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- ItemOn the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints(Berlin ; Heidelberg : Springer, 2021) Berthold, Holger; Heitsch, Holger; Henrion, René; Schwientek, JanWe present an adaptive grid refinement algorithm to solve probabilistic optimization problems with infinitely many random constraints. Using a bilevel approach, we iteratively aggregate inequalities that provide most information not in a geometric but in a probabilistic sense. This conceptual idea, for which a convergence proof is provided, is then adapted to an implementable algorithm. The efficiency of our approach when compared to naive methods based on uniform grid refinement is illustrated for a numerical test example as well as for a water reservoir problem with joint probabilistic filling level constraints.
- ItemConvergence bounds for empirical nonlinear least-squares(Les Ulis : EDP Sciences, 2022) Eigel, Martin; Schneider, Reinhold; Trunschke, PhilippWe consider best approximation problems in a nonlinear subset ℳ of a Banach space of functions (𝒱,∥•∥). The norm is assumed to be a generalization of the L 2-norm for which only a weighted Monte Carlo estimate ∥•∥n can be computed. The objective is to obtain an approximation v ∈ ℳ of an unknown function u ∈ 𝒱 by minimizing the empirical norm ∥u − v∥n. We consider this problem for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and is independent of the specified nonlinear least squares setting. Several model classes are examined and the analytical statements about the RIP are compared to existing sample complexity bounds from the literature. We find that for well-studied model classes our general bound is weaker but exhibits many of the same properties as these specialized bounds. Notably, we demonstrate the advantage of an optimal sampling density (as known for linear spaces) for sets of functions with sparse representations.
- ItemAnalysis and simulations for a phase-field fracture model at finite strains based on modified invariants(Berlin : Wiley-VCH, 2020) Thomas, Marita; Bilgen, Carola; Weinberg, KerstinPhase-field models have already been proven to predict complex fracture patterns for brittle fracture at small strains. In this paper we discuss a model for phase-field fracture at finite deformations in more detail. Among the identification of crack location and projection of crack growth the numerical stability is one of the main challenges in solid mechanics. Here we present a phase-field model at finite strains, which takes into account the anisotropy of damage by applying an anisotropic split of the modified invariants of the right Cauchy-Green strain tensor. We introduce a suitable weak notion of solution that also allows for a spatial and temporal discretization of the model. In this framework we study the existence of solutions and we show that the time-discrete solutions converge in a weak sense to a solution of the time-continuous formulation of the model. Numerical examples in two and three space dimensions illustrate the range of validity of the analytical results.
- ItemQuantitative Heat-Kernel Estimates for Diffusions with Distributional Drift(Dordrecht [u.a.] : Springer Science + Business Media B.V, 2022) Perkowski, Nicolas; van Zuijlen, Willem[For Abstract, see PDF]
- ItemOptimal control of multiphase steel production(Berlin ; Heidelberg : Springer, 2019) Hömberg, Dietmar; Krumbiegel, Klaus; Togobytska, NataliyaAn optimal control problem for the production of multiphase steel is investigated that takes into account phase transformations in the steel slab. The state equations are a semilinear heat equation coupled with an ordinary differential equation, that describes the evolution of the steel microstructure. The time-dependent heat transfer coefficient serves as a control function. Necessary and sufficient optimality conditions for the control problem are derived. For the numerical solution of the control problem, a reduced sequential quadratic programming method with a primal-dual active set strategy is developed. The numerical results are presented for the optimal control of a cooling line in the production of hot-rolled Mo–Mn dual phase steel. © 2019, The Author(s).
- ItemThe enhanced Sanov theorem and propagation of chaos(Amsterdam [u.a.] : Elsevier, 2017) Deuschel, Jean-Dominique; Friz, Peter K.; Maurelli, Mario; Slowik, MartinWe establish a Sanov type large deviation principle for an ensemble of interacting Brownian rough paths. As application a large deviations for the (-layer, enhanced) empirical measure of weakly interacting diffusions is obtained. This in turn implies a propagation of chaos result in a space of rough paths and allows for a robust analysis of the particle system and its McKean–Vlasov type limit, as shown in two corollaries.
