding initial-boundary value problem
Two-norm discrepancy and convergence of the stochastic gradient method with application to shape optimization
Get PDFThe present article is dedicated to proving convergence of the stochastic gradient method in case of random shape optimization problems. To that end, we consider Bernoulli's exterior free boundary problem with a random interior boundary. We recast this problem into a shape optimization problem by means of the minimization of the expected Dirichlet energy. By restricting ourselves to the class of convex, sufficiently smooth domains of bounded curvature, the shape optimization problem becomes strongly convex with respect to an appropriate norm. Since this norm is weaker than the differentiability norm, we are confronted with the so-called two-norm discrepancy, a well-known phenomenon from optimal control. We therefore need to adapt the convergence theory of the stochastic gradient method to this specific setting correspondingly. The theoretical findings are supported and validated by numerical experiments
Modeling and simulation of an isolated mini-grid including battery operation strategies under uncertainty using chance constraints
No full textThis paper addresses the challenge of handling uncertainties in mini-grid operation, crucial for achiev- ing universal access to reliable and sustainable energy, especially in regions lacking access to a national grid. Mini-grids, consisting of small-scale power generation systems and distribution infrastructure, offer a cost-effective solution. However, the intermittency and uncertainty of renewable energy sources poses chal- lenges, mitigated by employing batteries for energy storage. Optimizing the lifespan of the battery energy storage system is critical, requiring a balance between degradation and operational expenses, with battery operation strategies playing a key role in achieving this balance. Accounting for uncertainties in renewable energy sources, demand, and ambient temperature is essential for reliable energy management strategies. By formulating a probabilistic optimal control problem for minimizing the daily operational costs of stand- alone mini-grids under uncertainty, and exploiting the concept of joint chance constraints, we address the uncertainties inherent in battery dynamics and the associated operational constraints
Improving hp-variational physics-informed neural networks for steady-state convection-dominated problems
Get PDFThis paper proposes and studies two extensions of applying hp-variational physics-informed neural networks, more precisely the FastVPINNs framework, to convection-dominated convection-diffusion-reaction problems. First, a term in the spirit of a SUPG stabilization is included in the loss functional and a network architecture is proposed that predicts spatially varying stabilization parameters. Having observed that the selection of the indicator function in hard-constrained Dirichlet boundary conditions has a big impact on the accuracy of the computed solutions, the second novelty is the proposal of a network architecture that learns good parameters for a class of indicator functions. Numerical studies show that both proposals lead to noticeably more accurate results than approaches that can be found in the literature
Three-State P-SOS Models on Binary Cayley Trees
No full textWe consider a version of the solid-on-solid model on the Cayley tree of order two in which vertices carry spins of value or 2 and the pairwise interaction of neighboring vertices is given by their spin difference to the power p > 0. We exhibit all translation-invariant splitting Gibbs measures (TISGMs) of the model and demonstrate the existence of up to seven such measures, depending on the parameters. We further establish general conditions for extremality and non-extremality of TISGMs in the set of all Gibbs measures and use them to examine selected TISGMs for a small and a large p. Notably, our analysis reveals that extremality properties are similar for large p compared to the case p = 1, a case that has been explored already in previous work. However, for the small p, certain measures that were consistently non-extremal for p = 1 do exhibit transitions between extremality and non-extremality