Shape recognition in 3D point clouds
Tuesday, 25.10.22, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Used in many applications such as 3D-vision, logistics, and mapping, the point cloud has become one of the most important data types in recent years.\nHowever, due to their nature of being unstructured and unordered data, we face difficulties while processing them e.g. for shape recognition purposes.\nOur goal will be to develop techniques to classify point clouds into a predefined number of shapes. We will tackle this problem with a machine-learning approach and provide three types of neural networks operating on point clouds. Furthermore, we will prove a version of the Universal Approximation Theorem for neural networks operating on point clouds to mathematically prove the foundation of one of our neural network types.\n\nLastly, we will extract information on the data by describing some geometric invariances of the shapes to classify for. We will present our results, and the difficulties we faced as well as provide some tips on how to overcome them and give suggestions for future work and improvement. \n\nThe talk is suitable for anyone who has finished the basic mathematic lectures. It is of use to know the fundamentals of machine learning but we will briefly revise all the definitions and concepts necessary.
Rendering Models for Scattering from Specular Rough Surfaces
Tuesday, 8.11.22, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Wave optics can be used to describe linearly polarized light that propagates in free\nspace in the form of waves. Thus, it enables to explain many optical phenomena such\nas interference, diffraction, dispersion and coherence. Surfaces with structures the\nsize of the wavelength of the incident light lead to such effects, which are essential to\nthe natural appearance.\n\nIn the scope of this work, the Helmholtz equation endowed with the impedance\nboundary condition is used to model sunlight incident on rough metallic surfaces.\nAfter proving unique solvability of this electromagnetic scattering problem, the\nboundary integral equation method is used to calculate such solutions for micro\nsurface patches. Numerically, this is done by means of the boundary element method,\nfor which a GPGPU implementation is introduced. This setup allows the local\ndescription of the aforementioned wave-optical phenomena, which are presented in\nthe form of BRDFs. The results are then used to assess one particular prior work.\nThere, approximate wave optics are employed for which it is not entirely clear how\nthese simplifications affect the quality. Although our results contain systematic\ndifferences, the overall agreement is good, confirming the validity of the more efficient\nprior work.\n
Learning the time step size in Deep Neural Networks
Tuesday, 15.11.22, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
\n\nAbstract: Feature propagation in Deep Neural Networks (DNNs) can be associated to nonlinear discrete dynamical systems. Here, we are defining the discretization parameter (time step-size) to be an additional variable in the DNN. Hence, the time step-size can vary from layer to layer and is learned in an optimization framework. The proposed framework can be applied to any of the existing networks such as ResNet, DenseNet or Fractional-DNN. This framework is shown to help overcome the vanishing and exploding gradient issues. To illustrate the advantages, the proposed approach is applied to an ill-posed 3D-Maxwell's equation.
Polymorphic Uncertainty Quantification for the Additive Manufacturing of Elastic Rods.
Tuesday, 22.11.22, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We derive comprehensive models enabling an efficient uncertainty quantification of mechanical\nproperties for additively manufactured rod-shaped elastic solids Orod = (0, L) × hS(x1) in terms\nof errors introduced within the manufacturing process. Here, we consider the fused-filamentfabrication process where the main sources of uncertainties are given by variations of material\nproperties caused by fluctuation of material density and geometric deviations of the printed object from the designed object, see [CKB+18, PVB+19, KRJM+18]. The 3d-printed objects investigated in this work are made of polycarprolactone, a bioresorbable, biocompatible, polymer-based\nmaterial, which is used in the engineering of patient specific bone scaffolds, see [VDF+19].\nWe then deduce a comprehensive modelling approach in three space dimensions for determining\nthe effective mechanical properties of randomly perturbed elastic rods considering aleatoric and\nepistemic uncertainties in the representation of the random perturbations. To do so, we use the\npolymorphic uncertainty model of fuzzy structural analysis from [MGB00] which includes the\nrepresentation of random perturbations as fuzzy random fields (e.g. [PRZ93, Kwa78]) and MonteCarlo simulations (e.g. [KNP20, CGST11]) combined with finite element methods.\nFurthermore, we introduce an one-dimensional surrogate model for rod-shaped structures Orod =\n(0, L) × hS with a fixed cross-section S ⊂ R\n2\n. By this the problem can be reduced to an onedimensional optimization problem requiring only the solution of a system of ordinary differential equations. This leads to a marked reduction of computational effort compared to the threedimensional model concerning the computation of mechanical properties of randomly perturbed\nelastic rods.
