Abstract

Nuclear fusion reactor design crucially depends on numerical simulation. The plasma can usually be modeled using fluid equations (for mass, momentum and energy). However, the reactor also contains neutral (non-charged) particles (which are important in its operation), of which both the position and velocity distribution is important. This leads to a Boltzmann-type transport equation that needs to be discretised with a Monte Carlo method. One then obtains a coupled finite-volume/Monte-Carlo simulation, of which the results possess both a bias and a variance. In this talk, I introduce the problems associated with the simulation of the plasma edge region in a fusion reactor. I discuss how to couple a finite volume discretisation of the plasma equations with a Monte Carlo simulation of the neutral particles, and show how the Monte Carlo errors affect convergence of steady state computations and reliability of gradient computations (necessary during optimization).

Abstract

We consider the space of solutions to the continuity equation $d/dt~u(t,x) + div~j(t,x) = 0$ equipped with convex action functionals $A(u,j)$, which encode microstructure in e.g. porous media or road networks. In the limit over long distances, we compute the effective transport cost in terms of Gamma-convergence, with applications to the porous media equation.

Abstract

We consider isometric immersions of 3-dimensional domains into $\mathbb{R}^3$ at low regularity regimes, in particular in the case of a flat domain. A notion of second fundamental form can be defined when only $1/2$-fractional derivatives of the Gauss map are well-controlled. Through an analysis of the Codazzi system which - in a sense - survives at $2/3$-fractional differentiability, we can pass to a degenerate Monge-Ampère equation and extract valuable geometric information about the rigidity of these isometric immersions. Time permitting, we will discuss a conjecture of Gromov in this regard and some connected problems in nonlinear analysis and geometric function theory.

Abstract

In this survey talk we consider linear transport processes taking place along the edges of a metric graph. The processes are modelled by linear first order differential equations satisfying given boundary conditions in the vertices of the graph. We use methods from the theory of strongly continuous operator semigroups and present the conditions that yield the well-posedness of the problem. We also discuss some qualitative properties of the solutions.

Abstract

The talk will be devoted to the problem of decomposing a divergence-free vector measure into a family of measures induced by closed simple curves. We will discuss applications of such decompositions to rigidity properties of vector measures. Moreover, we will demonstrate that in the two-dimensional case such decomposition is possible for any divergence-free vector measure. Ultimately we will discuss some connections between such decompositions and uniqueness of solutions of Cauchy problem for continuity equation. The talk will be based on a recent joint work with P. Bonicatto.

Abstract

The concept of topological entropy plays a central role in the modern theory of Dynamical Systems. Originally introduced by Adler, Konheim and McAndrew in 1960’s, it was later reformulated and clarified by work of Rufus Bowen. It has roots in the work of Kolmogorov and Sinai. Positive topological entropy is usually taken as the mathematical definition of (topological) chaos, in fact fundamental results of Katok imply that, in low dimensions, positive topological entropy forces the presence of horseshoes.

After discussing this circle of ideas, the goal of this talk will be to understand how variational methods can be used to detect positive topological entropy in Hamiltonian systems. The prototypical example is that of a geodesic flow on a negatively curved surface. In a second step we will discuss how sophisticated variational methods - which go under the umbrella of Symplectic Field Theory and are based on elliptic PDEs - can be used to generalise the same kind of results to larger classes of Hamiltonian systems.

Abstract

I present our recent results on the convergence analysis of suitable finite volume methods for multidimensional Euler equations. We have shown that a sequence of numerical solutions converges weakly to a weak dissipative solution. The analysis requires only the consistency and stability of a numerical method and can be seen as a generalization of the famous Lax-equivalence theorem for nonlinear problems. The weak-strong uniqueness principle implies the strong convergence of numerical solutions to the classical solution as long as it exists.

On the other hand, if the classical solution does not exist we apply the so-called K-convergence and show how to compute effectively the observable quantities of a space-time parametrized measure generated by numerical solutions. Consequently, we derive the strong convergence of the empirical averages of numerical solutions to a weak dissipative solution. If time permits I will illustrate a connection to the concept of statistical convergence.

