# The DNA Seminar (spring 2019)

The DNA seminar is the seminar series of the DNA group (Differential Equations and Numerical Analysis). Seminars will usually be held on Wednesdays, 14:15-15:00 in room 734.

If you would like to give a talk, or have a guest who would like to give a talk, then please contact Markus Grasmair.

## Upcoming seminars:

Time/place: Tuesday 07th of May, 14:15–15:00, room 734
Speaker: Fabian Harang (Oslo)
Title: A multiparameter sewing lemma and applications
Abstract: We present an extension of the sewing lemma to multi parameter integrands. We apply this to study existence and uniqueness of hyperbolic equations driven by Hölder fields on hypercubes in a Young regime. Furthermore, we discuss the challenges of extending the results to a "rough path" regime, and we will propose further applications.

## Earlier this term:

Time/place: Thursday 04th of April, 14:15–15:00, room 656
Speaker: Didier Pilod (Bergen)
Title: Full family of flattening solitary waves for the mass critical generalized Korteweg–de Vries equation
Abstract: For the mass critical generalized Korteweg–de Vries equation on $\mathbb{R}$, we construct a family of non-scattering solutions describing flattening solitary waves for the full relevant range of exponents. The result and its proof are inspired by and complement recent blow-up result for this equation.
This talked is based on a joint work with Yvan Martel (École Polytechnique).
Time/place: Wednesday 27th of March, 14:15–15:00, room 734
Speaker: Nikita Kopylov (NTNU)
Title: Magnus-based geometric integrators for dynamical systems with time-dependent potentials
Abstract: The talk addresses the numerical integration of Hamiltonian systems with explicitly time-dependent potentials. These problems are common in mathematical physics as they appear in quantum, classical and celestial mechanics. The goal is to construct integrators for several important non-autonomous problems: the Schrödinger equation; the Hill and the wave equations, that describe oscillating systems; the Kepler problem with time-variant mass.
Proposed integrators are based on the composition and splitting methods and the Magnus expansion. The general idea is to recombine some simpler integrators to approximate the solution given by the Magnus expansion. For the linear Schrödinger equation with time-dependent potential, Magnus-based quasi-commutator-free integrators are build. Their efficiency is shown in numerical experiments with the Walker–Preston model of a molecule in an electromagnetic field. New Magnus-splitting methods are designed for the wave and the Hill equations. Their performance is demonstrated in numerical experiments with the Mathieu equation, the matrix Hill equation, the wave and the Klein–Gordon–Fock equations. The application of this approach to non-linear case is shown on the example of the Kepler problem with a decreasing mass.
Time/place: Wednesday 20th of March, 14:15–15:00, room 734
Speaker: Agnieszka Kałamajska (Warsaw)
Title: Strongly nonlinear multiplicative inequalities and their applications to PDEs
Abstract: We discuss the variants of the multiplicative interpolation inequality: $\int_\Omega |\nabla f(x)|^p h(f(x))\, dx \le C \int_\Omega \left( \sqrt{|\nabla^{(2)} f(x) {\mathcal{T}}_h (f(x))|}\right)^{p} h(f(x))\, dx,$ where $\Omega$ is a subset in $\mathbb{R}^n$, ${\mathcal{T}}_h (s)$ is certain transform of the function $h(\cdot )$ such that it retrieves the power functions up to the constant, under certain assumptions. Such inequalities were obtained in the series of my joint works written together with Jan Peszek, Tomasz Choczewski, Katarzyna Mazowiecka, Katarzyna Pietruska-Paluba, Alberto Fiorenza and Claudia Capone. They imply the classical Gagliargo-Nirenberg’s interpolation inequality when $p\ge 2$. In my talk I plan to show applications of that inequality to the regularity theory for degenerated PDEs of elliptic type, as well as mention some open problems and their possible applications in PDEs.
Time/place: Wednesday 13th of March, 14:15–15:00, room 734
Speaker: Geir Bogfjellmo (ICMAT)
Title: Symplectic integrators for atomistic spin dynamics
Abstract: For numerical integration of Hamiltonian systems, it is an advantage to use symplectic integrators. Standard symplectic integrators are well-known, but do not apply for non-flat symplectic spaces. In atomistic spin dynamics, the symplectic space $S_2^n\times T^* \mathbb{R}^m$ occurs as the state space of spin-lattice models of ferromagnetic materials. The topic of the talk is symplectic integrators on this space.
Time/place: Wednesday 06th of March, 14:15–15:00, room 734
Speaker: Andre Massing (NTNU)
Title: Unfitted finite element methods: discretizing geometry and partial differential equations
Abstract: Many advanced engineering problems require the numerical solution of multidomain, multidimension, multiphysics and multimaterial problems with interfaces. When the interface geometry is highly complex or evolving in time, the generation of conforming meshes may become prohibitively expensive, thereby severely limiting the scope of conventional discretization methods.

