The DNA Seminar (spring 2022)

The DNA seminar is the seminar series of the DNA group (Differential Equations and Numerical Analysis). We meet Mondays, 14:15–15:00, in room 734, SBII, Campus Gløshaugen and on zoom.

If you would like to give a talk or have a guest who would like to speak in the seminar, then please contact Kurusch Ebrahimi-Fard.

Upcoming seminars:

Time/place: Monday, June 13, 2022, 14:15-15:00, Zoom
Speaker: Salvador Ortiz-Latorre (UiO)
Title: tba
Abstract: tba \(\phantom{mmmmmmmmmmmmmm} \hspace{20cm}\)
Time/place: Monday, April 4, 2022, 14:15-15:00, Zoom
Speaker: Adeline Fermanian (Mines ParisTech, France)
Title: tba
Abstract: tba \(\phantom{mmmmmmmmmmmmmm} \hspace{20cm}\)
Time/place: Monday, March 21, 2022, 14:15-15:00, Zoom & Room 656, SBII, Campus Gløshaugen
Speaker: Joscha Diehl (Univ. of Greifswald, Germany)
Title: tba
Abstract: tba \(\phantom{mmmmmmmmmmmmmm} \hspace{20cm}\)
Time/place: Monday, March 7, 2022, 14:15-15:00, Zoom
Speaker: Christa Cuchiero (Univ. of Vienna, Austria)
Title: tba
Abstract: tba \(\phantom{mmmmmmmmmmmmmm} \hspace{20cm}\)
Time/place: Monday, February 21, 2022, 14:15-15:00, Zoom
Speaker: David Cohen (Chalmers Univ., Sweden)
Title: Efficient discretisations of stochastic Hamiltonian and Poisson systems
Abstract: We start by recalling classical results on time discretisations of (deterministic) Hamiltonian and Poisson systems. We will then randomly perturb such systems, and present and analyse various time integrators for an efficient simulation of stochastic Hamiltonian and Poisson systems.
The presentation is based on joint works with C-E. Bréhier, C. Chen, R. D'Ambrosio, K. Debrabant, T. Jahnke, A. Lang, A. Rößler and G. Vilmart.
Time/place: Monday, February 7, 2022, 14:15-15:00, Zoom
Speaker: Avi Mayorcas (Univ. of Cambridge, UK)
Title: tba
Abstract: tba \(\phantom{mmmmmmmmmmmmmm} \hspace{20cm}\)
Time/place: Monday, January 31, 2022, 14:15-15:00, Zoom & Room 656, SBII, Campus Gløshaugen
Speaker: Charles Curry (NTNU)
Title: Stochastic quantization: theory and practice
Abstract: We present an introduction to stochastic quantization, demonstrating how SDEs and SPDEs are useful tools in understanding phenomena ranging from analysis of Quantum Field Theories to solid state physics. In doing so we discuss aspects of renormalization and conclude with an outlook on numerical computations and challenges arising in generalizations of present techniques, particularly to gauge theories such as QED and QCD.

Earlier this term:

Time/place: Monday, January 24, 2022, 14:15-15:00, Zoom
Speaker: Milo Viviani (Scuola Normale Superiore, Pisa, Italy)
Title: Canonical Scale Separation in 2D Incompressible Hydrodynamics
Abstract: The fundamental rules governing a two-dimensional inviscid incompressible fluid are simple. Yet, to characterize the long-time behavior is a knotty problem. The fluid's motion is described by Euler's equations: a non-linear Hamiltonian system with infinitely many conservation laws. In both experiments and numerical simulations, coherent vortex structures, or blobs, emerge after an initial stage. These formations dominate the large-scale dynamics, but small scales also persist.
In his classical work, Kraichnan qualitatively describes a forward cascade of enstrophy into smaller scales and a backward cascade of energy into larger scales. Previous attempts to model Kraichnan's double cascade use filtering techniques that enforce separation from the outset. Here we show that Euler's equations possess an intrinsic, canonical splitting of the vorticity function. The splitting is remarkable in four ways:
(i) it is defined solely via the Poisson bracket and the Hamiltonian,
(ii) it characterizes steady flows,
(iii) without imposition it yields a separation of scales, enabling the dynamics behind Kraichnan's qualitative description,
(iv) it accounts for the "broken line” in the power law for the energy spectrum (observed in both experiments and numerical simulations).
The splitting originates from Zeitlin's truncated model of Euler's equations in combination with a standard quantum-tool: the spectral decomposition of Hermitian matrices. In addition to theoretical insight, the scale separation dynamics could be used for stochastic model reduction, where small scales are modelled by multiplicative noise. Preprint available at: https://arxiv.org/abs/2102.01451

Previous semesters:

Time/place: Monday, December 13, 2021, 14:15-15:00, Zoom & room 734, SBII, Campus Gløshaugen
Speaker: Karl K. Brustad (NTNU)
Title: One-dimensional Dirichlet problem for nonlocal equations
Abstract: Explicit solutions to the Dirichlet problem for a class of mean value equations on the real line are derived. This is a special case of a problem considered in joint work with Peter Lindqvist and Juan Manfredi, where existence, uniqueness, and uniform convergence of the nonlocal solutions towards the solution of the corresponding PDE was established.
Time/place: Monday, December 6, 2021, 14:15-15:00, Zoom
Speaker: Subbarao Venkatesh Guggilam (Old Dominion Univ., Norfolk, USA)
Title: Additive Static Feedback Connection: A Chen–Fliess Series Viewpoint
Abstract: Michel Fliess, in 1981, proved that an analytic dynamical system with input entering linearly has an input-output description written purely in terms of sum of weighted iterated integrals, called Chen-Fliess series. The expressions of iterated integrals can be symbolized by words formed out of noncommutative letters or indeterminates. Hence, the input-output description of such analytic vector fields can be described by a Chen-Fliess series whose underlying descriptor is a formal power series. The problem considered in the seminar is when does a Chen-Fliess series in an additive static feedback connection with a formal static map (definition of the map does not inherently depend on time) yield a closed-loop system with a Chen-Fliess series expansion? One can prove that such a closed-loop system always has a Chen-Fliess series representation. Furthermore, an algorithm based on the Hopf algebras for the shuffle group and the dynamic output feedback group is designed to compute the generating series of the closed-loop system. It is proved that the additive static feedback connection preserves local convergence and relative degree, but a counterexample shows that the additive static feedback does not preserve global convergence in general. The results presented are a part of the Ph.D. dissertation of the speaker. The talk will begin with a slight introduction to Chen-Fliess series.
Time/place: Thursday, December 2, 2021, Zoom & room 656 (SIMA-stuen)
14:15 - 15:00
Speaker: Olivier Ley (Rennes)
Title: Infinite and finite horizon stochastic Mean Field Games on networks
Abstract: The purpose of the talk is to describe several results obtained in collaboration with Yves Achdou (Paris), Manh-Khang Dao (Rouen) and Nicoletta Tchou (Rennes) about Mean Field Games on general networks. We study the associated coupled system of a Hamilton-Jacobi-Bellman PDE and a Fokker-Planck PDE. The system is complemented with Kirchhoff conditions for the HJB equations and dual transmission conditions for the FP equation at the vertices of the network. When dealing with general general Kirchhoff conditions, the value function associated with the stochastic control problem become discontinuous at the vertices, which is one the main difficulty. We prove the well-posedness of of the system both in the stationary and parabolic case.
15:15-16:00
Speaker: Annalisa Cesaroni (Padova)
Title: Fractional mean curvature flow of entire Lipschitz graphs.
Abstract: I will discuss some results on the fractional mean curvature flow of entire Lipschitz graphs, in particular regularity results, and long time asymptotics. By the level set method, it is possible to describe this evolution by looking at viscosity solutions of a fractional quasilinear parabolic problem. As in the classical case, in a suitable rescaled framework, if the initial graph is a sublinear perturbation of a cone, the evolution asymptotically approaches an expanding self-similar solution. We also discuss some results in the unrescaled case, such as stability of hyperplanes and stability of mean convex cones.
Time/place: Monday, November 29, 2021, 14:15-15:00, Zoom
Speaker: Markus Tempelmayr (MPI Leipzig)
Title: A multi-index based regularity structure for quasi-linear SPDEs
Abstract: We give an overview of the solution theory for singular SPDEs in case of a quasi-linear equation, following the recent approach of Otto, Sauer, Smith, Weber. The basic idea is to parametrize the model, which captures the local solution behavior, by partial derivatives w.r.t. the non-linearity. This allows for an efficient bookkeeping and an inductive construction of the model. Then we construct the structure group, which is needed to ``re-center'' the model, based on a Lie algebra consisting of infinitesimal generators of actions in the space of non-linearities. Although the approach is tree-free, we show morphism properties w.r.t. well-known tree-based structures in branched rough paths and regularity structures. Based on joint work with Pablo Linares and Felix Otto.