Abstract

This talk will survey some results for the two- or three-space dimensional compressible Euler equations, results both in theory and numerics. We shall present

• non-uniqueness results of weak entropy solutions for special initial data using convex integration
• introducing solution concepts beyond weak solutions that allows to show convergence to the incompressible limit of the compressible Euler equations with gravity
• the relationship between stationary preservation, maintaining vorticity, and asymptotic preserving numerical methods
• introduce a high order numerical method that holds promise to achieve this.

This is joint work among others with Simon Markfelder, Wasilij Barsukow, Eduard Feireisl and Phil Roe.

References

[1] W. Barsukow, J. Hohm, C. Klingenberg, and P. L. Roe. The active flux scheme on Cartesian grids and its low Mach number limit. Journal of Scientific Computing 81, pp. 594–622 (2019)

[2] E. Feireisl, C. Klingenberg, O. Kreml, and S. Markfelder. On oscillatory solutions to the complete Euler equations. Journal of Differential Equations 296 (2), pp. 1521-1543 (2020)

Abstract

By a classical result of Douglas, for any given $p\geq 0$ and configuration $\Gamma\subset \mathbb{R}^n$ of disjoint Jordan curves there exists an area minimizer among all compact surfaces of genus at most $p$ which span $\Gamma$. In the talk we will discuss a generalization of this theorem to singular configurations $\Gamma$ of possibly non-disjoint or self-intersecting curves. Furthermore, the talk will contain new existence results for regular configurations $\Gamma$ in more general ambient spaces such as Riemannian manifolds.

This is joint work with M. Fitzi.

Abstract

After Ricci flow, the mean curvature flow of submanifolds in a Riemannian manifold is perhaps the most natural and famous geometric flow, describing the gradient descent for the volume functional. It was therefore an exciting discovery when Knut Smoczyk demonstrated that in an ambient Ricci-flat Kähler manifold, the class of Lagrangian submanifolds is preserved under the flow. This phenomenon is now referred to as Lagrangian mean curvature flow.

Lagrangian mean curvature flow has since been shown to have properties and behaviour distinct from that of mean curvature flow in general. In this talk I will focus on the singular behaviour of the flow, highlight the differences and similarities to the general case, and bring attention to some open conjectures in the field.

Abstract

In this talk, we will focus on the Boltzmann equation modelling a polyatomic gas. Recent results on the existence and uniqueness theory and the derivation of the macroscopic models of six and fourteen moments will be presented.

Abstract

Helfrich (1973) and Canham (1970) introduced the following geometric curvature energy to model the shape of human red blood cells. The idea is that the two dimensional boundary layer $\Sigma\subset\mathbb{R}^3$ of such a cell minimises

$$\int_\Sigma |H-H_0|^2 dA$$

under suitable constraints on e.g. the enclosed volume and surface area. Here $H$ is the scalar mean curvature of $\Sigma$ and $H_0\in\mathbb{R}$ is a parameter called the spontaneous curvature, which represents an asymmetry in the boundary layer of the cell. This induces a prefered curvature of the cell. If this asymmetry is not desired, i.e. $H_0=0$, this Canham-Helfrich energy becomes a variant of the famous Willmore energy.

To show existence of such a minimiser, we will implement the direct method of the calculus of variations. Compactness for a minimising sequence under varifold convergence can be easily obtained. Unfortunately lower-semicontinuity of the Helfrich energy under this varifold convergence is in general not correct by a counterexample of Große-Brauckmann (1993). Nevertheless we can actually show a lower-semicontinuity estimate for the minimising sequence itself.

Throughout the talk we will highlight the main tools used from geometric measure theory. We explain these in some detail and how they are applied to our problem.

In the last part of the talk we will discuss some directions for future research in this area, i.e. some open problems and some modifications to the Canham-Helfrich energy itself.

Abstract

We investigate mean curvature flow of complete graphical hypersurfaces. These are complete hypersurfaces that can be written as graphs of functions defined on (bounded) domains and going to infinity at the boundary for completeness. We discuss two different curvature bounds for these flows, one that works below some arbitrary height and one that works above a cer- tain height and is dependent on the enveloping cylinder. We demonstrate what makes a uniform curvature bound difficult. Moreover, we give an example that behaves quite badly, illustrating issues that arise in this context. Lastly, we touch upon how one can obtain curvature bounds beyond singularities in the mean convex case.

Abstract

The discretization of parametric or stochastic PDEs and ODEs describing complex physical systems leads to high-dimensional problems, the solution of which requires a massive computational effort. Data analysis of large high-dimensional datasets or prediction tasks in spatio-temporal statistics may also require a huge computational effort and storage costs.