Effective toughness of brittle composite laminates
Thursday, 24.11.22, 11:30-12:30, Raum 226, Hermann-Herder-Str. 10
We consider a periodic layer of brittle elastic materials. As the layers become fine, the composite behaves elastically as a spatially homogeneous (averaged) material, whose stiffness modulus can be computed in terms of the relative volumes and the elastic modulus of the single layers. However, in the presence of a crack evolving through the layers it is still unclear if the quasi-static evolution is still represented in terms of a spatially homogeneous material with a crack. In particular not much is known on the effective (or averaged) toughness. Experimental measures, numerical simulations and theoretical estimates show surprising features of the effective toughness: it depends not only on the toughness and the size of the layers but also on their elastic moduli, and it may be even larger than the toughness of the layers (which is known as toughening effect).\nIn this framework, we provide a theoretical study and a couple of examples. We provide an abstract formula for the (possibly non-constant) effective toughness, then we prove convergence of the evolution and convergence of the energy identities, as the size of the layers vanishes. As a by-product we link the toughening effect to the micro-instabilities of the evolution, occurring at the interfaces between the layers of the composite. The two examples provide instead explicit calculations of the effective toughness, one of which presents a toughening effect.
Variational methods for a class of mixed local-nonlocal operators
Tuesday, 29.11.22, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Problems driven by operators of mixed local and nonlocal type have\nraised a certain interest in the last few years, for example in\nconnection with the study of optimal animal foraging strategies. From a\npure mathematical point of view, the superposition of local and\nnonlocal operators, such as the Laplacian and the Fractional Laplacian,\ngenerates a lack of scale invariance that can lead to unexpected\ncomplications. Our goal is to prove the existence of solutions of\nsemilinear elliptic problems governed by these operators and dependent\non a real parameter: when the parameter is sufficiently large, our\nexistence results are known or applications of standard variational\nmethods, but when the real parameter is too small, the situation\nsuddenly becomes more delicate, especially since the operator is no\nlonger positive-definite, the naturally associated bilinear form does\nnot induce a scalar product nor a norm, the variational spectrum may\nhave negative eigenvalues, and even the maximum principle may fail. In\nthis talk, I show how to overcome these difficulties and obtain the\nexpected existence results.\n
TBA
Tuesday, 6.12.22, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
TBA
Zero Temperature Surface Growth and Some Strange Fully Nonlinear Equations
Tuesday, 6.12.22, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
I will describe recent work on scaling limits of zero temperature (deterministic) surface growth models, motivated by KPZ universality and related to gradient \bphi interface models. Chatterjee (2021) and Chatterjee and Souganidis (2021) showed that a smooth choice of the dynamics leads to the deterministic KPZ equation. I will describe a class of examples with non-smooth dynamics, which, at large scales, are described by fully nonlinear parabolic equations with discontinuous coefficients. Joint work with P.E. Souganidis.\n
Non-Newtonian fluids with discontinuous-in-time stress tensor.
Tuesday, 13.12.22, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We consider the system of equations describing the flow of incompressible fluids in bounded domain. In the considered setting, the Cauchy stress tensor is a monotone mapping and has asymptotically \((s-1)\)-growth with the parameter \(s\) depending on the spatial and time variable. We do not assume any smoothness of \(s\) with respect to time variable and assume the log-H\b"{o}lder continuity with respect to spatial variable. Such a setting is a natural choice if the material properties are instantaneous, e.g. changed by the switched electric field. We establish the long time and the large data existence of weak solution provided that \(s \bge \bfrac{3d+2}{d+2}\).
