Analysis and Numerics of PDEs - University of Innsbruck

University of Innsbruck

Seminar on Analysis and Numerics of PDEs

The schedule includes talks in the seminar series as well as closely related events at the University of Innsbruck. If you would like to be added to our mailing list, please contact Heiko Gimperlein.

Unless noted otherwise, the seminar takes place in Room 609 in the building of the mathematics departments, Technikerstrasse 13, 6020 Innsbruck. From the city center you reach the campus in less than 15 minutes by tram lines 2, 5 (direction Technik West) or bus lines K (direction Kranebitten), T (direction Technik West) to the stop Technik.

Date	Speaker and title

04/06/25	Karsten Urban (Ulm)
	Model reduction for PDEs - Benefits, chances, limitations and challenges

	In many cases of practical interest, partial differential equations (PDEs) depend on parameters – one may think of coefficients, boundary conditions, material parameters or the geometry. Moreover, the PDE needs to be solved very often or in extremely short time for different parameter values. Such situations occur in optimization, uncertainty quantification (multi-query) or in situations where an approximate solution is needed extremely fast (realtime) or on devices with very limited capacity (embedded systems with low storage, CPU or energy demands). In such situations, the reduced basis method (RBM) offers the chance to determine a reduced system in an offline training phase and then to solve the reduced system online extremely fast. Both training phase and online certification rely on a posteriori analysis, which provide (1) a reduced system with guaranteed accuracy and (2) a reduced online solution with a certified upper bound for the error. In this talk, we will give an introduction into the RBM and show some applications where the RBM gives enormous speedup. We will also address the theoretical foundation leading to an understanding of chances and limitations of the RBM.


23/05/25	Lun Ji (Chinese Academy of Sciences)
	Bourgain techniques for low regularity error estimates

	Standard time-stepping techniques require a regularity constraint on the initial data u0 for dispersive equations. We introduce a class of low regularity integrators for problems where certain constraints are not satisfied. Moreover, when the regularity is critically low (u0 in Hs with s smaller or equal than d/2), the classical stability argument based on Sobolev spaces does not hold. We have developed a discrete Bourgain framework that overcomes this problem. In this talk, I will use several different dispersive models to summarize how these techniques are applied.


16/05/25	Marlis Hochbruck (Karlsruhe)
	Efficient higher-order time integration schemes for Maxwell's equations with discontinuous Galerkin discretizations on locally refined grids

	In this talk, we discuss the construction and analysis of higher-order time integration schemes for the full discretization of linear Maxwell equations on locally refined spatial grids. The schemes is based on a higher-order implicit method, e.g., an algebraically stable Runge-Kutta method. Our main contribution is to propose a preconditioned Krylov subspace method for solving the linear systems arising in each time step which is designed in such a way that its convergence only depends on the coarse mesh but not on the fine mesh. This is shown by approximation theory in the complex plane. The advantage of this approach is that it is applicable to any implicit scheme and also works for exponential integrators, where the action of the matrix exponential is approximated by rational Krylov subspace methods. It is even applicable to nonlinear problems, where such linear systems arise within the Newton iterations.


08/04/25	Alexandre Ern (Paris)
	Discontinuous Galerkin approximation of the Maxwell eigenvalue problem in first-order form

	The main result presented in this talk is that the discontinuous Galerkin approximation of the Maxwell eigenvalue problem written in first-order form is free of spurious eigenvalues. We include the case of heterogeneous material properties. The first part of the talk motivates this spectral problem and presents a literature review. The second part sketches the main analysis tools used in the proof. This is joint work with Jean-Luc Guermond.


01/04/25	Milos Tasic (Warwick)
15:00	Finite sized corrections for kinetic equations

	The derivation of kinetic equations from particle system is a well-established programme; a particularly well know-example is the justification of the non-linear Boltzmann equation as a scaling limit of hard sphere dynamics. By considering a simpler model of ballistically annihilating hard spheres and carefully studying cancellations associated with recollisions, we deduce much better estimates than the already known Gronwall estimates.



07-08/11/24	4th Austrian Calculus of Variations Day


05/11/24	Ignacio Labarca (Innsbruck)
13:30	Boundary Element Method for Dilute Colloidal Suspensions under Shear Flow

	We study dilute colloidal particle suspensions under a shear flow. To solve this problem, we propose boundary-integral formulations to study advection-diffusion equations in the stationary, frequency, and time-domain. The incompressible flow corresponds to a shear flow, for which a fundamental solution exists in the time-domain. The fundamental solution for the stationary and frequency-domain problems is approximated, and accurate discretization is achieved by a singularity subtraction technique based on the fundamental solution of the heat equation. Numerical experiments demonstrate the effectiveness of our discretization scheme for solving stationary and frequency-domain problems and capturing the non-analytic behavior of the stress tensor in the frequency-domain.