Time/place: Monday, November 22, 2021, 14:15-15:00, Zoom & room 734, SBII, Campus Gløshaugen
Speaker: Douglas Svensson Seth (NTNU)
Title: The three dimensional water wave problem with vorticity
Abstract: In 1981 Reeder and Shinbrot published the first rigorous existence result for the three dimensional water wave problem. Since then the theory has evolved and we will begin with a brief overview to highlight some of the differences between the two and the three dimensional problem. The rest of the talk will be dedicated to two more recent existence results for the three dimensional problem where the vorticity is nonzero. The first is based on the assumption that the velocity field of the water is a Beltrami field. In the other, the vorticity is given by an assumption that stems from magnetohydrodynamics. This is joint work with Erik Wahlén, Evgeniy Lokharu and Kristoffer Varholm.
Time/place: Monday, November 15, 2021, 14:15-15:00, Zoom & room 734, SBII, Campus Gløshaugen
Speaker: Susanne Solem (NMBU)
Title: A system of PDEs modelling noisy grid cells
Abstract: Grid cells are neurons which play an important role in the internal navigational system of mammals. Even though grid cell networks, and the patterns they create, have been extensively studied within several disciplines in the last years, understanding the effects of noise on the network remains a challenge. In this talk, I will discuss an upscaled noisy grid cell model in the form of a system of partial differential equations. Current results on the robustness of network activity patterns with respect to noise, in terms of bifurcations from unstable homogeneous states, will be presented. Based on joint work with José A. Carrillo (Oxford) and Helge Holden (NTNU).
Time/place: Monday, November 8, 2021, 14:15-15:00, Zoom & room 734, SBII, Campus Gløshaugen
Speaker: James Jackaman (NTNU)
Title: Inexact geometric linear solvers
Abstract: In this talk we will investigate the conservation properties of non-exact linear solvers before convergence. We shall study the famous GMRES (generalized residual method) and propose a modified algorithm which may preserve arbitrarily many conserved quantities. We will investigate this method for a variety of linear finite element discretisations and discuss the benefits and costs of this methodology. In addition, we discuss a user-friendly python implementation of this methodology.
Time/place: Monday, November 1, 2021, 14:15-15:00, Zoom
Speaker: Tien Truong (Lund Univ., Sweden)
Title: Solitary waves in a Whitham equation with small surface tension
Abstract: The talk attempts to connect a weakly dispersive toy model with the original two-dimensional gravity-capillary water wave problem; the proposal of Whitham in 1970 with the discovery of Kirchgässner in 1988. It is interesting because these two problems are structurally different, and the toolbox for nonlocal equations has recently seen a new development inspired by spatial dynamics. We show that the gravity-capillary Whitham equation features generalized solitary waves and modulated solitary waves. This agrees with the findings for the water wave problem. Finally, we discuss a missing ingredient for finding multipulse solitary waves.
Time/place: Monday, October 25, 2021, 14:15-15:00, Zoom
Speaker: William Salkeld (Université Cote d'Azur, France)
Title: Lions calculus and coupled Hopf algebras with applications to probabilistic rough paths
Abstract: In this talk, I will explain some of the foundation results for a new regularity structure developed to study interactive systems of equations and their mean-field limits. At the heart of this solution theory is a Taylor expansion using the so-called Lions measure derivative. This quantifies infinitesimal perturbations of probability measures induced by infinitesimal variations in a linear space of random variables. I will explore how basic properties of Lions derivatives evolve into the structures of a coupled Hopf algebra and describe the implications for Runge-Kutta methods and probabilistic rough paths. These lead to a new solution theory and simulation techniques for rough mean-field equations. This talk is based on ArXiv:2106.09801 and ongoing work with my supervisor Francois Delarue at Universite Cote d'Azur.
Time/place: Wednesday, October 20, 2021, 13:15-14:00, Zoom & room 734, SBII, Campus Gløshaugen
Speaker: Mario Maurelli
Title: Stochastic Euler equations: a geometric approach
Abstract: In their celebrated work [Ann. Math. 1970], Ebin and Marsden have shown local well-posedness of the incompressible Euler equations in any dimension by solving a smooth ODE on the infinite-dimensional space of volume-preserving Sobolev diffeomorphisms. In this talk, we will develop this approach for the incompressible Euler equations driven by an additive, stochastic force term: we will solve a stochastic ODE with smooth coefficients on the space of volume-preserving Sobolev diffeomorphisms and get in turn local well-posedness of the stochastic Euler equations. This approach is quite flexible and we believe it can be used for other stochastic PDEs. No particular prerequisite on geometry or stochastic analysis is needed, we will try to give the intuition behind the arguments. Based on the joint work with Klas Modin and Alexander Schmeding arXiv:1909.09982 .
PS: Mario will give short introduction to Regularisation by noise in ODEs and PDEs on Friday October 22, 11:15-12:15 and 13:15-14:00 in room 822 SB2.
Physical course only. Interested? Send an email to Espen.
Time/place: Monday, October 18, 2021, 14:15-15:00, Zoom & room 734, SBII, Campus Gløshaugen
Speaker: Luca Galimberti (NTNU)
Title: Neural Networks in Fréchet spaces
Abstract: In this talk we present some novel results obtained by Fred Espen Benth (UiO), Nils Detering (University of California Santa Barbara) and myself on abstract neural networks and deep learning. More precisely, we derive an approximation result for continuous functions from a Fréchet space \(X\) into its field \(\mathbb{F}, (\mathbb{F}\in\{\mathbb{R},\mathbb{C} \})\). The approximation is similar to the well-known universal approximation theorems for continuous functions from \(\mathbb{R}^n\) to \(\mathbb{R}\) with (multilayer) neural networks [Cyb,Hor,Fun,Les]. Similar to classical neural networks, the approximating function is easy to implement and allows for fast computation and fitting. Few applications geared toward derivative pricing and numerical solutions of parabolic partial differential equations will be outlined.
[Cyb] G. Cybenko. Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals and Systems, 2(4):303-314, 1989.
[Hor] K. Hornik, M. Stinchcombe, and H. White. Multilayer feedforward networks are universal approximators. Neural Networks, 2(5):359-366, 1989.
[Fun] K.-I. Funahashi. On the approximate realization of continuous mappings by neural networks. NeuralNetworks, 2(3):183-192, 1989.
[Les] M. Leshno, V. Y. Lin, A. Pinkus, and S. Schocken. Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Neural Networks, 6(6):861-867, 1993.
Time/place: Monday, October 11, 2021, 14:15-15:00, Zoom
Speaker: Ilya Chevyrev (Univ. of Edinburgh UK)
Title: Feature Engineering with Regularity Structures
Abstract: In this talk, I present a recent work in which we investigate models from the theory of regularity structures as features in machine learning tasks. A model is a polynomial function of a space-time signal designed to well-approximate solutions to partial differential equations (PDEs). Models can be seen as multi-dimensional generalisations of signatures of paths; this work therefore aims to extend the use of signatures in data science beyond the context of time-ordered data. I will introduce a flexible definition of a model feature vector and two algorithms which combine these features with supervised linear regression. I will also present several numerical experiments in which we use these algorithms to predict solutions to parabolic and hyperbolic PDEs with a given forcing and boundary conditions. Interestingly, in the hyperbolic case, the prediction power relies heavily on whether the boundary conditions are appropriately included in the model. Based on joint work with Andris Gerasimovics and Hendrik Weber.
Time/place: Monday, October 4, 2021, 14:15-15:00, Zoom & room 734, SBII, Campus Gløshaugen
Speaker: Kristoffer Varholm (NTNU)
Title: On the precise local behavior of extreme solutions of nonlocal dispersive equations
Abstract: We prove exact asymptotic behaviour at the origin for nontrivial solutions of a family of nonlocal equations. This family of equations includes those satisfied by the cusped highest steady waves for both the uni- and bidirectional Whitham equations. In particular, our results partially settle conjectures for such extreme waves posed in [1,2,3].
[1] M. Ehrnström and E. Wahlén, On Whitham’s conjecture of a highest cusped wave for a nonlocal dispersive equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 6, 1603–1637.
[2] M. Ehrnström, M. A. Johnson, and K. M. Claassen, Existence of a highest wave in a fully dispersive two-way shallow water model, Arch. Ration. Mech. Anal. 231 (2019), no. 3, 1635–1673.