For instance, the simulation of blood flow dynamics in vessel geometries requires a series of highly non-trivial steps to generate a high quality, full 3D finite element mesh from biomedical image data. Similar challenging and computationally costly preprocessing steps are required to transform geological image data into conforming domain discretizations which respect complex structures such as faults and large scale networks of fractures. Even if an initial mesh is provided, the geometry of the model domain might change substantially in the course of the simulation, as in, e.g., fluid-structure interaction and free surface flow problems, rendering even recent algorithms for moving meshes infeasible. Similar challenges arise in more elaborated optimization problems, e.g. when the shape of the problem domain is subject to the optimization process and the optimization procedure must solve a series of forward problems for different geometric configuration.

In this talk, we focus on recent unfitted finite element technologies as one possible remedy. The main idea is to design discretization methods which allow for flexible representations of complex or rapidly changing geometries by decomposing the computational domain into several, possibly overlapping domains. Alternatively, complex geometries only described by some surface representation can easily be embedded into a structured background mesh. In the first part of this talk, we briefly review how finite element schemes on cut and composite meshes can be designed by either using a Nitsche-type imposition of interface and boundary conditions or, alternatively, a partition of unity approach. Some theoretical and implementational challenges and their rectifications are highlighted. In the second part we demonstrate how unfitted finite element techniques can be employed to address various challenges from mesh generation to fluid-structure interaction problems, solving PDE systems on embedded manifolds of arbitrary co-dimension and PDE systems posed on and coupled through domains of different topological dimensionality.
Time/place: Thursday 28th of February, 14:15–15:00, room 656
Speaker: Jun Xue, Tianjin University
Title: Optimal small data scattering for the generalized derivative nonlinear Schrödinger equations
Abstract: We give the optimal small data scattering result for generalized derivative nonlinear Schrödinger equations. By defining a suitable working space, we obtain the closed estimation using tools of Strichartz’s estimates, maximal function estimates, smoothing effects, Leibniz and chain rule for fractional derivatives. This is the content of joint works with Ruobong Bai and Yifei Wu.
Time/place: Thursday 14th of February, 13:15–14:00, room 656
Speaker: Bruno Vergara (ICMAT)
Title: Convexity of Whitham's wave of extreme form
Abstract: In this talk I will discuss a conjecture of Ehrnström and Wahlén on the profile of travelling wave solutions of extreme form to Whitham's non-local dispersive equation. We will see that there exists a highest, cusped and periodic solution that is convex between consecutive crests, at which $C^{1/2}$-regularity has been shown to be optimal. The talk is based on joint work with A. Enciso and J. Gómez-Serrano.
Time/place: Wednesday 23rd of January, 14:15–15:00, room 734
Speaker: Nikolas Tapia (NTNU)
Title: Modification of branched rough paths
Abstract: Branched rough paths are a generalization of T. Lyons' geometric rough paths, introduced in 2010 by M. Gubinelli. They have played an important role in the solution theory of singular SPDEs and were one of the sources of inspiration for M. Hairer's theory of Regularity Structures. The problem of existence of a branched rough path above a fixed vector-valued Hölder path is, of course, important. We propose a solution to this problem based on an explicit form of the Baker-Campbell-Hausdorff formula due to Reutenauer, and on the Hairer-Kelly map. We introduce a new class of rough paths which we call anisotropic geometric rough paths. Our techniques also allow us to give an action of a Banach space of Hölder functions on branched rough paths.
This is joint work with L. Zambotti (Paris).

