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04.07.2022 15:00 Lukas Wessels:
Optimal Control of Stochastic Reaction-Diffusion Equationsvia ZOOM (Boltzmannstr. 3, 85748 Garching)

In this talk, we consider the following optimal control problem of stochastic reaction-diffusion equations. First we apply the spike variation method which relies on introducing the first and second order adjoint state. We give a novel characterization of the second order adjoint state as the solution to a backward SPDE on the space L2(Λ) ⊗ L2(Λ) ∼= L2(Λ2). Using this representation, we prove the maximum principle for controlled SPDEs. As another application of our characterization of the second order adjoint state, we derive additional necessary optimality conditions in terms of the value function. These results generalize a classical relationship between the adjoint states and the derivatives of the value function to the case of viscosity dif- ferentials. We also show how the necessary conditions lead us directly to a non-smooth version of the classical verification theorem in the framework of viscosity solutions. In the last part, we analyze an optimal control problem governed by the stochastic Nagumo model with a view towards efficient numerical approximations. We develop a gradient descent method for the approximation of optimal controls and present numerical examples. This talk is based on the following three joint works with Wilhelm Stannat: • W. Stannat and L. Wessels, Deterministic control of stochastic reaction- diffusion equations, Evol. Equ. Control Theory 10 (2021), pp. 701–722, https://doi.org/10.3934/eect.2020087. • W. Stannat and L. Wessels, Peng’s maximum principle for stochastic par- tial differential equations, SIAM J. Control Optim. 59 (2021), pp. 3552– 3573, https://doi.org/10.1137/20M1368057. • W. Stannat and L. Wessels, Necessary and sufficient conditions for optimal control of semilinear stochastic partial differential equations, submitted, https://arxiv.org/abs/2112.09639, 2022.

04.07.2022 16:30 Simone Floreani (TU Delft):
Hydrodynamics for the partial exclusion process in random environment2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

In this talk, I present a partial exclusion process in random environment, a system of random walks where the random environment is obtained by assigning a maximal occupancy to each site of the Euclidean lattice. This maximal occupancy is allowed to randomly vary among sites, and partial exclusion occurs. Under the assumption of ergodicity under translation and uniform ellipticity of the environment, we prove that the quenched hydrodynamic limit is a heat equation with a homogenized diffusion matrix. The first part of the talk is based on a joint work with Frank Redig (TU Delft) and Federico Sau (IST Austria).Finally, I will discuss some recent progresses in the understanding of what happens when removing the uniform ellipticity assumption. After recalling some results on the Bouchaud’s trap model, I will show that, when assuming that the maximal occupancies are heavy tailed and i.i.d., the hydrodynamic limit is the fractional-kinetics equation.The second part of the talk is based on an ongoing project with Alberto Chiarini (University of Padova) and Frank Redig (TU Delft).

06.07.2022 12:15 Anastasios Panagiotelis (University of Sydney, AUS):
Anomaly detection with kernel density estimation on manifoldsOnline: attendBC1 2.01.10 (Parkring 11, 85748 Garching)

Manifold learning can be used to obtain a low-dimensional representation of the underlying manifold given the high-dimensional data. However, kernel density estimates of the low-dimensional embedding with a fixed bandwidth fail to account for the way manifold learning algorithms distort the geometry of the underlying Riemannian manifold. We propose a novel kernel density estimator for any manifold learning embedding by introducing the estimated Riemannian metric of the manifold as the variable bandwidth matrix for each point. The geometric information of the manifold guarantees a more accurate density estimation of the true manifold, which subsequently could be used for anomaly detection. To compare our proposed estimator with a fixed-bandwidth kernel density estimator, we run two simulations with 2-D metadata mapped into a 3-D swiss roll or twin peaks shape and a 5-D semi-hypersphere mapped in a 100-D space, and demonstrate that the proposed estimator could improve the density estimates given a good manifold learning embedding and has higher rank correlations between the true and estimated manifold density. A shiny app in R is also developed for various simulation scenarios. The proposed method is applied to density estimation in statistical manifolds of electricity usage with the Irish smart meter data. This demonstrates our estimator's capability to fix the distortion of the manifold geometry and to be further used for anomaly detection in high-dimensional data.

06.07.2022 13:00 Jonathan Dawes:
Pattern formation with nonlocal terms, and Alan Turing’s later work on morphogenesis.Online: attend (Passcode 101816)

I will describe the influence of a nonlocal nonlinear term on the well-known dynamics of the model equation usually ascribed to Swift and Hohenberg (1977). Although a nonlocal term allows a continuous transition between purely local and completely global coupling there are interesting and perhaps unexpected aspects of the dynamics in intermediate cases.

