Spring 2017 - wiki.math.ntnu.no

Time/place: Wednesday 21st of June 2017, 10.15–11.00, room 734
Speaker: Monica Montardini (University of Pavia) Title: Isogeometric collocation method for free-boundary problems
Abstract: The purpose of this talk is to present a collocation-based, Newton-like scheme that can be used to find solutions of free-boundary problems, i.e. problems where also the domain of definition is unknown. Particular attention is devoted to the type of collocation used, which is based on an optimal-order choice of collocation points. Working in the frame of isogeometric analysis, we introduce the numerical method and present some numerical results we obtained. Finally, we show preliminary results obtained by applying this method to the Euler’s equations: We focus on travelling periodic solutions and present a first investigation of their bifurcation points and critical layers. This is a joint work with F. Remonato and G. Sangalli.

Time/place: Wednesday 21st of June 2017, 10.15–11.00, room 734

Speaker: Monica Montardini (University of Pavia)
Title: Isogeometric collocation method for free-boundary problems

Abstract: The purpose of this talk is to present a collocation-based, Newton-like scheme that can be used to find solutions of free-boundary problems, i.e. problems where also the domain of definition is unknown. Particular attention is devoted to the type of collocation used, which is based on an optimal-order choice of collocation points. Working in the frame of isogeometric analysis, we introduce the numerical method and present some numerical results we obtained. Finally, we show preliminary results obtained by applying this method to the Euler’s equations: We focus on travelling periodic solutions and present a first investigation of their bifurcation points and critical layers.
This is a joint work with F. Remonato and G. Sangalli.

Time/place: Tuesday 13th of June 2017, 14.15–15.00, room 734
Speaker: Marta Lewicka (University of Pittsburgh) Title: On the Monge-Ampère equation via prestrained elasticity
Abstract: In this talk, I will present results regarding the regularity and rigidity of solutions to the Monge-Ampère equation, inspired by the role played by this equation in the context of prestrained elasticity. We will show how the Nash-Kuiper convex integration can be applied here to achieve flexibility of Hölder solutions, and how other techniques from fluid dynamics (the commutator estimate, yielding the degree formula in the present context) find their parallels in proving the rigidity of Hölder, Sobolev or Besov solutions. A prestrained elastic body is a three dimensional elastic object, modeled in its reference configuration by an open domain and a Riemannian metric. This metric, by the main ansatz, is induced by mechanisms such as growth, plasticity, thermal expansion etc, and is the cause of the elastic deformation that determines the shape of the body. One can seek this deformation through a variational minimization principle, as the best possible immersion of the given Riemannian manifold into the flat space. For thin films, the characteristics of the original prestrain metric induce the distinct nonlinear theories in the singular limit of vanishing thickness, that determine the above mentioned minimizing deformation. A particular aspect of these limit theories are the curvature constraints that are manifested as the Monge-Ampère equation in the appropriate energy regimes.

Time/place: Tuesday 13th of June 2017, 14.15–15.00, room 734

Speaker: Marta Lewicka (University of Pittsburgh)
Title: On the Monge-Ampère equation via prestrained elasticity

Abstract: In this talk, I will present results regarding the regularity and rigidity of solutions to the Monge-Ampère equation, inspired by the role played by this equation in the context of prestrained elasticity. We will show how the Nash-Kuiper convex integration can be applied here to achieve flexibility of Hölder solutions, and how other techniques from fluid dynamics (the commutator estimate, yielding the degree formula in the present context) find their parallels in proving the rigidity of Hölder, Sobolev or Besov solutions.

A prestrained elastic body is a three dimensional elastic object, modeled in its reference configuration by an open domain and a Riemannian metric. This metric, by the main ansatz, is induced by mechanisms such as growth, plasticity, thermal expansion etc, and is the cause of the elastic deformation that determines the shape of the body. One can seek this deformation through a variational minimization principle, as the best possible immersion of the given Riemannian manifold into the flat space. For thin films, the characteristics of the original prestrain metric induce the distinct nonlinear theories in the singular limit of vanishing thickness, that determine the above mentioned minimizing deformation. A particular aspect of these limit theories are the curvature constraints that are manifested as the Monge-Ampère equation in the appropriate energy regimes.

Time/place: Friday 09th of June 2017, 13.15–14.00, room 734
Speaker: Laurel Ohm (University of Minnesota) Title: An error analysis framework for slender body theory
Abstract: An increasingly important question in microbiology involves understanding how microorganisms use thin cilia to propel themselves through viscous fluid. Exploring how these thin structures interact with the surrounding fluid often requires numerical simulations of large numbers of fibers. Slender body theory simplifies these simulations by treating each fiber as a one-dimensional curve of point forces. Despite the many numerical models based on slender body theory, very little analytical work has been done to quantify the error introduced by this approximation. In this talk, we construct a PDE framework for understanding the slender body problem and derive the first such error estimate. In addition to rigorously justifying a common modeling approximation, our work provides a framework for answering related questions in microfluidics.

Time/place: Friday 09th of June 2017, 13.15–14.00, room 734

Speaker: Laurel Ohm (University of Minnesota)
Title: An error analysis framework for slender body theory

Abstract: An increasingly important question in microbiology involves understanding how microorganisms use thin cilia to propel themselves through viscous fluid. Exploring how these thin structures interact with the surrounding fluid often requires numerical simulations of large numbers of fibers. Slender body theory simplifies these simulations by treating each fiber as a one-dimensional curve of point forces. Despite the many numerical models based on slender body theory, very little analytical work has been done to quantify the error introduced by this approximation. In this talk, we construct a PDE framework for understanding the slender body problem and derive the first such error estimate. In addition to rigorously justifying a common modeling approximation, our work provides a framework for answering related questions in microfluidics.