12/09/24	Vu Thai Luan (Texas Tech)
	Recent Advancements in Exponential Integrators for Multiple Time Scale Problems


20/08/24	Jörg Nick (ETH Zürich)
13:00	Time-dependent electromagnetic scattering from dispersive materials

	The talk discusses time-dependent electromagnetic scattering from metamaterials that are described by dispersive material laws. We consider the numerical treatment of a scattering problem in which a dispersive material law, for a causal and passive homogeneous material, determines the wave-material interaction in the scatterer. The resulting problem is nonlocal in time inside the scatterer and is posed on an unbounded domain. Well-posedness of the scattering problem is shown using a formulation that is fully given on the surface of the scatterer via a time-dependent boundary integral equation. Discretizing this equation by convolution quadrature in time and boundary elements in space yields a provably stable and convergent method that is fully parallel in time and space. Under regularity assumptions on the exact solution we derive error bounds with explicit convergence rates in time and space. Numerical experiments illustrate the theoretical results and show the effectiveness of the method.


28/05/24	Caroline Lasser (TU Munich)
16:00	Colloquium: Numerical analysis for watching chemistry happen


28/05/24	Nuutti Hyvönen (Aalto)
11:00	The linearised inverse conductivity problem: reconstruction and Lipschitz stability for infinite-dimensional spaces of perturbations

	The linearised inverse conductivity problem is investigated in a two-dimensional bounded simply connected domain with a smooth enough boundary. After extending the linearised problem for square integrable perturbations, the space of perturbations is orthogonally decomposed and Lipschitz stability, with explicit Lipschitz constants, is proven for each of the infinite-dimensional subspaces. The stability estimates are based on using the Hilbert-Schmidt norm for the Neumann-to-Dirichlet boundary map and its Frechet derivative with respect to the conductivity coefficient. A direct reconstruction method that inductively yields the orthogonal projections of a conductivity coefficient onto the aforementioned subspaces is devised and numerically tested with data simulated by solving the original nonlinear forward problem.


16-17/05/24	Austrian Numerical Analysis Day 2024


07/05/24	Apala Majumdar (Strathclyde)
11:00	Solution Landscapes in the Landau-de Gennes theory for Nematic Liquid Crystals: Analysis, Computations and Applications

	Nematic liquid crystals are classical examples of partially ordered materials that combine fluidity with the order of crystalline solids. They are the working material of a range of electro-optic devices i.e. in the liquid crystal display industry and more recently, they are used in sensors, actuators, elastomers, security applications and pathological studies. We review the celebrated Landau-de Gennes theory for nematic liquid crystals and focus on the modelling of nematics confined to thin quasi-2D systems, with reference to 2D polygons. We perform asymptotic analysis in certain distinguished limits, encoded in terms of geometrical, material and temperature-dependent parameters, accompanied by exhaustive numerical studies of solution landscapes that include stable and unstable solution branches for these systems. There are several numerical challenges associated with the numerical computation of the unstable solution branches and their unstable directions, for which we use the powerful High-Index Shrinking Optimisation Dimer Method. In the last leg of the talk, we discuss the mathematical modelling of some recent experiments on nematic shells and doped bent-core liquid crystals, to illustrate the synergistic links between theory, experiment and novel applications. All collaborations will be acknowledged throughout the talk.


22/01/24	Roland Donninger (Wien)
12:30	Self-similar blowup for wave maps

	Wave maps are the fundamental example of geometric wave equations and they have been studied extensively in the last 30 years. One particular line of research is concerned with singularity formation in finite time which often manifests itself by the presence of self-similar solutions. I will present recent progress in the study of the stability of self-similar wave maps at the critical regularity level. This is based on joint work with David Wallauch.


15/01/24	Helmut Harbrecht (Basel)
11:30	Wavelet compressed, modified Hilbert transform in the space-time discretization of the heat equation


11/12/23	Michael Feischl (TU Wien)
11:30	Stochastic collocation for dynamic micromagnetism

	We consider the stochastic Landau-Lifschitz-Gilbert equation, an SPDE model for dynamic micromagnetism. We first convert the problem to a (highly nonlinear) PDE with parametric coefficients using the Doss-Sussmann transform and the Levy-Ciesielsky parametrization of the Brownian motion. We prove analytic regularity of the parameter- to-solution map and estimate its derivatives. These estimates are used to prove convergence rates for piecewise-polynomial sparse grid methods. Moreover, we propose novel time-stepping methods to solve the underlying deterministic equations.