[3] T. Truong, E. Wahlén, and M. H. Wheeler, Global bifurcation of solitary waves for the Whitham equation, arXiv:2009.04713 (2020).
Time/place: Monday, September 20, 2021, 14:15-15:00, Zoom & room 734, SBII, Campus Gløshaugen
Speaker: Adrien Laurent (Bergen)
Title: Exotic aromatic B-series for sampling the invariant measure of ergodic stochastic differential equations in \(R^d\) and on manifolds
Abstract: For sampling the invariant measure of ergodic systems in large dimensions, it is known that the order conditions are different from those for the standard weak order conditions over short time, and they can be intricate to compute. We propose a new methodology for creating high order integrators for sampling the invariant measure of ergodic SDEs in \(R^d\) and on manifolds. In the particular case of overdamped Langevin dynamics, we obtain the order conditions for a class of Runge-Kutta methods, thanks to a new extension of the Butcher-series, called exotic aromatic B-series. To illustrate the methodology, an integrator of order two is introduced, and numerical experiments on the sphere, the torus and the special linear group confirm the theoretical findings.
Time/place: Monday, September 13, 2021, 14:15-15:00, Zoom & room 734, SBII, Campus Gløshaugen
Speaker: Artur Rutkowski (NTNU)
Title: Dirichlet problem, extension and trace, and Douglas identities for nonlocal operators
Abstract: We give the existence and uniqueness of the variational solutions for the Dirichlet problem governed by general nonlocal operators built on symmetric Lévy measures. We also discuss the extension and trace problem for the related Sobolev-type spaces, motivated by the problem of determining minimal possible assumptions for the exterior condition. In the process, we obtain a nonlocal Douglas identity which connects the energy of the harmonic function with a certain energy of its exterior condition.
Time/place: Wednesday, September 1, 2021, 13:15-14:00, Zoom & room 734, SBII, Campus Gløshaugen
Speaker: Helge Glockner (Paderborn Univ., Germany)
Title: Flows, parameter dependence, and diffeomorphism groups
Abstract: Flows of complete time-dependent vector fields give rise to curves in diffeomorphism groups and thus furnish a link between solutions to ordinary differential equations and infinite-dimensional Lie groups. This sheds a new light on the parameter-dependence of solutions. Notably, it enables approximations to be studied which arise when general vector fields are replaced by vector fields which are piecewise constant in time. The general context of the studies are evolution equations on infinite-dimensional Lie groups (regularity theory). I shall give an introduction to the topic and examples.
Time/place: Friday, June 25, 2021, 14:00–14:45, Zoom
Speaker: Matthew Tandy (NTNU)
Title: Lipschitz Stability for the Hunter-Saxton Equation
Abstract: We study the Lipschitz stability of solutions to the Hunter-Saxton equation, \begin{equation} u_t(x,t) + u(x,t)u_x(x,t) = \frac{1}{4}\left(\int_{-\infty}^{x} u_x^2(y, t)\ dy - \int_x^{+\infty} u_x^2(y,t)\ dy\right). \end{equation} This equation was introduced by Hunter and Saxton in 1991 as a model for the nonlinear instability in the director field of a nematic liquid crystal. Solutions of this equation are of interest because they may develop singularities, called wave breaking, in finite time. That is, the derivative \(u_x \to -\infty\) spatially pointwise, while \(u\) remains bounded and continuous. Different solutions then exist depending on how much of the energy is lost at these singularities, with the loss of energy defined by some given \(\alpha\). In this work we take \(\alpha\) to be a constant between \(0\) and \(1\). To construct solutions, we will transform from the Eulerian formulation to a Lagrangian formulation using a generalised method of characteristics. We will then define a metric in the Lagrangian formulation, which is Lipschitz continuous in time. Using this, we define a metric in Eulerian coordinates that is also Lipschitz continuous in time. This is a joint work with Katrin Grunert.
Time/place: Friday, June 18, 2021, 14:00–14:45, Zoom
Speaker: Ola Isaac Høgåsen Mæhlen (NTNU)
Title: One-sided Hölder conditions for solutions of weakly dispersive equations
Abstract: The majority of dispersive equations in one space-dimension can be realised as dispersive perturbations of the Burgers equation \begin{equation}u_t + u u_x = L u_x,\end{equation} where \(L\) is a local or nonlocal symmetric operator. For sufficiently weak dispersion, the Burgers’ nonlinearity dominates and classical solutions break down due to shock-formation/wave-breaking. Using hyperbolic techniques we establish global existence and uniqueness of entropy solutions for weakly dispersive equations, but our main focus will be on a new generalization of the classical Oleinik estimate for Burgers’ equation. We obtain one-sided Hölder conditions for the solutions, which in turn controls their height and provides a novel bound of the lifespan of classical solutions based on their initial skewness.
This is joint work with Jun Xue (NTNU).
Time/place: Friday, June 11, 2021, 14:00–14:45, Zoom
Speaker: Davide Murari (NTNU)
Title: Lie group integrator’s approach to the N-fold pendulum
Abstract: Lie group integrators have been proven to be effective in many applications, where the preservation of geometrical properties is of interest. In the talk, after a brief introduction to these integrators, this approach to the mechanical system of the N-fold 3D pendulum is proposed. We introduce the main points in the intrinsic derivation of the system on a manifold and present a transitive group action which allows us to reframe it into the Lie group integrators setting. This problem can be considered a toy model to understand how to workout more intricate multi-body systems.
Time/place: Friday, June 4, 2021, 14:00–14:45, Zoom
Speaker: Olav Ersland (NTNU)
Title: On numerical approximations of fractional and nonlocal Mean Field Games
Abstract: In this talk we present a recent work, where we construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove several convergence results: Convergence along subsequences to viscosity-very weak solutions for (i) degenerate equations in dimension d=1 and (ii) nondegenerate equations in arbitrary dimensions. We also give results on full convergence and convergence to classical solutions. Numerical tests are implemented for a range of different nonlocal diffusions and support our analytical findings.
This is a joint work with Indranil Chowdhury and Espen R. Jakobsen.
Time/place: Friday, May 28, 2021, 14:00–14:45, Zoom
Speaker: Esten Nicolai Wøien (NTNU)
Title: A PDE-based Method for Shape Registration
Abstract: In the square root velocity framework, the computation of shape space distances and the registration of curves requires solution of a non-convex variational problem. In the talk, we present a new PDE-based method for solving this problem numerically. The method is constructed from numerical approximation of the Hamilton-Jacobi-Bellman equation for the variational problem, and has quadratic complexity and global convergence for the distance estimate. In conjunction, we propose a backtracking scheme for approximating solutions of the registration problem, which additionally can be used to compute shape space geodesics. The methods have linear numerical convergence, and improved efficiency compared previous global solvers.
Time/place: Friday, May 21, 2021, 14:00–14:45, Zoom
Speaker: Evelyn Buckwar (Johannes Kepler Univ. Linz, Austria)
Title: A couple of ideas on splitting methods for SDEs
Abstract: We discuss developing splitting methods for stochastic differential equations. Splitting methods are a well-known type of numerical methods in the context of Geometric Numerical Integration of ordinary differential equations, in particular, they are known to be structure preserving schemes in various situations. Extensions of these methods to the case of stochastic differential equations exist for a considerable time already and they currently appear to become quite popular. In this talk I will present examples illustrating some benefits of splitting methods for SDEs. Illustrative examples include SDEs employed in neuroscience and computation-based inference.
Time/place: Friday, May 14, 2021, 14:00–14:45, Zoom
Speaker: Karolina Kropielnicka (Univ. of Gdańsk, Poland)
Title: Solving the linear semiclassical Schrödinger equation on the real line
Abstract: The numerical solution of a linear Schrödinger equation in the semiclassical regime is very well understood in a torus \(T^d\). A raft of modern computational methods are precise and affordable while conserving energy and resolving high oscillations very well. This, however, is far from the case with regard to its solution in \(\mathbb{R}^d\), a setting more suitable for many applications.
In this talk we will be concerned with the extension of the theory of splitting methods to this end. We start our journey with numerical analysis on the real line. The main idea is to derive the solution using a spectral method from a combination of solutions of the free Schrödinger equation and of linear scalar ordinary differential equations, in a symmetric Zassenhaus splitting method. This necessitates a detailed analysis of certain orthonormal spectral bases on the real line and their evolution under the free Schrödinger operator.
This talk is based on results obtained with Arieh Iserles, Marcus Webb and Katharina Schratz.