Non-local effects and degenerate Cahn-Hilliard equation
Tuesday, 13.12.22, 15:00-16:00, Raum 226, Hermann-Herder-Str. 10
I will discuss several situations when one has to perform\nlimit passage from non-local to local operators in the context of the\ndegenerate Cahn-Hilliard equation. This includes kinetic derivation of\nthe equation (arXiv:2208.01026, with C. Elbar, M. Mason, B. Perthame),\nfairly classical problem of passage to the limit from non-local to\nlocal equation (arXiv:2208.08955, with C. Elbar) and the same problem\nfor aggregation-diffusion system (in progress, together with J. A.\nCarrillo, C. Elbar). Not all of these problems are fully understood and\nto some of them, solutions are available only on the torus.
The wave equation with acoustic boundary conditions on non-locally reacting surfaces
Tuesday, 10.1.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The subject of the talk are some recent results on the wave equation posed in a suitably regular open domain of R\nN , supplied\nwith an acoustic boundary condition on a part of the boundary and an homogeneous Neumann boundary condition on the\n(possibly empty) remaining part of it, contained in the recent joint book with Delio Mugnolo.\n\nIn this talk we shall focus on non–locally reacting boundaries without any Dirichlet boundary conditions. We first give well–\nposedness results in the natural energy space and regularity results. Hence we shall give precise qualitative results for solutions\nwhen Ω is bounded and r = 2, ρ = Const., k, δ ≥ 0. In particular we shall exhibit some physically inexplicable trivial solutions\nwhich make the problem not asymptotically stable, even with an effective damping, while the problem is asymptotically stable\nwhen the initial data are restricted to a 1-codimensional subspace, which is invariant under the flow.\nThis mathematical result motivated a re-thinking of the physical derivation of the Acoustic Wave Equation, found in most\ntexbooks in Theoretical Acoustic and Classical Mechanics, and of the specific Acoustic Boundary Conditions. The main outcome\nof this detailed analysis is described as follows. In both the Eulerian and the Lagrangian frameworks, due to Hooke’s law, the\nPDEs appearing in it need to be integrated with the integral constraint found in the stability analysis in order to correctly model\nthe physical problem. This fact was never observed in the existing literature
Das Babuška-Paradoxon
Tuesday, 17.1.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Das Babuška-Paradoxon ist beim linearen Biegen von Platten zu beobachten. Es tritt bei einer stückweise linearen Approximation eines gekrümmten Plattenrandes. Die Lösung auf dem somit definierten Gebiet konvergiert demnach nicht gegen die „echte“ Lösung. Eine Rolle spielt es somit insbesondere beim Berechnen numerischer Lösungen z.B. mit der Finite Elemente Methode. In diesem Vortrag werden wir eine mathematische Herleitung des Paradoxon nachvollziehen und versuchen dessen Auftreten durch numerische Experimente zu verifizieren. Hierbei werden wir jedoch einige Diskrepanzen zwischen unseren Ergebnissen und etablierter Theorie feststellen.\n\n
A variational approach to the regularity of optimal transportation
Tuesday, 24.1.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
In this talk I want to present a purely variational approach to the regularity theory for the Monge-Ampère equation, or rather optimal transportation, introduced by Goldman—Otto. Following De Giorgi’s strategy for the regularity theory of minimal surfaces, it is based on the approximation of the displacement by a harmonic gradient, which leads to a one-step improvement lemma, and feeds into a Campanato iteration on the C^{1,\balpha}-level for the displacement.\n\nThe variational approach is flexible enough to cover general cost functions by importing the concept of almost-minimality: if the cost is quantitatively close to the Euclidean cost function |x-y|^2, a minimiser for the optimal transport problem with general cost is an almost-minimiser for the one with quadratic cost. This allows us to reprove the C^{1,\balpha}-regularity result of De Philippis—Figalli, while bypassing Caffarelli’s celebrated theory. (This is joint work with F. Otto and M. Prod’homme)
Algorithmic Solution of Elliptic Optimal Control Problems with Control Constraints by Means of the Semismooth Newton Method“
Tuesday, 24.1.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
I will consider the standard elliptic optimal control problem governed by the Poisson equation where the control space \(U\) is chosen to be either \(L^2(\bOmega)\) or \(H^1_0(\bOmega)\). In particular, I will introduce an abstract framework that provides q-superlinear convergence of the semismooth Newton method and which can be applied to get this convergence rate for any of the choices of \(U\). Further, semismoothness results and characterizations of the elements in the generalized differential will be done. The talk will focus on the infinite dimensional setting, but error estimates and numerical results will also be provided.