27/11/23	Simon Blatt (Salzburg)
11:30	Analyticity of solutions to fractional partial differential equations

	In this talk, we explore the topic of analyticity of solutions to elliptic equations in a fresh context, tracing back to Hilbert's 19th problem. While Bernstein's work in 1904 yielded early results for classical elliptic partial differential equations, fractional and non-local equations have remained less explored, with limited results available, mostly restricted to special cases like the Hartree-Fock and Boltzmann equations. We will present known results before delving into our recent discoveries, spanning from unique fractional equations related to knot energies to semi-linear and completely non-linear integro-differential equations, encompassing s-minimal surfaces. Central to our findings is the use of novel refined estimates that account for the long-range interactions within these equations.


20/11/23	Gianluca Ceruti (Innsbruck)
11:30	A retraction perspective on dynamical low-rank approximation

	The availability of more powerful hardware has opened up new opportunities for exploring the time-evolution of high-dimensional problems that describe phenomena of interest. However, from a discrete perspective, tackling these challenges comes with increased computational and storage complexity, leading to bottlenecks. This makes time-dependent model order reduction strategies highly attractive in this context. Among these strategies, variational approaches like Dirac-Frenkel have garnered significant interest. In this contribution, we delve into one such approach: Dynamical Low-Rank (DLR) approximation. We provide an overview of the main elements, examine the limitations in terms of time accuracy, and introduce a retraction perspective on the topic. We demonstrate how retractions can be employed to explore well-known DLR techniques, such as the KSL and KLS algorithms, and to devise novel approaches. In particular, we show that these retractions are well-suited for constructing curve approximations on manifolds, with a local truncation error of order three. This introduces new perspectives on addressing current time-accuracy constraints in recent robust DLR algorithms.


10/11/23	Workshop on PDEs and Uncertainty Quantification
	Claudia Garetto (London), Michael Kunzinger (Wien), Edoardo Patelli (Glasgow)


17/10/23	Phan Thanh Nam (Munich)
12:30	Critical mass in nuclear fission and isoperimetric inequalities

	I will discuss the connection from the critical mass in nuclear fission described via the liquid drop model and the classical isoperimetric inequalities. In particular, I will address some recent results on the existence/nonexistence of minimizers, as well as several open questions. The talk is based on joint work with Rupert Frank.


04/07/23	Michael Oberguggenberger (Innsbruck)
12:30	Solutions to semilinear wave equations of very low regularity

	The talk reports about recent joint work with Heiko Gimperlein. First, we exhibit new phenomena concerning wellposedness and propagation of singularities for semilinear wave equations with p-th power nonlinearity for initial data of very low Sobolev-regularity. In one space dimension, solutions whose singular support propagates along any ray outside the light cone are obtained. These solutions exist for any Sobolev exponent s < 1/2 in space, while the singular support of any solution of higher regularity is known to be contained in the light cone. Second, we establish a new wellposedness result for semilinear wave equations with Sobolev data whose Fourier transform is supported in a half-line. The result improves the wellposedness results for Sobolev data without the support condition and, in some cases, gives Sobolev-wellposedness for certain s < 0. Extensions to higher space dimensions as well as some observations illuminating the phenomena are given.


16/05/23	Erich Wehrle (Collins Aerospace), industrial seminar
12:30, HSB6	Design optimization including dynamics and nonlinearity


25/04/23	Ruma Maity (Aalto, Helsinki)
12:30	Finite element methods for reduced Landau-De Gennes minimization problems of nematic liquid crystals and ferronematic systems


21/03/23	Akansha Sanwal (Innsbruck)
12:30	Low regularity well-posedness for dispersion generalised KP-I equations

	The talk concerns new well-posedness results for the dispersion generalised KP-I equations in R2 with initial data in anisotropic Sobolev spaces. For strong dispersion, we show global well-posedness in L2. This is achieved by exploiting transversality in the resonant case via bilinear Strichartz estimates and nonlinear Loomis-Whitney inequality. For small dispersion, the equations cannot be solved by Picard iteration and we use frequency-dependent time localisation. This is based on joint work with Robert Schippa (Karlsruhe Institute of Technology, Germany).


14/03/23	Gissell Estrada-Rodriguez (UPC Barcelona / Oxford)
12:30	Macroscopic limits of kinetic equations for swarming


24/01/23	Maciej Maliborski (Vienna)
13:15	Soliton resolution for critical equivariant Yang-Mills equation outside a ball

	Inspired by the recent work of Jendrej and Lawrie on soliton resolution for equivariant wave maps, we consider the 4-dimensional equivariant Yang-Mills equation outside a ball, for which solutions exist globally. We provide numerical evidence supporting the soliton resolution conjecture for our toy model: asymptotically, the solution approaches a superposition of rescaled kinks and radiation. Based on joint work with Piotr Bizon and Bradley Cownden.