## Previous semester:

Time/place: Friday 14th of December, 14:15-15:00, room 734
Speaker: Evgueni Dinvay (Bergen)
Title: Global well-posedness for a dispersive system of the Whitham–Boussinesq type
Abstract: A linearly fully dispersive Boussinesq system modelling surface waves of an inviscid incompressible fluid is under consideration. In this talk, I will give a proof of the local well-posedness of the initial value problem for the system in classical Sobolev spaces implementing a dispersive estimate of Strichartz type with the fixed point argument. Conservation of Hamiltonian allows us to extend globally well-posedness at least for small initial data.
The work is done in collaboration with Achenef Tesfahun Temesgen (UiB).
Time/place: Friday 07th of December, 14:15-15:00, room 734
Speaker: Eero Ruosteenoja (NTNU)
Title: Tug-of-war games and the normalized $p$-Laplacian
Abstract: Tug-of-war games are two-player zero-sum stochastic games that have connections to $p$-Laplace and infinity Laplace equations. These games were discovered by Peres, Schramm, Sheffield and Wilson in 2006-2008. I will explain the connection and discuss some properties of the normalized $p$-Laplace equation arising from the games.
Time/place: Tuesday 04th of December, 14:15-15:00, room 734
Speaker: Mathew A. Johnson (University of Kansas)
Title: On the Stability of Roll Waves
Abstract: Roll-waves are a well observed hydrodynamic instability occurring in inclined thin film flow, often mathematically described as periodic traveling wave solutions of either the viscous or inviscid St. Venant (i.e. shallow water) system. In this talk, I will discuss recent progress concerning the stability of both viscous and, if time allows, inviscid roll-waves in a variety of asymptotic regimes, including near the onset of hydrodynamic instability and large-Froude number analysis.
This is joint work with Blake Barker (BYU), Pascal Noble (University of Toulouse), L. Miguel Rodrigues (University of Rennes), Zhao Yang (IU) and Kevin Zumbrun (IU).
Time/place: Friday 09th of November, 14:15-15:00, room 734
Speaker: Antti Hannukainen (Aalto University)
Title: Application of the LOD Method to the Eddy Current Problem
Abstract: Computational electromechanics aims to predict the behavior of a given electrical machine, e.g., to compute eddy current losses or torque from a set of known inputs. This involves numerically solving the time-dependent magnetic field inside a device from a suitably selected approximation of the Maxwell's equations. The properties of electrical machines: complex geometry, non-linear and rapidly varying material parameters, a rotating computational domain as well as the coupling between mechanical and magnetic fields, make the problem challenging to study.
In this talk, I discuss approximate solution of the magnetic field in electrical machine parts with rapidly varying material properties. Examples of such parts are, e.g., the laminated core of an electrical motor or the coils of a transformer. I consider the eddy current problem, that is often used to model low-frequence electrical machines, and apply the localized orthogonal decomposition-method (LOD) to treat the varying material parameters.
The LOD-method is based on decomposition of the solution space into space of slowly and rapidly oscillating functions, see [1]. A projection operator is used to obtain a problem for the slowly oscillating component. A special care has to be placed in the construction of the spaces, so that the action of the projection can be locally approximated. A suitable construction for the eddy current problem is obtained by application of the Helmholtz decomposition. I discuss the theoretical properties of the LOD-method in the context of Poisson equation, finite element solution of the eddy current problem and the application of the LOD-method to the eddy current problem.

