# The DNA Seminar (spring 2020)

The DNA seminar is the seminar series of the DNA group (Differential Equations and Numerical Analysis). Seminars will be usually held on Wednesdays at 14:15–15:00 in room F2 (gamle fysikk).

**Due to the current health situation the seminar will be held on Zoom.**

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

## Upcoming seminars:

Time/place: Wednesday, June 24, 14:15–15:00, Zoom |
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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). |

## Earlier this term:

Time/place: Wednesday, June 17, 14:15–15:00, Zoom |
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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, 14:15–15:00, Zoom |
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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, 14:15–15:00, Zoom |
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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, 14:15–15:00, Zoom |
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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, 14:15–15:00, Zoom |
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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, 14:15–15:00, Zoom |
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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, 14:15–15:00, room 656 |
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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, 14:15–15:00, room F2 |
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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, 14:15–15:00, room 734 |
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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, 14:15–15:00, room 734 |
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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 |
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Time/place: Thursday, February 06, 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, 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, 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, 14:15–15:00, room F2 |
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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, 14:15–15:00, room F2 |
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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, 15:15–16:00, room 734 |
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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). |

## Previous semester:

Time/place: Tuesday 03rd of December, 14:15–15:00, room 734 |
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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, 10:15–11:00, room 656 |
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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, 14:15–15:00, room 734 |
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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, 14:15–15:00, room 734 |
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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, 14:15–15:00, room F3 (gamle fysikk) |
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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, 15:15–16:00, room F3 (gamle fysikk) |
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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. |