In these and also in many other cases the data can be conceptually arranged in the format of a tensor of high degree, and stored in some truncated or lossy compressed format. We look at some common post-processing tasks which are too time and storage consuming in the uncompressed data format and not obvious in the compressed format, as such huge data sets can not be stored in their entirety, and the value of an element is not readily accessible through simple look-up.

The tasks we consider are finding the location of maximum or minimum, or finding all elements in some interval (level sets), or the number of elements with a value in such a level set, the probability of an element being in a particular level set, and the mean and variance of the total collection.

The algorithms to be described are fixed point iterations of particular point-wise functions of the data, which will then exhibit the desired result. We allow the actual computational representation to be a lossy compression, and we allow the algebra operations to be performed in an approximate fashion, to maintain a high compression level. One such example format which is addressed explicitly and described in some detail is the representation of the data as a tensor with compression in the form of a low-rank representation.

Abstract

We consider the problem of the derivation of an effective model for viscous dilute suspensions. A previous work by D. Gérard-Varet and M. Hillairet showed that, if a second order Stokes effective approximation exists then the mean value of the second order correction for the effective viscosity is given by a mean-field limit that can be studied and computed under further assumptions on the particle configurations. We extend this result by identifying the second order correction in the general case and show the convergence to the limit effective model as soon as the mean field limit exists. In particular we recover the mean-field analysis considered by D. Gérard-Varet and M. Hillairet in their paper for the homogeneous case of periodic and random stationary particle configurations. This is a joint work with D. Gérard-Varet.

Abstract

We examine Serrin’s classical overdetermined problem under a perturbation of the Neumann boundary condition. The solution of the problem for constant Neumann boundary condition exists provided that the underlying set is a ball. The question arises whether for a perturbation of the constant there still are sets admitting solutions to the problem. Furthermore, one may ask whether solutions to the perturbed problem are close to the original solution, i.e. whether the solution for constant Neumann boundary condition is a stable one. We prove the existence and uniqueness of solutions for small perturbations using a modified implicit function theorem. Furthermore, we arrive at linear stability estimates. This is work in progress and joint work with Michiaki Onodera.

Abstract

Focussing on models of springy wires, we study framed curves whose evolution is driven both by the bending energy and the twisting energy. The latter tracks the rotation of the frame about the centerline of the curve. In order to prevent topology changes during the evolution, we add a self‐avoiding term, namely the tangent‐point energy. We discuss the discretization of this model and present some numerical simulations. This is joint work with Sören Bartels.

Abstract

In this talk I will discuss some aspects of the potential theory, fine properties and boundary behaviour of the solutions to the Total Variation Flow. Instead of the classical Euclidean setting, we intend to work mostly in the general setting of metric measure spaces. During the past two decades, a theory of Sobolev functions and BV functions has been developed in this abstract setting. A central motivation for developing such a theory has been the desire to unify the assumptions and methods employed in various specific spaces, such as weighted Euclidean spaces, Riemannian manifolds, Heisenberg groups, graphs, etc. The total variation flow can be understood as a process diminishing the total variation using the gradient descent method. This idea can be reformulated using parabolic minimizers, and it gives rise to a definition of variational solutions. The advantages of the approach using a minimization formulation include much better convergence and stability properties. This is a very essential advantage as the solutions naturally lie only in the space of BV functions. Our main goal is to give a necessary and sufficient condition for continuity at a given point for proper solutions to the total variation flow in metric spaces. This is joint work with Vito Buffa and Juha Kinnunen.

Abstract

I will talk about a new phase field approximation of the Willmore energy. I start from a diffuse perimeter approximation considered by Amstutz-van Goethem and motivate from this an approximation of the Willmore energy. I show a $\Gamma$-$\limsup$ estimate for the approximation and justify by a formal asymptotic expansion that a corresponding $L^2$-Gradient Flow converges to the Willmore Flow. I will also consider another model proposed by Karali-Katsoulakis and motivate an approximation of the Willmore energy based on it. I will outline the similarities between the models concerning the formal asymptotics of the $L^2$-Gradient Flow. Joint work with M. Röger, TU Dortmund.