[1] A. Målqvist and D. Peterseim, “Localization of elliptic multiscale problems,” Mathematics of Computation, vol. 83, no. 290, pp. 2583–2603, 2014.
Time/place: Friday 02nd of November 2018, 14:15-15:00, room 734
Speaker: Joscha Diehl (MPI Leipzig)
Title: Invariants of the iterated-integral signature
Abstract: Recently the iterated-integral signature has found applications in statistics and machine learning. In many situations, there is a group acting on data that one wants to "mod out". One example is the accelerator data coming from a mobile phone. The orientation of the phone in a user's pocket is usually unknown. Preprocessing of the data then tries to calculate features that are invariant to the action of $SO(3)$. I describe how invariant features (to $SO$, $GL$ and permutations) can be found in the signature. This is joint work with Jeremy Reizenstein (University of Warwick).
Time/place: Wednesday 24th of October 2018, 14:15-15:00, room 656
Title: Conservative Galerkin methods as adaptive algorithms for Hamiltonian PDEs
Abstract: Here we investigate the construction of conservative methods for Hamiltonian PDEs, a large class of PDEs endowed with physically relevant geometric structures. Namely, these problems conserve an underlying Hamiltonian functional over time. They arise from a variety of areas, not least meteorological, such as the semi-geostrophic equations, and oceanographical, such as Korteweg-de Vries (KdV) type equations and the nonlinear Schrodinger equations. We describe a general methodology for the construction of spatial finite element discretisations which conserve a discrete Hamiltonian functional. Restricting our focus to KdV type equations we present a conservative spatial finite element method, which we couple with a discrete gradient method in time. We go on to investigate the effects of spatial adaptivity over time on both stability and the conservative properties of our numerical scheme.
Time/place: Friday 19th of October 2018, 14:15-15:00, room 734
Speaker: Bernd Kawohl (Cologne)
Title: An Analyst's View at Image Processing
Abstract: A standard goal in mathematical image processing is to remove noise and to sharpen contours. There are two analytical approaches to this problem. A variational one due to Mumford and Shah, and a PDE approach using forward-backward diffusion equations connected to the names Perona and Malik. In my lecture I will try to show what these seemingly unrelated approaches have in common and that they lead to uncovered territory in mathematics.
Time/place: Tuesday 16th of October 2018, 14:15-15:00, room 734
Speaker: Geir Bogfjellmo (ICMAT)
Title: Interpolation on non-flat spaces
Abstract: Cubic spline interpolation on Euclidean space is a standard topic in numerical analysis, with countless applications in science and technology. The generalization of cubic splines to manifolds is not self-evident, with several distinct approaches. One possibility is to mimic De Casteljau's algorithm, which leads to generalized Bézier curves. To construct $C^2$-splines from such curves is a complicated non-linear problem, until now lacking numerical methods.
We have derived an iterative algorithm for $C^2$-splines on Riemannian symmetric spaces, and proven convergence. In terms of numerical tractability and computational efficiency, the new method surpasses those based on Riemannian cubics. We demonstrate the algorithm for three geometries of interest: the $n$-sphere, complex projective space, and the real Grassmannian.
Time/place: Friday 05th of October 2018, 14:15-15:00, room 734
Speaker: Tommi Brander (NTNU)
Title: Variable exponent Calderón's problem in one dimension
Abstract: We consider Calderón's problem of recovering an unknown weight function (conductivity) in an elliptic PDE from measurements at the boundary. In the one-dimensional case this amounts to recovering the conductivity of a wire from measurements of voltage and current; in the usual linear case, only the total conductance can be recovered. We use the nonlinear $p(x)$-Laplace equation as a forward model and show that the conductivity can be recovered in the coarsest (smallest) sigma-algebra that makes the exponent $p(x)$ measurable. Conditional expectation and multiplicative system theorem may also make an appearance.
Time/place: Friday 28th of September 2018, 14:15-15:00, room 734
Speaker: Hao Ni (University College London)
Title: Modelling the Effects of Data Streams using the signature feature and its Applications