Time/place: Friday, May 7, 2021, 14:00–14:45, Zoom
Speaker: Samuel Walsh (Univ. of Missouri, USA)
Title: Orbital stability of internal waves
Abstract: In this talk, I will discuss new results on the nonlinear stability of capillary-gravity waves propagating along the interface dividing two immiscible fluid layers of finite depth. The motion in both regions is governed by the incompressible and irrotational Euler equations, with the density of each fluid being constant but distinct. We prove that for supercritical surface tension, all known small-amplitude localized waves are (conditionally) orbitally stable in the natural energy space. Moreover, the trivial solution is shown to be conditionally stable when the Bond and Froude numbers lie in a certain unbounded parameter region. For the near critical surface tension regime, we show that one can infer conditional orbital stability or orbital instability of small-amplitude traveling waves solutions to the full Euler system from considerations of a dispersive PDE similar to the steady Kawahara equation.
These results are obtained by reformulating the problem as an infinite-dimensional Hamiltonian system, then applying a version of the Grillakis–Shatah–Strauss method recently developed with K. Varholm and E. Wahlén. A key part of the analysis consists of computing the spectrum of the linearized augmented Hamiltonian at a shear flow or small-amplitude wave. For this, we generalize an idea used by Mielke to treat capillary-gravity water waves beneath vacuum. This is joint work with R. M. Chen.
Time/place: Friday, April 30, 2021, 14:00–14:45, Zoom
Speaker: Matthias Ehrhardt (Univ. of Bath, UK)
Title: Bilevel Learning for Inverse Problems
Abstract: Variational regularization techniques are dominant in the field of inverse problems. A drawback of these techniques is that they are dependent on a number of parameters which have to be set by the user. This issue can be approached by machine learning where we estimate these parameters from data. This is known as "Bilevel Learning" and has been successfully applied to many tasks, some as small-dimensional as learning a regularization parameter, others as high-dimensional as learning a sampling pattern in MRI. While mathematically appealing this strategy leads to a nested optimization problem which is computationally difficult to handle. In this talk, we discuss a number of applications of bilevel learning for imaging as well as new computational approaches. There are a number of open problems in this relatively recent field of study, some of which I will highlight along the way.
Time/place: Friday, April 23, 2021, 14:00–14:45, Zoom
Speaker: Catherine Higham (Univ. of Glasgow, UK)
Title: Applications of Deep Learning for Quantum Technologies
Abstract: Deep learning is being applied and obtaining impressive results in many novel emerging Quantum Technologies. In this talk, I will introduce some machine learning approaches, in a range of quantum physics photonics applications (single-pixel camera/video/LiDAR), where neural networks including convolutional autoencoders and generative adversarial networks are being used to solve experimental optimization, inverse and classification/regression problems.
Time/place: Friday, April 16, 2021, 14:00–14:45, Zoom
Speaker: Desmond Higham (Univ. of Edinburgh, UK)
Title: A Brief Introduction to Deep Learning for Applied Mathematicians
Abstract: Multilayered artificial neural networks are becoming a pervasive tool in a host of application fields. At the heart of this deep learning revolution are familiar concepts from calculus, approximation theory, optimization and linear algebra. I will provide a very brief introduction to the basic ideas that underlie deep learning from the perspective of a numerical analyst. I will also highlight areas where applied and computational mathematicians are well-placed to make research contributions.
Time/place: Friday, April 9, 2021, 14:00–14:45, Zoom
Speaker: Lars Ruthotto (Emory Univ., USA)
Title: Numerical Analysis Perspectives on Deep Neural Networks
Abstract: The resurging interest of deep learning are commonly attributed to advances in hardware and growing data sizes and less so to new algorithmic improvements. However, cutting edge numerical methods are needed to tackle ever larger and more complex learning problems. In this talk, I will illustrate the use of numerical analysis tools for improving the effectiveness of deep learning algorithms. With a focus on deep neural networks that can be modeled as differential equations, I will highlight the importance of choosing an adequate time integrator. I will also compare, using a numerical example, the difference of the first-discretize-then-optimize and the first-optimize-then-discretize paradigms for training residual neural networks. Finally, I show that exploiting the separable structure of most learning problems can increase the efficiency and the accuracy of training.
Time/place: Friday, March 26, 2021, 14:00–14:45, Zoom
Speaker: Nils Berglund (Univ. d'Orléans, France)
Title: Metastable dynamics of stochastic Allen-Cahn PDEs on the torus
Abstract: Allen-Cahn equations are parabolic PDEs modelling phase separation. While their solution theory is the same as for the \(\Phi^4\) model of Quantum Field Theory, their dynamics when forced by weak space-time white noise is very different, because of the phenomenon of metastability: solutions tend to spend exponentially long (in the noise intensity) times near local minima of the energy function. The exponent of the mean transition time can be deduced from a large-deviation principle, but computing the subexponential prefactor requires more work.
Using a potential-theoretic approach, we show that in one spatial dimension, the prefactor can be expressed in terms of a Fredholm determinant. In two spatial dimensions, a formal computation of the prefactor by the same approach fails to converge. However, taking into account results on renormalisation of singular PDEs, we show that the prefactor involves a renormalised Carleman-Fredholm determinant.
Based on joint works with Barbara Gentz, and with Giacomo Di Gesù and Hendrik Weber.
Time/place: Friday, March 19, 2021, 14:00–14:45, Zoom
Speaker: Antoine Lejay (INRIA, Univ. de Lorraine, France)
Title: General rough differential equations through flow approximations: a constructive tool
Abstract: The theory of rough paths allows one to consider differential equations driven by rough signals. Many alternative constructions now exist, all relying on the various forms of so-called sewing lemma. Among them, the non-linear sewing lemma transforms a family of approximations which is "close to" be flow, defined in term of composition, to a real flow related to the rough differential equation. This approach also quantifies the quality of numerical schemes.
In this talk, we explore some approximations based on regular flows parametrized by objects living in a suitable algebraic structure that determines the nature of the driving path. To apply the non-linear sewing lemma, we control their compositions by repeated use of the Newton formula. To work with minimal assumptions on the regularity of the flow, a key tool is to establish a multivariate Taylor formula with an integral remainder. This latter mixes both analytic and algebraic considerations.
We then give several examples and applications to rough differential equations driven by geometric rough paths, branched rough paths and also by paths living in possibly other structures.
Time/place: Friday, March 12, 2021, 14:00–14:45, Zoom
Speaker: Nikolas Tapia (WIAS and TU Berlin)
Title: Numerical schemes for Rough Partial Differential Equations
Abstract: In the first part of the talk, I will introduce the basic tools from rough analysis including Davie expansions. Then, I will show how a previous result, joint with C. Bellingeri, A. Djurdjevac and P. K. Friz (arXiv:2002.10432), can be applied to obtain a numerical scheme for the transport equation driven by irregular multiplicative noise. Finally, I will concentrate on ongoing work with A. Djurdjevac and C. Bayer on a Finite Element Method for linear parabolic PDEs with multiplicative noise.
Time/place: Friday, March 5, 2021, 14:00–14:45, Zoom
Speaker: Elisabeth Köbis (NTNU)
Title: Introduction to Set Optimization
Abstract: Set optimization is concerned with finding minimal solutions of set-valued mappings, where the outcome sets are compared by binary relations among sets, so-called set relations. In this talk, we give a characterization of set relations by means of nonlinear scalarization functionals. We will in addition formulate two notions of approximate solutions and give conditions for approximate minimal solutions by means of single inequalities. Furthermore, we propose a generalized Jahn-Graef-Younes method to compute the set of (approximate) minimal elements. Our methods do not rely on any convexity assumptions on the considered sets. Moreover, an application to uncertain programming, in particular, robustness, is presented.
Time/place: Friday, Feb. 19, 2021, 14:00–14:45, Zoom
Speaker: Carina Geldhauser (Lund Univ., Sweden)
Title: Space-discretizations of reaction-diffusion SPDEs
Abstract: In this talk we will discuss two different viewpoints on a space-discrete reaction-diffusion equation with noise: First, as an interacting particle system in a bistable potential, and second, as a lattice differential equation. Each viewpoint sheds light on a different phenomenon, which will be highlighted in the talk. Based on joint works with A. Bovier (Bonn) and Ch. Kuehn (TU Munich).
Time/place: Friday, Feb. 12, 2021, 14:00–14:45, Zoom
Speaker: James Jackaman (NTNU)
Title: Geometric space-time finite element discretizations for multi-symplectic PDEs
Abstract: In this talk we shall focus on the motivation, design and properties of geometric space-time finite element methods, utilising multisymplectic partial differential equations as prototypical example.