- ItemDensity of convex intersections and applications(London : Royal Society, 2017) Hintermüller, M.; Rautenberg, C.N.; Rösel, S.In this paper, we address density properties of intersections of convex sets in several function spaces. Using the concept of Γ-convergence, it is shown in a general framework, how these density issues naturally arise from the regularization, discretization or dualization of constrained optimization problems and from perturbed variational inequalities. A variety of density results (and counterexamples) for pointwise constraints in Sobolev spaces are presented and the corresponding regularity requirements on the upper bound are identified. The results are further discussed in the context of finite-element discretizations of sets associated with convex constraints. Finally, two applications are provided, which include elasto-plasticity and image restoration problems.
- ItemOn periodic solutions for one-phase and two-phase problems of the Navier–Stokes equations(Basel : Springer, 2020) Eiter, Thomas; Kyed, Mads; Shibata, YoshihiroThis paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier–Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasilinear systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value 0, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal Lp–Lq regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, which is obtained by the systematic use of R-solvers developed in Shibata (Diff Int Eqns 27(3–4):313–368, 2014; On the R-bounded solution operators in the study of free boundary problem for the Navier–Stokes equations. In: Shibata Y, Suzuki Y (eds) Springer proceedings in mathematics & statistics, vol. 183, Mathematical Fluid Dynamics, Present and Future, Tokyo, Japan, November 2014, pp 203–285, 2016; Comm Pure Appl Anal 17(4): 1681–1721. https://doi.org/10.3934/cpaa.2018081, 2018; R boundedness, maximal regularity and free boundary problems for the Navier Stokes equations, Preprint 1905.12900v1 [math.AP] 30 May 2019) to the resolvent problem for the linearized equations and the transference theorem obtained in Eiter et al. (R-solvers and their application to periodic Lp estimates, Preprint in 2019) for the Lp boundedness of operator-valued Fourier multipliers. These approaches are the novelty of this paper. © 2020, The Author(s).
- ItemBeyond just “flattening the curve”: Optimal control of epidemics with purely non-pharmaceutical interventions(Berlin ; Heidelberg : Springer, 2020) Kantner, Markus; Koprucki, ThomasWhen effective medical treatment and vaccination are not available, non-pharmaceutical interventions such as social distancing, home quarantine and far-reaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptible-exposed-infectious-recovered) model and continuous-time optimal control theory, we compute the optimal non-pharmaceutical intervention strategy for the case that a vaccine is never found and complete containment (eradication of the epidemic) is impossible. In this case, the optimal control must meet competing requirements: First, the minimization of disease-related deaths, and, second, the establishment of a sufficient degree of natural immunity at the end of the measures, in order to exclude a second wave. Moreover, the socio-economic costs of the intervention shall be kept at a minimum. The numerically computed optimal control strategy is a single-intervention scenario that goes beyond heuristically motivated interventions and simple “flattening of the curve”. Careful analysis of the computed control strategy reveals, however, that the obtained solution is in fact a tightrope walk close to the stability boundary of the system, where socio-economic costs and the risk of a new outbreak must be constantly balanced against one another. The model system is calibrated to reproduce the initial exponential growth phase of the COVID-19 pandemic in Germany. © 2020, The Author(s).
- ItemModelling and simulation of flame cutting for steel plates with solid phases and melting(Berlin ; Heidelberg : Springer, 2020) Arenas, Manuel J.; Hömberg, Dietmar; Lasarzik, Robert; Mikkonen, Pertti; Petzold, ThomasThe goal of this work is to describe in detail a quasi-stationary state model which can be used to deeply understand the distribution of the heat in a steel plate and the changes in the solid phases of the steel and into liquid phase during the flame cutting process. We use a 3D-model similar to previous works from Thiébaud (J. Mater. Process. Technol. 214(2):304–310, 2014) and expand it to consider phases changes, in particular, austenite formation and melting of material. Experimental data is used to validate the model and study its capabilities. Parameters defining the shape of the volumetric heat source and the power density are calibrated to achieve good agreement with temperature measurements. Similarities and differences with other models from literature are discussed. © 2020, The Author(s).