Abstract: Regression analysis aims to use observational data from multiple observations to develop a functional relationship relating explanatory variables to response variables, which is important for much of modern statistics, and econometrics, and also the field of machine learning. In this talk, we consider the special case where the explanatory variable is a data stream. We provide an approach based on identifying carefully chosen features of the stream which allows linear regression to be used to characterize the functional relationship between explanatory variables and the conditional distribution of the response; the methods used to develop and justify this approach, such as the signature of a stream and the shuffle product of tensors, are standard tools in the theory of rough paths and provide a unified and non-parametric approach with potential significant dimension reduction. To further improve the efficiency of the signature method, we can combine the non-linear regression method (e.g. neural network) with the signature feature set. Numerical examples are provided to show the superior performance of the proposed method. Lastly I will show that the signature based method have achieved the state-of-the-art results in online handwritten text recognition and action recognition.
Time/place: Tuesday 18th of September 2018, 14:15-15:00, room 734
Speaker: Valeriya Naumova (Simula Research Laboratory)
Title: A machine learning approach to optimal regularization: affine manifolds
Abstract: Despite a variety of methods and techniques for parameter choice, the issue of parameter and model selection, in general, still remains a challenge for many applications. The main difficulty lies in constructing a rule, allowing to choose the parameter from a given noisy dataset without relying either on any a priori knowledge of the solution or on the noise level.
In this talk, combining advances from statistical learning theory with insights from regularisation theory, we propose a novel approach to approximate the high-dimensional function, mapping noisy data into a good approximation to the optimal parameter in Tikhonov and elastic-net regularisation. Our assumptions are that solutions of the problem are statistically distributed in a concentrated manner on (lower dimensional) linear subspaces and the noise is sub-gaussian. We provide explicit error bounds on the accuracy of the approximated parameter and the corresponding regularization solution. Furthermore, we present an efficient algorithm for the computation of an approximate optimal parameter from a given training data. We also compare our approach to the state-of-the-art parameter selection criteria and illustrate its superiority in terms of accuracy and computational time on a number of simulated and real data.
Time/place: Tuesday 28th of August 2018, 14:15-15:00, room 734
Speaker: Olga Trichtchenko (Western University, Canada)
Title: Stability of periodic travelling wave solutions to Kawahara and related equations
Abstract: In this talk, we explore the simplest dispersive equation that exhibits high-frequency instabilities, the Kawahara equation. We show how to derive criteria for an instability to occur and relate it to a generalised resonance condition. We proceed by examining how the instability changes and grows in time as a function of the underlying solution we perturb about. We conclude by commenting on what happens with a different dispersion relation or different nonlinearity in the underlying equation.
Time/place: Thursday 30th of August 2018, 10:15-11:00, room KJL23 (kjelhuset)
Speaker: Giancarlo Sangalli (Pavia)
Title: Isogeometric Analysis: A valuable high-order method
Abstract: The concept of k-refinement was proposed as one of the key features of isogeometric analysis, ”a new, more efficient, higher-order concept”, in the seminal work [1]. The idea of using high-degree and continuity splines (or NURBS, etc.) as a basis for a new high-order method appeared very promising from the beginning and received confirmations from the next developments. The k-refinement leads to several advantages: Higher accuracy per degree-of-freedom, improved spectral accuracy, the possibility of structure-preserving smooth discretization's are the most interesting features that have been studied actively in the community. At the same time, the k-refinement brings significant challenges at the computational level: using standard finite element routines, its computational cost grows with respect to the degree, making degree raising computationally expensive. However, recent ideas allow a computationally efficient k-refinement: I present in this talk the results of [2].

References:
[1] T.J.R. Hughes, J.A. Cottrell, and Y. Bazilevs: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods Applied Mechanics and Engineering, 194: 4135-4195 (2005).
[2] G. Sangalli and M. Tani: Matrix-free isogeometric analysis: The computationally efficient k- method, arXiv:1712.08565, pp. 1–21 (2017).
[3] F. Calabrò, G. Sangalli, and M. Tani: Fast formation of isogeometric Galerkin matrices by weighted quadrature, Computational Methods in Applied Mechanics and Engineering, 316: 606–622 (2017).