A multisymplectic PDE in one time and one space dimension is a PDE which can be written as a first order system in the form \begin{equation} \nonumber K\,\mathbf{z}_t+L\,\mathbf{z}_x=\nabla S(\mathbf{z}) \end{equation} \(K\) and \(L\) are skew-symmetric matrices, \(\mathbf{z}\in \mathbb{R}^d\), \(S(\mathbf{z})\) smooth. Historically multisymplectic methods are space-time approximations of such PDEs, often based on finite differences. These methods are designed to preserve a discrete variant of the conservation law of multisymplecticity. Unfortunately, discretizations of multisymplectic PDEs in strong form (e.g. multisymplectic discretizations) can sometimes be not well defined locally, and/or globally, or not have solutions/unique solutions.
We shall investigate the preservation of discrete local momentum and energy conservation laws, deriving space-time finite element integrators with a focus on preserving these invariants. We conclude with the presentation of numerical experiments and an outlook on how the proposed finite element method may be utilised for geometric space-time adaptive algorithms.
Time/place: Friday, Feb. 5, 2021, 14:00–14:45, Zoom
Speaker: Fabian Harang (UiO)
Title: Pathwise regularization by noise for SDEs and SPDEs with multiplicative noise
Abstract: In this talk we will discuss two results relating to pathwise regularization by noise for stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) with multiplicative noise. We will begin with the wellposedness of an SDE controlled by a fractional Brownian motion with H>1/2. Similarly, as for the case of ODEs, it is well known that existence and uniqueness for such equations are granted under the assumption of Lipschitz diffusion coefficients. There have been few results that extend this assumption to a wider class of coefficients. Based on the recent progress in pathwise regularization by noise, we investigate the effect that perturbations of the SDE by a continuous path may have on the well-posedness of the equation. We find that that for a large class of continuous processes perturbing the equation, the SDE is well-posed even when the diffusion coefficient is only a distribution. This extends the pathwise regularization techniques developed by Catellier and Gubinelli (2016), to the case of stochastic differential equations with multiplicative noise. This result is based on joint work with Lucio Galeatii (university of Bonn). We will continue with a discussion of a recent result which is similar in spirit. We investigate the nonlinear stochastic heat equation with multiplicative noise. The multiplicative noise considered here is only spatially dependent, and we investigate the effects of a perturbation of the equation by a measurable path. We will see that also in this case, with a suitable choice of perturbation, the equation is well-posed, even when the nonlinear coefficient is only a distribution, and the multiplicative noise is considered to be a spatial white noise. This work is based on a recent preprint together with Rémi Catellier (University of Nice Sophia Antipolis).
Time/place: Friday, Jan. 29, 2021, 14:00–14:45, Zoom
Speaker: Olivier Verdier (Huawei)
Title: Automatic Differentiation on Manifolds
Abstract: In order to solve an optimisation problem, one has to compute gradients. A very popular and convenient way to do this is using automatic differentiation.
I will explain what automatic differentiation is in general, discuss some of its implementation details, and how it can be adapted to work for optimization problems on manifolds. I will focus a little on problems involving deformations (image matching, template reconstruction in tomography), where optimization is carried out on the group of diffeomorphisms.
Time/place: Wednesday, Dec. 16, 2020, 10:15–11:00, Zoom
Speaker: Brynjulf Owren (NTNU)
Title: Deep neural networks as structure preserving optimal control problems
Abstract: A deep neural network model consists of a large number of layers, each with a number of parameters or controls associated to them. In supervised learning, these parameters are optimised to match training data in the best possible way. The data are propagated through the layers by nonlinear transformations, and in an important subclass of models (ResNet) the transformation can be seen as the numerical flow of some continuous vector field. Ruthotto and Haber (2017), as well as Cheng et al., have experimented in using a different type of vector fields to improve the deep learning model. In particular, it is of interest that the trained model has good long time behaviour and is stable in the deep limit when the number of layers tends to infinity. The models presented in the literature have certain built-in structural properties, they can for instance be gradient flows or Hamiltonian vector fields. A difficulty is however that the models are not autonomous and therefore it is less clear what their flows actually preserve. Starting from such ResNet vector fields, we shall discuss their properties and derive some new nonlinear stability bounds. The long time behaviour of these neural ODE flows is important in the generalisation mode, i.e. after the model has been trained. But also in the training algorithm itself, structure preserving numerical schemes are important. In deep learning models, the use of gradient flows for optimisation is prevalent, and there exists a number of different algorithms that can be used, some of them can be interpreted as approximations of the flow of certain vector fields with dissipations, such as conformal Hamiltonian systems. If time permits, we will briefly discuss also these algorithms and in particular the need for and efficiency of regularisation.
Joint work with: Martin Benning, Elena Celledoni, Matthias Ehrhardt, Christian Etmann, Robert McLachlan, Carola-Bibiane Schönlieb and Ferdia Sherry.
Time/place: Wednesday, Dec. 9, 2020, 10:15–11:00, Zoom
Speaker: Gilles Vilmart (Univ. Genève)
Title: Second kind explicit stabilized integrators for ergodic and stiff stochastic (partial) differential equations
Abstract: For large dimensional and stiff diffusion problems, explicit stabilized integrators are an efficient alternative to implicit or semi-implicit methods to avoid the severe timestep restriction faced by standard explicit time integrators.
We present a family of explicit stabilized integrators for ergodic and stiff problems, based on second kind Chebyshev polynomials, that yield an optimal size of extended mean-square stability domain that grows at the same quadratic rate as the optimal family for deterministic problems. We also show that the new explicit stabilized schemes converge in the strong sense when applied to stochastic semilinear diffusion partial differential equations.
Based on joint works with A. Abdulle (Lausanne), I. Almuslimani (Rennes), and C.-E. Bréhier (Lyon).
Time/place: Wednesday, Dec. 2, 2020, 10:15–11:00, Zoom
Speaker: Fabian Harang (Univ. Oslo)
Title: Infinitely regularizing paths, and regularization by noise
Abstract: In this talk we will discuss regularization by noise from a pathwise perspective using non-linear Young integration, and discuss the relations with occupation measures and local times. This methodology of pathwise regularization by noise was originally proposed by Gubinelli and Catellier (2016), who use the concept of averaging operators and non-linear Young integration to give meaning to certain ill-posed SDEs.
In a recent work together with Prof. Nicolas Perkowski at Freie University, Berlin, we show that there exists a class of paths with exceptional regularizing effects on ODEs, using the framework of Gubinelli and Catellier. In particular, we prove existence and uniqueness of ODEs perturbed by such a path, even when the drift is given as a Scwartz distribution. Moreover, the flow associated to such ODEs are proven to be infinitely differentiable. Our analysis can be seen as purely pathwise, and is only depending on the existence of a sufficiently regular occupation measure associated to the path added to the ODE.
As an example, we show that a certain type of Gaussian process has infinitely differentiable local times, whose paths then can be used to obtain the infinitely regularizing effect on ODEs. This gives insight into the powerful effect that noise may have on certain equations. If time permits, we will also discuss an ongoing extension of these results towards regularization of certain PDE/SPDEs by noise.​
Time/place: Wednesday, Nov. 25, 2020, 10:15–11:00, Zoom
Speaker: Pranav Singh (Univ. Bath)
Title: Convergence of Magnus based methods for Schrödinger equations
Abstract: Magnus expansion based methods are an efficient class of integrators for solving Schrödinger equations that feature time dependent potentials such as lasers. These methods have been found to be highly effective in computational quantum chemistry since the pioneering work of Tal Ezer, Kosloff and Cerjan in the early 90s.
The convergence of the Magnus expansion, however, is understood only for ODEs and traditional analysis suggests a much poorer performance of these methods than observed experimentally. It was not till the work of Hochbruck and Lubich in 2003 that a rigorous analysis justifying the application to PDEs with unbounded operators, such as the Schrödinger equation, was presented.
In this talk I will present the extension of this analysis to the semiclassical regime, where the highly oscillatory solution conventionally suggests large errors and a requirement for very small time steps.
Time/place: Wednesday, Nov. 11, 2020, 10:15–11:00, Zoom
Speaker: Adrian M. Ruf (ETH Zürich)
Title: Convergence rates of numerical methods for conservation laws with discontinuous flux
Abstract: The subject of conservation laws with discontinuous flux has been an active re-search area during the last several decades. Many different selection criteria to single out a unique weak solution have been proposed and several numerical schemes have been designed and analyzed in the literature. Surprisingly, the preexisting literature on convergence rates for such schemes is practically nonexistent. In this talk, focusing on so-called adapted entropy solutions, I will present recent developments in this direction and prove convergence rates for finite volume and front tracking methods. As an application, I will briefly describe how these results can be used for uncertainty quantification in two-phase reservoir simulations for reservoirs with varying geological properties.