11/01/23	Rolf Stenberg (Aalto University)
16:30	Stabilized Finite Element Methods for Contact Problems


16/12/22	Lukas Eigentler (Dundee / Bielefeld)
14:00	Modelling dryland vegetation patterns

	Vegetation patterns are a ubiquitous feature of semi-arid regions and are a prime example of a self-organisation principle in ecology. In this talk, I present bifurcation analyses of two PDE models to (i) investigate the effects of nonlocal seed dispersal, and (ii) identify a mechanism that enables species coexistence despite competition for a limiting resource. First, I present a nonlocal model in which plant dispersal is modelled by a nonlocal convolution term, motivated by empirical data. Asymptotic analysis of the model is possible due to a scale difference between plant dispersal and water transport. I show that a condition for pattern onset in the model can be derived analytically, which indicates that long-range seed dispersal inhibits the onset of spatial patterns. Results on pattern existence and stability, obtained via a numerical continuation method, further show a change in the type of stability boundaries in the pattern's stability regions as dispersal distance is varied. This suggests increased resilience of patterns to reductions in precipitation due to long dispersal distances. Stability results further propose a resolution of a mismatch between previous mathematical models predicting movement of vegetation patterns and some field studies reporting stationary patterns. Second, I reveal that the vegetation's self-organisation principle also acts as a coexistence mechanism. I present a multispecies model for two plant species that interact with a sole limiting resource. A stability analysis of the system's single-species patterns, performed through a calculation of their essential spectra, provides an insight into the onset of coexistence states. I show that coexistence solution branches bifurcate off single-species solution branches as the single-species states lose their stability to the introduction of a second species. Moreover, I present a comprehensive existence and stability analysis to establish key conditions, including a balance between the species' local competitive abilities and their colonisation abilities, for species coexistence in the model.


06/12/22	Lukas Neumann (Innsbruck)
13:15	Initial boundary value problems for 1D pressureless gas dynamics

	It is well known that solutions to the pressureless Euler system develop singularities in finite time also for very smooth initial data. Analytical difficulties generated by this fact become even more severe if one looks at the initial boundary value problem. We extended a potential method originally proposed by F. Huang and Z. Wang to the boundary value problem in the quarterspace. This allows to construct solutions that satisfy the inflow boundary conditions in an appropriate sense. If time allows we will discuss extensions to dual boundaries and uniqueness issues specific to the boundary value problem. This talk is based on joint work with Abhrojyoti Sen, Michael Oberguggenberger and Manas. R. Sahoo.


22/11/22	Heiko Gimperlein (Innsbruck, inaugural lecture)
16:00	Anomalous diffusion and complex materials:
	From models to simulations in space and time


15/11/22	Christian Klein (Bourgogne)
13:15	Multi-domain spectral methods for dispersive PDEs

	We discuss numerical approaches for various nonlinear dispersive PDEs, in particular the nonlinear Schrödinger equation, the Korteweg-de Vries equation and the Benjamin-Ono equation. In particular we are interested in numerical methods on the compactified real line thus avoiding non-reflecting boundary conditions. As examples we consider the stability of breathers and the solutions for initial data not vanishing at infinity.


08/11/22	Alexander Ostermann (Innsbruck)
13:15	Bourgain techniques for error estimates at low regularity

	Standard numerical integrators such as splitting methods or exponential integrators suffer from order reduction when applied to semi-linear dispersive problems with non-smooth initial data. In this talk, we focus on the cubic nonlinear Schrödinger equation with periodic boundary conditions. For such problems, we present and analyze (filtered) Fourier integrators that exhibit superior convergence rates at low regularity. Numerical examples illustrating the analytic results will be given. This is joint work with Frederic Rousset (Paris-Saclay), Katharina Schratz (Sorbonne, Paris), Yifei Wu (Tianjin University, China) and Fangyan Yao (South China University, Guangzhou)


11/10/22	Sebastien Court (Innsbruck)
12:15	Feedback stabilization of a surface tension system modeling the motion of a two-dimensional soap bubble

	The aim of this work is to design a feedback operator for stabilizing in infinite time horizon a system modeling the interactions between a viscous incompressible fluid and the deformation of a soap bubble. The latter is represented by an interface separating a bounded domain of $\R^2$ into two connected parts filled with viscous incompressible fluids. The interface is a smooth perturbation of the 1-sphere, and the surrounding fluids satisfy the incompressible Stokes equations in time-dependent domains. The mean curvature of the surface defines a surface tension force which induces a jump of the normal trace of the Cauchy stress tensor. The response of the fluids is a velocity trace on the interface, governing the time evolution of the latter, via the equality of velocities. The data are assumed to be sufficiently small, in particular the initial perturbation, that is the initial shape of the soap bubble is close enough to a circle. The control function is a surface tension type force on the interface. We design it as a finite-dimensional feedback operator, which enables us to define a control operator that stabilizes locally the soap bubble to a circle with an arbitrary exponential decay rate, up to non-contact with the outer boundary.