ODE methods in non-local equations
Dienstag, 16.6.20, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
In this talk, I will introduce some ODE-type methods to find radially symmetric solutions for non-local PDEs. I will provide an overview and some improvements of the main ideas behind these methods. The main underlying idea is understanding a non-local PDE problem as an infinite dimensional ODE system. These methods were first developed in the non-local gluing scheme to find solutions for the fractional Yamabe problem, which are singular along a non-zero dimensional submanifold. I will also show some new applications.\n\nIn particular, I will present a variation of constants formula for fractional Hardy operators, a suitable extension in the spirit of Caffarelli–Silvestre and an equivalent formulation as an infinite system of second order constant coefficient ODEs. Moreover, I will show that, like in classical ODEs, quantities such as the Hamiltonian and Wronskian may then be used. As an example of applications, we obtain a Frobenius theorem and a new proof for the non-degeneracy of the fast-decay singular solution of the fractional Lane-Emden equation. We also establish new Pohozaev identities.\n\nThis is a joint work with Weiwei Ao, Hardy Chan, Marco Fontelos, María del Mar González and Juncheng Wei.
Shape optimization of convex, rotational symmetric domains for an eigenvalue problem arising in optimal insulation
Dienstag, 30.6.20, 14:15-15:15, virtueller SR 226, HH-10
I will present the main results from my master thesis about shape optimization for an eigenvalue problem arising in optimal insulation. Under the assumption of convexity, the existence of an optimal domain can be proven, but due to the difficulties of approximating convex domains in \(\bmathbb{R}^3\), the constraint to rotational symmetric domains is used to reduce the problem to a two-dimensional setting, allowing the numerical approximation of optimal domains.
Multiscale simulation of bone tissue regeneration
Dienstag, 7.7.20, 14:15-15:15, Raum 226, virtuell Euwe
The development of multiscale methods and technologies is of vital importance not only in the field of mechanobiology, but also in Mechanical and Biological Engineering in general. Therefore, the desire of the M2BE group is to extend these multiscale methodologies to different application areas of engineering, for example, bone tissue engineering.\n\nLarge bone defects represent a clinical challenge for which regenerative therapies and tissue engineering strategies aim at offering treatment alternative to conventional replacement approaches by metallic implants (Pobloth et al., 2018). Additionally, 3D printing technologies provide directly printed porous scaffolds with designed shape, controlled chemistry and interconnected porosity, where bone will regenerate (Valainis et al., 2019). In the last decades a strong effort has been made to develop computer tools to optimize scaffold designs. In fact, computer techniques and mathematically based models are becoming very useful tools for material engineers and biologists to advance the understanding of the scaffold behaviour under different environments (Metz et al., 2020). Bone tissue regeneration using scaffolds in vivo inherently works on two well-differentiated spatio-temporal scales: the tissue level and the pore scaffold.\n\nMathematical modelling of tissue regeneration within scaffolds has been traditionally restricted to one of these scales. I will present a micro-macro mathematical approach for bone tissue engineering regeneration within scaffolds (Sanz-Herrera et al., 2009). At the tissue level, the macroscopic mechanical, diffusive and flow properties are derived by means of the asymptotic homogenization theory (Sanz-Herrera et al., 2008). At the microscopic scale, bone tissue regeneration at the scaffold microsurface is simulated using a certain bone growth model based on a bone remodeling theory (Beaupre et al., 1990). \n\nAdditionally, a new approach will be presented modelling tissue regeneration but restricted to one of these scales (tissue scale). Our latest mechano-driven computational model of bone ingrowth where the local mechanical environment and the regenerative potential of an individual host regulate the mineralization process within a porous scaffold.
On fluctuations in particle systems and their links to partial differential equations
Dienstag, 14.7.20, 14:15-15:15, virtual SR226 (Euwe)
Partial differential equations often arise as a scaling limit of particle models. We present a link that allows to determine the evolution operator of a class of parabolic equations directly from particle data. As a preliminary step, we will explain how transport coefficients in these equations can be computed from particle simulations. The argument relies on a suitable representation of the governing PDE as gradient flow of the entropy. \n \nSpecifically, we study particle systems and analyse their fluctuations. Fluctuations can often provide useful information on underlying processes, for example in form of fluctuation-dissipation relations. These fluctuations can be described by stochastic differential equations or variational formulations related to large deviations. The link between finite systems and their many-particle limit will be analysed in this formulation.\nThe talk will focus on a class of stochastic equations which are often called equations of fluctuating hydrodynamic and study some relatively simple incarnations of these equations.\n \nThe talk will close with a discussion of analytic results. As will be explained, the (non-)existence theory of the underlying stochastic differential equations is delicate and surprising. Existence for a regularised model will be sketched. \n \nThis is joint work with X. Li, P. Embacher, N. Dirr and C. Reina (numerical work) and Federico Cornalba and Tony Shardlow (analytic work).
Discretizations of the total variation for singular functions
Dienstag, 21.7.20, 14:15-15:15, virtual SR226 (Euwe)
This talk will be about works in collaboration with Corentin Caillaud (CMAP) and Thomas Pock (TU Graz) on the numerical analysis of discretizations of the total variation functional, when solutions are discontinuous. We will discuss error bounds and (an)isotropy properties.
Phase field approaches for tumour growth
Dienstag, 28.7.20, 14:15-15:15, virtual SR226 (Euwe)
Modelling of tumour growth is one of the challenging frontiers of applied mathematics. In the last years, phase field models for tumour growth have been studied intensively. Alike classical free boundary models they use a continuum approach to describe the growth of tumours. However, an advantage to free boundary models is that phase field models allow for topology changes like break up and coalescence. In addition, phase field methods can be used numerically without an explicit tracking of the interface which is necessary for free boundary models. In my talk I will introduce a macroscopic model for tumour growth in which cell-cell adhesion effects are taken into account with the help of a Ginzburg–Landau type energy. The resulting evolution equation is a Cahn-Hilliard equation taking source and sink terms into account. In addition, nutrient diffusion is incorporated by a coupling to a reaction-diffusion equation. I will show existence, uniqueness and regularity results.\n\nFinally, using optimisation theory and reduced order modelling I will describe how parameter in the system can be estimated in a patient specific way. Properties of solutions will be illustrated with the help of numerical simulations.