Time/place: Thursday, Nov. 5, 2020, 13:15–14:00, Zoom
Speaker: Massimiliano Gubinelli (Univ. Bonn)
Title: Grassmann stochastic analysis and stochastic quantisation of Euclidean Fermions
Abstract: This talk is about the extension of some probabilistic construction to the case of Grassmann valued random variables, i.e. random variables which anticommute. This requires setting up the problem in the context of non-commutative probability. Moreover we study some simple stochastic differential equations for Grassmann variables and derive information on their invariant states.
Joint work with S. Albeverio, L. Borasi and F. de Vecchi. Based on the paper: Albeverio, Sergio, Luigi Borasi, Francesco C. De Vecchi, and Massimiliano Gubinelli. ‘Grassmannian Stochastic Analysis and the Stochastic Quantization of Euclidean Fermions’. arXiv:2004.09637
Time/place: Wednesday, Oct. 28, 2020, 10:15–11:00, Zoom
Speaker: Yuya Suzuki (NTNU)
Title: A quasi-Monte Carlo method combined with operator splitting for time-dependent Schrödinger equations
Abstract: In the first half of this talk I will give a brief introduction of quasi-Monte Carlo (QMC) methods which are originally used for numerical integration over a multidimensional unit cube. I will show some known results including the classical Koksma–Hlawka inequality and also modern results in reproducing kernel Hilbert space settings.
In the second half of the talk, I will apply one important branch of QMC methods, namely rank-1 lattice rules, for numerically solving time-dependent Schrödinger equations in multidimensional settings. We combine operator splitting (time discretization) and a pseudospectral method on rank-1 lattice (space discretization). We show the theoretical error bound of the scheme and also numerical results which confirm the theory. This second part is based on joint work with Dirk Nuyens and Gowri Suryanarayana.
Time/place: Wednesday, Oct. 21, 2020, 10:15–11:00, Zoom
Speaker: Ben Tapley (NTNU)
Title: Simulating slender particles in viscous flow
Abstract: A slender particle is a particle that is long but with a small cross sectional radius. Being able to accurately and cheaply simulate slender particles in viscous flow is essential for our understanding of micro swimming organisms, for example. In this presentation, we present a new model for calculating the forces and torques on slender particles. The model gives rise to a Fredholm integral equation whose result needs to be integrated to find the forces and torques on the particle. We then propose and analyse a fast numerical method for solving these integral equations. Finally, numerical experiments are given to validate the model and method.
Time/place: Thursday*, Oct. 15, 2020, 13:15–14:00, Zoom
Speaker: Lorenzo Zambotti (Sorbonne Univ. Paris, France)
Title: Hairer's Reconstruction Theorem without Regularity Structures
Abstract: This talk, based on joint work with Francesco Caravenna 2005.09287, is devoted to Martin Hairer's Reconstruction Theorem, which is one of the cornerstones of his theory of Regularity Structures [Hairer 2014]. Our aim is to give a new self-contained and elementary proof of this Theorem and of some applications. We present it as a general result in the theory of distributions that can be understood without any knowledge of Regularity Structures themselves, which we do not even need to define.
*Note: the talk was moved -exceptionally- to Thursday due to the unavailability of the speaker on Wednesdays.
Time/place: Wednesday, Oct. 7, 2020, 10:15–11:00, Zoom
Speaker: Weinan E (Princeton Univ., USA)
Title: Machine Learning and Numerical Analysis
Abstract: The heart of machine learning is the approximation of functions using finite pieces of data. This is one of the main pillars of numerical analysis. Thus it is not surprising that the success of machine learning in dealing with functions in very high dimensions has opened up some brand new territories in computational mathematics, with potentially unprecedented impact for years to come.
In the first part of this talk, I will review some of the most exciting advances of using machine learning to address problems in scientific computing and computational science.
In the second part of this talk, I will discuss how machine learning can be formulated as a problem in numerical analysis and how ideas from numerical analysis can be used to understand machine learning as well as construct new machine learning models and algorithms.
Time/place: Wednesday, Sept. 30, 2020, 10:15–11:00, Zoom
Speaker: Tale Bakken Ulfsby (NTNU)
Title: Stabilized cut discontinuous Galerkin methods for advection-diffusion-reaction problems on surfaces
Abstract: Advection-diffusion-reaction (ADR) problems on surfaces with complicated or moving geometries appear in many important problems in science and engineering, including the modeling of cell motility in cell biology and the study of transport phenomena in fractured porous media. Standard numerical approaches such as the finite element method demand the construction of meshes fitted to the geometry. If the domain geometry is very complicated or undergoes large deformations, the required generation of fitted meshes can be very time consuming and computationally expensive, and account for a large percentage of the overall simulation time.
As a potential remedy, new generations of unfitted or cut finite element methods have been developed in the last decade. For surface problems, these methods use a fixed background mesh while the embedded surface is described implicitly. In this talk, we present a new Cut Discontinuous Galerkin (CutDG) method for the discretization of advection-dominant problems on surfaces. The discrete function space is defined as the restriction of the basis functions associated with the underlying background mesh to the surface. Geometrically robustness problems usually caused by troublesome cut configurations are avoided by adding carefully designed CutFEM stabilizations. As a result, we can prove geometrically robust stability and optimal a priori error estimates. Finally, numerical results are presented to illustrate our theoretical findings.
Time/place: Wednesday, Sept. 23, 2020, 10:15–11:00, Zoom
Speaker: Torstein Nilssen (UiA)
Title: Rough path variational principles for fluid equations
Abstract: In this presentation we will first recall the insight of Arnold that Euler's equation can be understood as a geodesic equation on an infinite dimensional manifold. Using Lagrange multipliers, this yields a natural framework for understanding the structure of random/irregular perturbations of fluid equations. In this presentation, we will consider rough path perturbations and the corresponding variational principles.
Time/place: Wednesday, Sept. 16, 2020, 10:15–11:00, Zoom
Speaker: Sondre Tesdal Galtung (NTNU)
Title: A semi-discrete Camassa–Holm system based on variational principles
Abstract: In this talk, based on joint works with Katrin Grunert and Xavier Raynaud (2003.03114, 2006.15562). I will present a discretization of a Camassa-Holm system (2CH). This system is an extension of the well-known Camassa–Holm equation and serves as a model for shallow water waves. The discretization is based on a derivation of 2CH from variational principles in Lagrangian coordinates: Discretizing the energy functionals, an analogous procedure yields a semi-discrete, Lagrangian 2CH system with its own conserved quantities. Some of the main challenges encountered in the study of the discrete system will be highlighted, and I will indicate how we prove convergence to solutions of 2CH. Finally, some numerical examples illustrate the performance of the discretization when paired with explicit time integrators.
Time/place: Wednesday, June 24, 2020, 14:15–15:00, Zoom
Speaker: Sebastian Riedel (TU Berlin)
Title: Runge-Kutta methods for rough differential equations
Abstract: In the 60s, J.C. Butcher used a power series expansion over rooted trees for ordinary differential equations to study numerical methods of a high order. Nowadays, such series are called B-series. We derive a B-series representation for rough differential equations and calculate the local order of the truncated expansion. En passant, we introduce Gubinelli's concept of a branched rough path which is closely related. As a result, we obtain a class of numerical methods that can be used to solve random rough differential equations numerically. They apply, for instance, to stochastic differential equations driven by a fractional Brownian motion. Joint work with Martin Redmann (Halle).
Time/place: Wednesday, June 17, 2020, 14:15–15:00, Zoom
Speaker: Miłosz Krupski (NTNU)
Title: Control of the jump process in non-local mean field games PART 2
Abstract: We study a version of non-local "mean field game" (MFG) system of equations, derived from a stochastic game model, in which players control the underlying Lévy process.
The MFG system consists of two equations: in type of Hamilton-Jacobi-Bellman and Fokker-Planck-Kolmogorov. We prove well-posedness of the two equations individually, and then study solutions of the MFG system itself.
Part 1 (presented by Indranil Chowdhury) will introduce the MFG model of our interest, starting from a stochastic game, and then focus on the results on the Fokker-Planck equation.
Part 2 (presented by Miłosz Krupski) will describe the stochastic model in more detail and develop the results in regard of the MFG system as a whole.
Time/place: Tuesday, June 16, 2020, 14:15–15:00, Zoom
Speaker: Indranil Chowdhury (NTNU)
Title: Control of the jump process in non-local mean field games PART 1
Abstract: We study a version of non-local "mean field game" (MFG) system of equations, derived from a stochastic game model, in which players control the underlying Lévy process.