Abstract

Tomographic imaging aims at recovering information about the interior of a body from multiple measurements taken from outside. On an abstract level tomography can be modeled as an inverse problem with operator valued measurements. In this talk we address two main difficulties arising in the numerical solution of such problems: ill-posedness and high-dimensionality. The first issue can be resolved by application of regularization methods and in order to reduce the computational complexity, we develop a problem-adapted model order reduction strategy that allows to reconstruct tomographic images from perturbed measurements in a stable an extremely efficient manner.

Abstract

In this talk I will recall the classic theory of transport equations due to DiPerna, P.-L. Lions and Ambrosio and then show to recent result on ill-posedness in the case that the DiPerna-Lions integrability condition fails. Using the classic theory and recent extensions of it I will show how this result implies the failure of almost-everywhere-uniqueness for ODEs with Sobolev coefficients. (Joint work with S. Modena and L. Székelyhidi.)

Watch the seminar by clicking here.

Abstract

The classical Łojasiewicz inequality (cf. [1]) describes the particular behavior of a real analytic function $\mathcal{E}\colon \mathbb{R}^{d}\to \mathbb{R}$ near a critical point $\bar{u}$: For some $C > 0$ and $\theta \in (0,\frac{1}{2}]$ we have the estimate

$$ \vert \mathcal{E}(u)-\mathcal{E}(\bar{u})\vert^{1-\theta} \leq C \Vert \nabla\mathcal{E}(u)\Vert \qquad (1) $$

for all $u$ in a neighborhood of $\bar{u}$. In his pioneering work [3], L. Simon extended (1) to the infinite dimensional setting and deduced convergence results for the associated gradient flow of $\mathcal{E}$. Since then, the Łojasiewicz-Simon gradient inequality has been widely used as a powerful tool to analyze asymptotic properties of PDEs with a gradient flow structure. For many scientific models, it is natural to require that certain quantities remain conserved during the evolution process. This imposes some constraints on the model, e.g. the conservation of total mass. In order to apply the gradient inequality to the associated “constrained” gradient flow, one needs to prove a suitable version of (1) on a manifold, modelling the constraint. We present sufficient conditions for the Łojasiewicz-Simon gradient inequality to hold on a submanifold of a Banach space and discuss the optimality of our assumptions. As an application, we deduce smooth convergence for the length preserving elastic flow of curves (joint work with Adrian Spener).

References

[1] S. Lojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, Les Équations aux Dérivées Partielles (1963), pp. 87-89, CNRS, Paris.

[2] F. Rupp, On the Łojasiewicz-Simon gradient inequality on submanifolds, arXiv:1907.09292 (2019).

[3] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 (1983), no. 3, pp. 525-571.

Abstract

We present an effective adaptive procedure for the numerical approximation of the steady-state Gross-Pitaevskii equation which consists of a combination of gradient flow iterations and adaptive finite element mesh refinements. The mesh-refinement is solely based on energy minimization. Numerical tests show that this strategy is able to provide highly accurate results, with optimal convergence rates with respect to the number of freedom.

Abstract

In many physical systems dispersion plays an important role. If the systems are nonlinear, it is possible that concentration effects balance dispersive effects such that structures of permanent form like solitons can be observed. A famous nonlinear PDE which models this phenomenon is the Nonlinear Schrödinger (NLS) equation. Moreover, the NLS equation can be formally derived as an approximation equation for the dynamics of the envelopes of oscillating wave packets in complicated nonlinear dispersive systems. The so-called NLS approximation has various applications in science and technology, for example, in hydrodynamics, optics or spintronics. To understand to which extent this approximation yield correct predictions of the qualitative behavior of the original systems it is important to justify the validity of the approximation by estimates of the approximation errors in the physically relevant length and time scales. In this talk, we give an overview on the NLS approximation, its applications and its justifications. Concerning the justifications, we will put special emphasis on the most challenging case, namely, the proof of error estimates for the NLS approximation being valid for surface water waves with and without surface tension. These estimates are obtained by parametrizing the two-dimensional surface waves by arc length, which enables us to derive error bounds that are uniform with respect to the strength of the surface tension, as the height of the wave packet and the surface tension go to zero.