Intriguingly, it turns out that the analysis of this model problem is closely related to the problem Alan Turing worked on after the publication of his well-known 1952 paper in mathematical biology. Unfinished, unpublished archive material reveals fascinating insights into his attempt to tackle a much more complex mathematical pattern formation problem. I will survey this material and show how it goes far beyond the 1952 paper in both mathematical content and ambition.

11.07.2022 15:00 Thomas Richthammer (Universität Paderborn):
TBA2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

TBA

11.07.2022 15:00 Jacopo Vittadello:
Dynamics of Nonautonomous Measure Driven Differential Equations and Applications to Resiliencevia ZOOM (Boltzmannstr. 3, 85748 Garching)

Autonomous dynamical systems and their attractors have been extensively studied in mathematical literature. In particular, in recent times persistence of attractors has been investigated under the lenses of many different resilience indicators, which try to capture the ability of the attractor to endure different kinds of perturbations. These may include variation of the initial condition, or of the differential equation itself. In this work we first develop a rigorous theory of measure driven differential equations, a generalization of classical ODEs in which the Lebesgue measure is substituted by some other signed measure, possibly introducing an impulsive component and thus discontinuities in the solutions. These notions are largely based on Schmaedeke's and Das and Sharma's works. Then, we extend a resilience indicator, Meyer and McGehee's intensity of attraction, to the nonautonomous measure driven setting, generalizing the relevant results accordingly. In particular, we will see that by controlling the effect of time--dependent bounded perturbations of the differential equation we also gain meaningful information about nonautonomous bounded perturbations of such equation. We stress that the indicator we obtain is very general, as it can be applied to any measure driven dynamical system, but it can in particular be employed on autonomous systems in order to impose a greater variety of perturbations. More specifically, thanks to the measure driven component, it is possible to express initial data perturbations in the form of perturbations of the driving force of the differential equation. This allows to quantitatively compare our measure driven intensity with other autonomous resilience indicators whose perturbations assume the form of a variation of the initial condition (possibly repeatedly), namely the distance to threshold and the flow--kick resilience.

11.07.2022 16:30 Olga Aryasova (Universität Jena):
TBA2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

TBA

18.07.2022 15:00 Peter van Heijster:
Travelling waves in a reaction-diffusion model with nonlinear forward-backward-forward diffusionMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Reaction–diffusion equations (RDEs) are often derived as continuum limits of lattice-based discrete models. Recently, a discrete model which allows the rates of movement, proliferation and death to depend upon whether the agents are isolated has been proposed, and this approach gives various RDEs where the diffusion term is convex and can become negative (Johnston et al., 2017), i.e. forward–backward–forward diffusion. Numerical simulations suggest these RDEs support both smooth and shock-fronted travelling waves. In this talk, I will formalise these preliminary numerical observations by analysing the smooth and shock-fronted travelling waves

18.07.2022 16:30 Ruizhe Sun (LMU):
Epidemics on Random Intersection GraphsB 252 (Theresienstr. 39, 80333 München)

In this thesis, we develop a Reed-Frost model based on random intersection graphs. Our interest is the size of the set of ultimately recovered individuals as population size grows to infinity. Several branching processes will be constructed as approximating processes to serve this purpose. Eventually, benefiting from the clique-based structure provided by the random intersection graph, we will discuss the exact distribution of the quantity of interest in both small and large outbreaks.

19.07.2022 12:15 Tobias Boege (Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig):
t.b.a.Online: attendBC1 2.01.10 (Parkring 11, 85748 Garching)

t.b.a.

21.07.2022 16:30 Tobias Dyckerhoff (Uni Hamburg):
NNOnline: attend (Meeting-ID: 913-2473-4411; Password: StatsCol22)A 027 (Theresienstr. 39, 80333 München)

25.07.2022 15:00 Elisabetta Brocchieri:
Cross-diffusion systems: existence and uniqueness of strong solutionsvia ZOOM (Boltzmannstr. 3, 85748 Garching)

Cross-diffusion systems are non-linear parabolic systems with relevant applications in biology and ecology. In this talk, we study the existence of strong solutions for a triangular cross-diffusion system with reaction terms which include the Lotka-Volterra type. The main idea consists in analysing an auxiliary system in a non-divergence form which is equivalent to the cross- diffusion system, by introducing a convenient change of variable. Then, we regularize the auxiliary system, we prove the existence of strong solutions by a fixed-point theorem and we pass to the limit. Moreover, we also investigate the regularity and the uniqueness of the solution. In particular, we prove that the solution is bounded in L∞((0, T ) × Ω), with T > 0 and the space domain Ω ⊂ RN , provided that N ≤ 3, and it is unique if N ≤ 2.

25.07.2022 16:30 Viktor Bezborodov (Universität Göttingen):
TBA2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

TBA