The MFG system consists of two equations: in type of Hamilton-Jacobi-Bellman and Fokker-Planck-Kolmogorov. We prove well-posedness of the two equations individually, and then study solutions of the MFG system itself.
Part 1 (presented by Indranil Chowdhury) will introduce the MFG model of our interest, starting from a stochastic game, and then focus on the results on the Fokker-Planck equation.
Part 2 (presented by Miłosz Krupski) will describe the stochastic model in more detail and develop the results in regard of the MFG system as a whole.
Time/place: Wednesday, May 27, 2020, 14:15–15:00, Zoom
Speaker: Yvain Bruned (Univ. of Edinburgh, UK)
Title: Resonance based schemes for dispersive equations via decorated trees PART II
Abstract: We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the PDE and to approximate with high order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular SPDEs with Regularity Structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations (which encode the dominant frequencies). The structure proposed in this work is new and gives a variant of the Butcher-Connes-Kreimer Hopf algebra on decorated trees.
Part I: We will introduce the general concept of the numerical method and present some examples.
Part II: We will develop the algebraic structures and present the decorated trees needed for writing the general numerical scheme.
Time/place: Tuesday, May 26, 2020, 14:15–15:00, Zoom
Speaker: Katharina Schratz (Heriot-Watt Univ., Edinburgh, UK, & Sorbonne Univ. Paris, France)
Title: Resonance based schemes for dispersive equations via decorated trees PART I
Abstract: We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the PDE and to approximate with high order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular SPDEs with Regularity Structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations (which encode the dominant frequencies). The structure proposed in this work is new and gives a variant of the Butcher-Connes-Kreimer Hopf algebra on decorated trees.
Part I: We will introduce the general concept of the numerical method and present some examples.
Part II: We will develop the algebraic structures and present the decorated trees needed for writing the general numerical scheme.
Time/place: Wednesday, April 22, 2020, 14:15–15:00, Zoom
Speaker: Nikolas Tapia (TU Berlin and WIAS)
Title: Transport and continuity equations with (very) rough noise
Abstract: We show existence and uniqueness of strong solutions (in the rough paths sense) to the stochastic transport equation with general Hölder-continuous noise. We also study existence and uniqueness of the associated continuity equation. (Based on joint work with C. Bellingeri, A. Djurdjevac, P. K. Friz.)
Time/place: Wednesday, April 15, 2020, 14:15–15:00, Zoom
Speaker: Peter Pang (NTNU)
Title: The Hunter-Saxton equation with noise
Abstract: This talk will be on some results derived for a stochastic Hunter-Saxton equation. The overarching theme will be an existence theory. Rather than dwelling heavily on technicalities, the intention is to give a relatively self-contained talk centred around phenomena exhibited by solutions. In particular, we shall look at stochastic blow-up/wave-breaking, the stopping times for which explicit laws can be derived. We shall also discuss continuation beyond blow-up. The primary tool is a stochastic method-of-characteristics. Results are based on joint work with Helge Holden and Kenneth H. Karlsen (https://arxiv.org/abs/2003.13984).
Time/place: Tuesday, March 10, 2020, 14:15–15:00, room 656
Speaker: Benjamin Fahs (Imperial College, London)
Title: Sine-kernel determinant on two large intervals
Abstract: The sine-kernel determinant on a set \(A\) represents the probability that there are no eigenvalues in the set \(A\) in the bulk scaling limit of the Gaussian Unitary Ensemble.
The asymptotic study of the determinant when \(A\) is a single large interval was initiated by Dyson in 1962 and was solved 50 years later through the work of numerous authors.
We consider the case where \(A\) is composed of 2 large intervals. In this case, a study of the asymptotics was started by Widom. The most detailed formula up to now is due to Deift, Its, Zhou in 1997, who obtained a formula up to an unknown multiplicative constant. We present a formula for the multiplicative constant, and additionally the uniform transition between 1 and 2 intervals.
Time/place: Wednesday, February 19, 2020, 14:15–15:00, room F2
Speaker: Jørgen Endal (NTNU)
Title: The one-phase fractional Stefan problem
Abstract: We study the existence, properties of solutions, and free boundaries of the one-phase Stefan problem with fractional diffusion posed in \(\mathbb{R}^N\). The equation for the enthalpy \(h\) reads \(\partial_t h+ (-\Delta)^{s}\Phi(h) =0\) where the temperature \(u:=\Phi(h):=\max\{h-L,0\}\) is defined for some constant \(L>0\) called the latent heat, and \((-\Delta)^{s}\) is the fractional Laplacian with exponent \(s\in(0,1)\). We prove the existence of a continuous and bounded selfsimilar solution of the form \(h(x,t)=H(x\,t^{-1/(2s)})\) which exhibits a free boundary at the change-of-phase level \(h(x,t)=L\) located at \(x(t)=\xi_0 t^{1/(2s)}\) for some \(\xi_0>0\). This special solution will be an important tool to obtain that the temperature has finite speed of propagation while the enthalpy has infinite speed, and that the support of the temperature never recedes. Other interesting properties like e.g. \(L\to0^+\) and \(L\to\infty\) will also be discussed, and the theory itself is illustrated by convergent finite-difference schemes.
This is a joint work with Félix del Teso and Juan Luis Vázquez.
Time/place: Tuesday, February 18, 2020, 14:15–15:00, room 734
Speaker: Emmanuel Chasseigne (Tours)
Title: Ergodic problems for viscous Hamilton-Jacobi equations with inward drift
Abstract: I will first give a general overview of ergodic-type problems for viscous Hamilton-Jacobi equations in the whole space. We will also study brifely the associated stochastic control optimization problem as well as the links between the elliptic and parabolic cases.

Then I will focus on the existence of critical (additive) eigenvalues and their qualitative properties depending on the potential/forcing term appearing in the equation.

Finally, I will discuss some more recent advances in presence of an additional inward drift.
Time/place: Thursday, February 13, 2020, 14:15–15:00, room 734
Speaker: Jacek Jendrej (Paris)
Title: Strongly interacting kink-antikink pairs for scalar fields on a line
Abstract: I will present a recent joint work with Michał Kowalczyk and Andrew Lawrie. A nonlinear wave equation with a double-well potential in 1+1 dimension admits stationary solutions called kinks and antikinks, which are minimal energy solutions connecting the two minima of the potential. We study solutions whose energy is equal to twice the energy of a kink, which is the threshold energy for a formation of a kink-antikink pair. We prove that, up to translations in space and time, there is exactly one kink-antikink pair having this threshold energy. I will explain the main ingredients of the proof.
Mini-workshop on numerical analysis
Time/place: Thursday, February 06, 2020, 11:15–12:00, room F4
Speaker: Geir Bogfjellmo (NTNU)
Title: Interpolation on curved 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. We 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 methods based on Riemannian cubics. The algorithm is demonstrated for three geometries of interest: the \(n\)-sphere, complex projective space, and real Grassmannians.
Time/place: Thursday, February 06, 2020, 14:15–15:00, room R10
Speaker: Nick Trefethen (Oxford)
Title: From the Faraday cage to lightning Laplace and Helmholtz solvers
Abstract: We begin with the story of the Faraday cage used for shielding electrostatic fields and electromagnetic waves. Feynman in his Lectures claims the shielding is exponential with respect to the gap between wires and that it works with wires of infinitesimal radius. In fact, the shielding is much weaker than this and requires wires of finite radius (which is why it's hard to see into your microwave oven). How can we compute the field inside a 2D cage? This brings us to the numerical part of the talk. When the boundaries are smooth, series expansions (going back to Runge in 1885) converge exponentially. When there are corners and associated singularities, the new technique of lightning Laplace and Helmholtz solvers, depending on rational or Hankel functions with poles exponentially clustered near the corners, converges root-exponentially. The name "lightning" comes from the fact that this method exploits the same mathematics that makes lightning strike at sharp points. Lightning solvers and the related AAA approximation algorithm are bringing in a new era of application of rational functions and their relatives to PDEs, conformal mapping, and other numerical problems.
Time/place: Thursday, February 06, 2020, 15:15–16:00, room R10
Speaker: André 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.