tba
Tuesday, 31.1.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Multiscale computer model of bone regeneration within scaffolds in T2DM
Tuesday, 31.1.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
\nBone has the fascinating ability to self-regenerate. However, under certain conditions, such as\nlarge bone defects, this ability is impaired. The treatment of large bone defects is a major\nclinical challenge. There are a number of existing treatments such as the addition of growth\nfactors or autologous bone grafting, but with many associated side effects (Roddy et al., 2018).\n3D printed scaffolds might help to overcome these challenges by providing guiding cues for\ntissue regeneration (Pobloth et al., 2018), however their design remains challenging and mainly\nbased on an experimental trial and error approach. Moreover, the treatment of large bone defects\ngets even more challenging when comorbid with Type 2 diabetes mellitus (T2DM). T2DM is a\nchronic metabolic disease known by the presence of elevated blood glucose levels that is\nassociated with reduced bone regeneration, high fracture risk and non-union. Currently, the\ntreatment of bone defects mostly depends on clinical imaging with some references to patients’\nphysiology and medical history (Wiss, D.A. and W.B. Stetson. 1996), but lacks details about\nthe patient-specific healing potential. Omics offer quantifiable biological indication for\npatients’ intrinsic bone regenerative capacity. In this study, in silico computer modelling\napproaches are used to 1) investigate the effect of scaffold design on the bone regeneration\noutcome, 2) understand the underlying mechanisms behind impaired bone regeneration in\nT2DM and 3) investigate the potential of using an omics informed computer model to predict\npatient-specific bone regeneration.\nA multiscale in silico approach was used that combines finite element and agent based models,\nwhich allow to quantify the mechanical environment within the defect region and to simulate\nthe cellular response. Using these models, bone regeneration was investigated in healthy and\nT2DM conditions, as well as within different scaffold designs. Moreover, the potential of\nomics-driven cellular activity rates to predict bone regeneration was investigated.\nOur validated models suggest that scaffolds with strut-like architecture are beneficial over\nscaffolds with gyroid architecture promoting cell migration towards the scaffold core, both in\nhealthy and T2DM conditions. Impaired MSC proliferation, MSC migration and osteoblast\ndifferentiation were identified as key players behind impaired bone regeneration in T2DM.\nOur results show that bone regeneration is influenced by scaffold architecture agreeing with\nexperimental studies showing different healing outcomes for different scaffold designs. The\nidentification of the key cellular activities behind impaired bone regeneration in T2DM should\nallow the optimization of the scaffold design to promote bone regeneration in co-morbid\npatients.
Tuesday, 7.2.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
TBA
An averaged space-time discretization of the stochastic p-Laplace system
Tuesday, 7.2.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
\nAbstract: In this talk we discuss the stochastic p-Laplace system. In\ngeneral non-linear as well as stochastic equations have limited\nregularization properties. Thus, the solution does not enjoy arbitrary\nhigh regularity. This leads to difficulties in the numerical\napproximation. We propose a new numerical scheme based on the\napproximation of time averaged values of the (unknown) solution.\nAdditionally, we provide a sampling algorithm to approximate the\nstochastic input. We verify optimal convergence of rate 1/2 in time and\n1 in space. This is a joint work with Lars Diening (Bielefeld) and\nMartina Hofmanová (Bielefeld).
TBA
Tuesday, 7.2.23, 15:00-16:00, Raum 226, Hermann-Herder-Str. 10
TBA
Discrete hyperbolic curvature flow in the plane
Tuesday, 14.3.23, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
\n\nAbstract: Hyperbolic curvature flow is a geometric evolution equation that in the plane\ncan be viewed as the natural hyperbolic analogue of curve shortening flow.\nIt was proposed by Gurtin and Podio-Guidugli to model certain wave\nphenomena in solid-liquid interfaces. We propose a semidiscrete finite difference method\nfor the approximation of hyperbolic curvature flow and prove error bounds in natural norms.\nWe also present numerical simulations, including the onset of singularities starting\nfrom smooth strictly convex initial data. This is joint work with Robert N\b"urnberg (Trento).