Abstract

In this talk we will focus on variational estimates for singular integral operators defined on AD regular measures satisfying some geometric condition. In particular, I will present the following result, which is a joint work with Xavier Tolsa: let $0<n<d$ be integers and let $\mu$ be an $n$-dimensional AD regular measure in $\mathbb R^d$. Then, $\mu$ is uniformly $n$-rectifiable if and only if the variation for the Riesz transform with respect to $\mu$ is a bounded operator in $L^2(\mu)$. This result is related to an important open problem, posed by David and Semmes, about the equivalence between uniform rectifiability and $L^2$ boundedness of the Riesz transforms.

Key words

Geometric flows, parabolic linear systems, elliptic functions, elliptic curves

Abstract

This talk is motivated by questions for existence and uniqueness of strong solutions to a degenerate quasilinear fourth-order non-Newtonian thin-film equation. Originating from a non-Newtonian Navier–Stokes system, the equation is derived by lubrication theory and under the assumption that capillarity is the only driving force. The fluid’s shear-thinning rheology is described by the so-called Ellis constitutive law. Regarding the existence and uniqueness of strong solutions, it turns out that there is a qualitative difference between flow behaviour exponents $α ∈ (1, 2)$ and those larger than or equal $2$. If we associate to the equation an abstract quasilinear Cauchy problem, it turns out that in the case $α ≥ 2$ the differential operator and the right-hand side are Lipschitz continuous in an appropriate sense and the classical Hölder theory of Eidel’man (1969) as well as the abstract theory of Amann (1993) are applicable. The situation is more delicate for flow behaviour exponents $α ∈ (1, 2)$. In this regime the operator and the right-hand side are only $(α −1)$-Hölder continuous, whence there is no hope to obtain existence and uniqueness by Banach’s fixed point theorem. For this reason we prove an abstract existence result for quasilinear parabolic problems of fourth order with Hölder continuous coefficients. This result provides existence of strong solutions to the non-Newtonian thin-film problem in the setting of fractional Sobolev spaces and (little) Hölder spaces. Uniqueness of strong solutions is derived by energy methods and by using the particular structure of the equation.

Abstract

Minimizing a convex function of a signed measure is a typical problem in signal processing or machine learning arising e.g., in gridless spikes deconvolution or two-layer neural networks training. This is a difficult problem: convex algorithms that are traditionally studied often exhibit undesirable behavior (high complexity or low precision). In this talk, I will present an alternative non-convex approach: discretize the measure into “particles” and run gradient descent on their positions and weights. This is simple to implement and often efficient in practice, but the theoretical analysis is challenging due to non-convexity. I will first present a general consistency result: when properly initialized, this method converges to global minimizers as the number of particles goes to infinity. I will then present quantitative convergence results for problems with a sparsity-inducing regularization. The analysis involves tools and ideas from unbalanced optimal transport and Wasserstein gradient flows theories. This talk is based on joint work with Francis Bach.

References

https://arxiv.org/abs/1805.09545
https://arxiv.org/abs/1907.10300

Abstract

Relative entropy is a rather coarse but flexible tool for studying stability properties of systems of hyperbolic conservation laws. In particular, it provides weak strong uniqueness. It is commonly applied to problems on the whole of Euclidean space or on the flat torus, since its interaction with boundary conditions is not straightforward. We will discuss to which extent it can be extended to hyperbolic balance laws on networks.

Systems of hyperbolic balance laws on networks play a significant role in applications such as traffic flow or gas networks. In problems of this type, the networks are described by graphs with (spatially one dimensional) hyperbolic systems on each edge and (algebraic) coupling conditions at the nodes. A frequently studied example are isothermal Euler equations on the edges. In this case, one obvious coupling condition is conservation of mass at the nodes but further coupling conditions are not obvious - indeed, conservation of momentum cannot be expected due to interaction of the flow with the pipe walls and a lack of information on junction geometry.

A few years ago, energy consistent coupling conditions have been proposed by Reigstad. We will show that energy consistent coupling conditions do not ensure that a relative entropy stability framework is available for such systems. This is due to a lack of control on sums of relative entropy fluxes at the nodes.

We will discuss how (certain aspects of) a relative entropy framework can be recovered, when attention is restricted to subsonic solutions of the isothermal Euler equations. This is, indeed, a relevant setup for the operation of natural gas networks. In this setting, the relative entropy guarantees that Lipschitz solutions depend continuously on their initial data and it can be used for studying low Mach/large friction limits. At the same time, this variant of relative entropy no longer provides weak entropic-strong uniqueness.