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    Matlab code to solve state and control constrained linear quadratic Mean-Field Games

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    oai:archive.wias-berlin.de:wias_mods_00007478This repository provides a Matlab code to approximate a state and control constrained linear quadratic Mean-Field Game. This code was developed with Matlab version R2023a in the context of the Ph.D. thesis of Mike Theiß. The theoretical background can be found in the thesis with the name "First Order Mean-Field Games and Mean-Field Optimal Control with State and Control Constraints", which is available at the bibliography of the Humboldt University

    Sharp-interface problem of the Ohta--Kawasaki model for symmetric diblock copolymers

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    The Ohta-Kawasaki model for diblock-copolymers is well known to the scientific community of diffuse-interface methods. To accurately capture the long-time evolution of the moving interfaces, we present a derivation of the corresponding sharp-interface limit using matched asymptotic expansions, and show that the limiting process leads to a Hele-Shaw type moving interface problem. The numerical treatment of the sharp-interface limit is more complicated due to the stiffness of the equations. To address this problem, we present a boundary integral formulation corresponding to a sharp interface limit of the Ohta-Kawasaki model. Starting with the governing equations defined on separate phase domains, we develop boundary integral equations valid for multi-connected domains in a 2D plane. For numerical simplicity we assume our problem is driven by a uniform Dirichlet condition on a circular far-field boundary. The integral formulation of the problem involves both double- and single-layer potentials due to the modified boundary condition. In particular, our formulation allows one to compute the nonlinear dynamics of a non-equilibrium system and pattern formation of an equilibrating system. Numerical tests on an evolving slightly perturbed circular interface (separating the two phases) are in excellent agreement with the linear analysis, demonstrating that the method is stable, efficient and spectrally accurate in space

    A stochastic gradient method for a class of nonlinear PDE-constrained optimal control problems under uncertainty

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    The study of optimal control problems under uncertainty plays an important role in scientific numerical simulations. This class of optimization problems is strongly utilized in engineering, biology and finance. In this paper, a stochastic gradient method is proposed for the numerical resolution of a nonconvex stochastic optimization problem on a Hilbert space. We show that, under suitable assumptions, strong or weak accumulation points of the iterates produced by the method converge almost surely to stationary points of the original optimization problem. Measurability and convergence rates of a stationarity measure are handled, filling a gap for applications to nonconvex infinite dimensional stochastic optimization problems. The method is demonstrated on an optimal control problem constrained by a class of elliptic semilinear partial differential equations (PDEs) under uncertainty

    Nonlinear Wasserstein distributionally robust optimal control

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    This paper presents a novel approach to addressing the distributionally robust nonlinear model predictive control (DRNMPC) problem. Current literature primarily focuses on the static Wasserstein distributionally robust optimal control problem with a prespecified ambiguity set of uncertain system states. Although a few studies have tackled the dynamic setting, a practical algorithm remains elusive. To bridge this gap, we introduce an DRNMPC scheme that dynamically controls the propagation of ambiguity, based on the constrained iterative linear quadratic regulator. The theoretical results are also provided to characterize the stochastic error reachable sets under ambiguity. We evaluate the effectiveness of our proposed iterative DRMPC algorithm by comparing the closed-loop performance of feedback and open-loop on a mass-spring system. Finally, we demonstrate in numerical experiments that our algorithm controls the propagated Wasserstein ambiguity

    On the existence of generalized solutions to a spatio-temporal predator-prey system

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    In this paper we consider a pair of coupled non-linear partial differential equations describing the interaction of a predator-prey pair. We introduce a concept of generalized solutions and show the existence of such solutions in all space dimension with the aid of a regularizing term, that is motivated by overcrowding phenomena. Additionally, we prove the weak-strong uniqueness of these generalized solutions and the existence of strong solutions at least locally-in-time for space dimension two and three

    A porous-media model for reactive fluid-rock interaction in a dehydrating rock

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    We study the GENERIC structure of models for reactive two-phase flows and their connection to a porous-media model for reactive fluid-rock interaction used in Geosciences. For this we discuss the equilibration of fast dissipative processes in the GENERIC framework. Mathematical properties of the porous-media model and first results on its mathematical analysis are provided. The mathematical assumptions imposed for the analysis are critically validated with the thermodynamical rock data sets

    Convergence to self-similar profiles in reaction-diffusion systems

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    We study a reaction-diffusion system on the real line, where the reactions of the species are given by one reversible reaction pair satisfying the mass-action law. We describe different positive limits at both sides of infinityand investigate the long-time behavior. Rescaling space and time according to the parabolic scaling, we show that solutions converge exponentially to a similarity profile when the scaled time goes to infinity. In the original variables, these profiles correspond to asymptotically self-similar behavior describing the phenomenon of diffusive mixing of the different states at infinity.Our method provides global exponential convergence for all initial states with finite relative entropy. For the case with equal stoichiometric coefficients, we can allow for self-similar profiles with arbitrary equilibrated states,while in the other case we need to assume that the two states atinfinity are sufficiently close such that the self-similar profile is relative flat

    Sharp deviation bounds and concentration phenomenon for the squared norm of a sub-gaussian vector

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    Let \Xv be a Gaussian zero mean vector with \Var(\Xv) = \BBH . Then \| \Xv \|^{2} well concentrates around its expectation \dimA = \E \| \Xv \|^{2} = \tr \BBH provided that the latter is sufficiently large. Namely, ¶\bigl( \| \Xv \|^{2} - \tr \BBH > 2 \sqrt{\xx \tr(\BBH^{2})} + 2 \| \BBH \| \xx \bigr) \leq \ex^{-\xx} and ¶\bigl( \| \Xv \|^{2} - \tr \BBH < - 2 \sqrt{\xx \tr(\BBH^{2})} \bigr) \leq \ex^{-\xx} ; see \cite{laurentmassart2000}. This note provides an extension of these bounds to the case of a sub-gaussian vector \Xv . The results are based on the recent advances in Laplace approximation from \cite{SpLaplace2022}

    The emergence of a giant component in one-dimensional inhomogeneous networks with long-range effects

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    We study the weight-dependent random connection model, a class of sparse graphs featuring many real-world properties such as heavy-tailed degree distributions and clustering. We introduce a coefficient, (deltaf), measuring the effect of the degree-distribution on the occurrence of long edges. We identify a sharp phase transition in (deltaf) for the existence of a giant component in dimension (d=1)

    Mixed Laplace approximation for marginal posterior and Bayesian inference in error-in-operator model

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    Laplace approximation is a very useful tool in Bayesian inference and it claims a nearly Gaussian behavior of the posterior. \cite{SpLaplace2022} established some rather accurate finite sample results about the quality of Laplace approximation in terms of the so called effective dimension p under the critical dimension constraint p3≪n. However, this condition can be too restrictive for many applications like error-in-operator problem or Deep Neuronal Networks. This paper addresses the question whether the dimensionality condition can be relaxed and the accuracy of approximation can be improved if the target of estimation is low dimensional while the nuisance parameter is high or infinite dimensional. Under mild conditions, the marginal posterior can be approximated by a Gaussian mixture and the accuracy of the approximation only depends on the target dimension. Under the condition p2≪n or in some special situation like semi-orthogonality, the Gaussian mixture can be replaced by one Gaussian distribution leading to a classical Laplace result. The second result greatly benefits from the recent advances in Gaussian comparison from \cite{GNSUl2017}. The results are illustrated and specified for the case of error-in-operator model

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