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: Wednesday, February 05, 2020, 14:15–15:00, room F2
Speaker: Cristopher Salvi (Oxford)
Title: Capturing similarities between streams: the warping \(p\)-variation distance
Abstract: We introduce a new metric on the space of unparameterized paths which originates from fundamental results in rough paths theory. We describe why natural signals can be described as rough paths and how the new distance is well-designed to capture (dis)similarities of streams. We then turn to the challenge of designing an efficient algorithm to compute the inroduced metric, which consists in the solution of a min-max problem. As a solution to this probem, we propose a branch-and-bound algorithm that drastically reduces the number of nodes to explore and therefore increases computational performance. We compare our metric to other distances classically used in data science, such as dynamic time warping, for two simple classification and a clustering tasks, and show experimentally that the new metric achieve better performance.
Time/place: Wednesday, January 22, 2020, 14:15–15:00, room F2
Speaker: Rinaldo M. Colombo (Brescia)
Title: Conservation Laws: Analysis & Modeling
Abstract: Conservation Laws are first order partial differential equations whose analytic developments has always been interwined with the questions posed by their applications. Originally, fluid dynamics and more recently vehicular traffic and crowd dynamics have provided an incredible amount of problems. The present talk will describe a few recent developments: a "purely analytic" one and others inspired by applications. The latter also pose new optimization and game theoretic questions.
Time/place: Tuesday, January 14, 2020, 15:15–16:00, room 734
Speaker: Luca Galimberti (Oslo)
Title: Stochastic continuity equations on Riemannian manifolds: renormalization and uniqueness
Abstract: We are given a \(d\)-dimensional (\(d \ge 1\)) smooth closed manifold \(M\), endowed with a smooth Riemannian metric \(h\). We study the Cauchy problem for the following stochastic continuity equation \[ (1) \qquad d\rho + \operatorname{div}_h (\rho u)dt + \sum_{i=1}^N \operatorname{div}_h (\rho a_i) \circ dW^i (t) = 0 \quad\text{ on }\ [0, T ] \times M, \] and \(\rho(0) = \rho_0\) on \(M\). Typically, \(u \colon [0, T ] \times M \to T M\) is a time-dependent irregular vector field on \(M\) which is interpreted as a velocity field, while \(\rho = \rho(\omega, t, x)\) is a density for a mass distribution. \(a_1,\ldots,a_N\) are arbitrary smooth vector fields on \(M\), \(W^1,\cdots,W^N\) are independent real Brownian motions, and the symbol \(\circ\) means that the equation is understood in the Stratonovich sense.
After introducing a proper concept of weak solution, we prove a very delicate renormalization theorem, in the spirit of DiPerna and Lions (1989), which at once provides us with some a priori estimates for the solution \(\rho\) as well as with an easy uniqueness result, under the mild assumption \[\operatorname{div}_h u \in L^1(0,T;L^\infty(M)).\] Crucial ingredients in this renormalization argument are the introduction of an ad hoc regularization procedure for mixed tensor fields of arbitrary order on the manifold \(M\), as well as a “second order” commutator argument, which is necessary because of the appearance of a second order differential operator in the equivalent Itô formulation of (1).
Time/place: Tuesday 03rd of December, 2019, 14:15–15:00, room 734
Speaker: Hung Le (NTNU)
Title: On the existence and instability of solitary waves with a finite dipole
Abstract: In this talk, we consider the existence and stability properties of two-dimensional solitary waves traversing an infinitely deep body of water. We assume that above the water is air and that the waves are acted upon by gravity with surface tension effects on the air-water interface. In particular, we study the case where there is a finite dipole in the bulk of the fluid, that is, the vorticity is a sum of two weighted \(\delta\)-functions. Using an implicit function theorem argument, we construct a family of solitary waves solutions for this system that is exhaustive in a neighborhood of 0. Our main result is that this family is conditionally orbitally unstable. This is proved using a modification of the Grillakis—Shatah–Strauss method recently introduced by Varholm, Wahlén, and Walsh.
Time/place: Thursday 28th of November, 2019, 10:15–11:00, room 656
Speaker: Balázs Kovács (Tübingen)
Title: A convergent algorithm for mean curvature flow with and without forcing
Abstract: We will sketch a proof of convergence for semi- and full discretizations of mean curvature flow of closed two-dimensional surfaces. The proposed and studied numerical method combines evolving surface finite elements, whose nodes determine the discrete surface like in Dziuk's algorithm proposed in 1990, and linearly implicit backward difference formulae for time integration. The proposed method differs from Dziuk's approach in that it discretizes Huisken's evolution equations (from [Huisken (1984)]) for the normal vector and mean curvature and uses these evolving geometric quantities in the velocity law projected to the finite element space. This numerical method admits a convergence analysis, which combines stability estimates and consistency estimates to yield optimal-order \(H^1\)-norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix–vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. We will also present various numerical experiments to illustrate and complement the theoretical results. Furthermore, we will give an outlook towards forced mean curvature flow, that is for problems coupling mean curvature flow with a surface PDE.
The talk is based on joint work with B. Li (Hong Kong) and Ch. Lubich (Tübingen).
Time/place: Tuesday 8th of October, 2019, 14:15–15:00, room 734
Speaker: Gunnar Taraldsen (NTNU)
Title: The spectrum of a random operator is a random set
Abstract: The theory of random sets is demonstrated to prove useful for the theory of random operators. A random operator is here defined by requiring the graph to be a random set. It is proved that the spectrum and the set of eigenvalues of random operators are random sets. These results seem to be a novelty even in the case of random bounded operators. The main technical tools are given by the measurable selection theorem, the measurable projection theorem, and a characterisation of the spectrum by approximate eigenvalues of the operator and the adjoint operator. A discussion of some of the existing definitions of the concept of a random operator is included at the end of the paper.
Time/place: Tuesday 1st of October, 2019, 14:15–15:00, room 734
Speaker: Mark Groves (Saarland University)
Title: Solitary-wave solutions to the full dispersion Kadomtsev-Petviashvili equation
Abstract: The KP-I equation \[u_t + m(D) u_x - 2uu_x = 0,\] where \(m(D)\) is the Fourier multiplier operator with multiplier \[m(k)=1+\frac{k_2^2}{2k_1^2} + \frac{1}{2}(\beta-{\textstyle\frac{1}{3}})k_1^2,\] arises as a weakly nonlinear model equation for gravity-capillary waves with strong surface tension (Bond number \(\beta>\frac{1}{3}\)). This equation admits – as an explicit solution – a "fully localised" or "lump" solitary wave which decays to zero in all spatial directions.
Recently there has been interest in the full dispersion KP-I equation \[u_t + \tilde{m}(D) u_x - 2uu_x = 0\] obtained by retaining the exact dispersion relation from the water-wave problem, that is, replacing \(m\) by \[\tilde{m}(k)=\left((1+\beta|k|^2)\frac{\tanh |k|}{|k|}\right)^{1/2}\left(1+\frac{k_2^2}{k_1^2}\right).\] In this talk I show that the full dispersion KP-I equation also has a fully localised solitary-wave solution. The existence theory is variational and perturbative in nature.
This project is joint work with Mats Ehrnström (NTNU, Norway).
Time/place: Thursday 26th of September, 2019, 14:15–15:00, room F3 (gamle fysikk)
Speaker: David Ambrose (Drexel University, Philadelphia)
Title: Existence Theory for a Mean Field Games Model of Household Wealth
Abstract: We study a nonlinear system of partial differential equations arising in macroeconomics which utilizes a mean field approximation. This equation together with the corresponding data, subject to two moment constraints, is a model for debt and wealth across a large number of similar households, and was introduced in a recent paper of Achdou, Burea, Lasry, Lions, and Moll. We introduce a relaxation of their problem, generalizing one of the moment constraints; any solution of the original model is a solution of this relaxed problem. We prove existence and uniqueness of strong solutions to the relaxed problem, under the assumption that the time horizon is small. Since these solutions are unique and since solutions of the original problem are also solutions of the relaxed problem, we conclude that if the original problem does have solutions, then such solutions must be the solutions we prove to exist. Furthermore, for some initial data and for sufficiently small time horizons, we are able to show that solutions of the relaxed problem are not solutions of the original problem. In this way we demonstrate nonexistence of solutions for the original problem in certain cases.
Time/place: Thursday 26th of September, 2019, 15:15–16:00, room F3 (gamle fysikk)
Speaker: Miles Wheeler (University of Bath)
Title: New exact solutions to the steady 2D Euler equations
Abstract: We present a large class of explicit "hybrid" equilibria for the 2D Euler equations, consisting of point vortices embedded in a smooth sea of "Stuart-type" vorticity. Mathematically, these are singular solutions of the elliptic Liouville equation satisfying some additional constraints at each singularity.
This is joint work with Vikas Krishnamurthy, Darren Crowdy, and Adrian Constantin.
2022-01-26, Kurusch Ebrahimi-Fard