Time/place: Wednesday 7th of June 2017, 13.15–14.00, room 734
Speaker: Gustaf Söderlind (Lund University) Title: Zero stability on nonuniform grids. A Toeplitz operator approach
Abstract: For multistep ODE methods, zero stability is a necessary condition for convergence, as proved by Lax and Dahlqvist in 1955 and 1956, respectively. For adaptive methods, however, little is known about zero stability. A few results are known, but they appear pessimistic when it comes to how fast step size ratios can increase. In this talk we take a new approach based on Toeplitz operators. We show that every multistep method on a nonuniform grid can be factorized into an extraneous operator and a simple integration on a uniform grid. While the latter is always stable, the problem with zero stability is entirely due to the extraneous operator. Here we develop this theory to pave the way for proving convergence of multistep methods on smooth nonuniform grids. The conclusion of the talk is that it may now — finally! — be possible to prove that adaptive methods, as parameterized by Arévalo, are convergent in a mathematical sense on smooth nonuniform grids.

Time/place: Wednesday 7th of June 2017, 13.15–14.00, room 734

Speaker: Gustaf Söderlind (Lund University)
Title: Zero stability on nonuniform grids. A Toeplitz operator approach

Abstract: For multistep ODE methods, zero stability is a necessary condition for convergence, as proved by Lax and Dahlqvist in 1955 and 1956, respectively. For adaptive methods, however, little is known about zero stability. A few results are known, but they appear pessimistic when it comes to how fast step size ratios can increase.
In this talk we take a new approach based on Toeplitz operators. We show that every multistep method on a nonuniform grid can be factorized into an extraneous operator and a simple integration on a uniform grid. While the latter is always stable, the problem with zero stability is entirely due to the extraneous operator.
Here we develop this theory to pave the way for proving convergence of multistep methods on smooth nonuniform grids. The conclusion of the talk is that it may now — finally! — be possible to prove that adaptive methods, as parameterized by Arévalo, are convergent in a mathematical sense on smooth nonuniform grids.

Time/place: Thursday 1st of June 2017, 10.15–11.00, room 656
Speaker: W. Steven Gray (ODU, Norfolk, Virginia) Title: Discrete-Time Approximations of Fliess Operators
Abstract: A convenient way to represent a nonlinear input-output system in control theory is via a Chen-Fliess functional expansion or Fliess operator. This is a weighted series of iterated integrals involving the control input functions. The general goal of this talk is to describe how to approximate Fliess operators with iterated sums and to provide accurate error estimates for two different scenarios, one where the series coefficients are growing at a local convergence rate, and the other where they are growing at a global convergence rate. In each case, it is shown that the error estimates are asymptotically achievable for certain worst case inputs. Of particular interest is the special case where the operators are rational, i.e., where they have rational generating series, and thus are realizable in terms of bilinear ordinary differential state equations. Specifically, it is shown here that a discretization of the state equation via a kind of Euler approximation coincides exactly with the discrete-time Fliess operator approximator of the continuous-time rational operator.

Time/place: Thursday 1st of June 2017, 10.15–11.00, room 656

Speaker: W. Steven Gray (ODU, Norfolk, Virginia)
Title: Discrete-Time Approximations of Fliess Operators

Abstract: A convenient way to represent a nonlinear input-output system in control theory is via a Chen-Fliess functional expansion or Fliess operator. This is a weighted series of iterated integrals involving the control input functions. The general goal of this talk is to describe how to approximate Fliess operators with iterated sums and to provide accurate error estimates for two different scenarios, one where the series coefficients are growing at a local convergence rate, and the other where they are growing at a global convergence rate. In each case, it is shown that the error estimates are asymptotically achievable for certain worst case inputs. Of particular interest is the special case where the operators are rational, i.e., where they have rational generating series, and thus are realizable in terms of bilinear ordinary differential state equations. Specifically, it is shown here that a discretization of the state equation via a kind of Euler approximation coincides exactly with the discrete-time Fliess operator approximator of the continuous-time rational operator.

Time/place: Wednesday 24th of May 2017, 10.15–11.00, room KJL4
Speaker: Laura Spinolo (Pavia) Title: On the nonlocal to local limit for conservation laws
Abstract: I will discuss a question posed by Amorim, Colombo and Teixeira, which can be loosely speaking formulated as follows. Consider a family of continuity equations where the velocity field is given by the convolution of the solution with a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law: can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one in general does not have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law. This is joint work with Maria Colombo and Gianluca Crippa.

Time/place: Wednesday 24th of May 2017, 10.15–11.00, room KJL4

Speaker: Laura Spinolo (Pavia)
Title: On the nonlocal to local limit for conservation laws

Abstract: I will discuss a question posed by Amorim, Colombo and Teixeira, which can be loosely speaking formulated as follows. Consider a family of continuity equations where the velocity field is given by the convolution of the solution with a regular kernel. In the singular limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a conservation law: can we rigorously justify this formal limit? We exhibit counterexamples showing that, despite numerical evidence suggesting a positive answer, one in general does not have convergence of the solutions. We also show that the answer is positive if we consider viscous perturbations of the nonlocal equations. In this case, in the singular local limit the solutions converge to the solution of the viscous conservation law. This is joint work with Maria Colombo and Gianluca Crippa.

Time/place: Monday 08th of May 2017, 11.15-12.00, room 734
Speaker: Douglas N. Arnold (University of Minnesota) Title: The Regge family of finite elements: structure-preserving elements for metrics
Abstract: Over the past decade there has been a great deal of interest in compatible or structure-preserving discretizations of PDE, that is, discretizations which exactly retain certain key geometric or algebraic properties of the continuous problem at the discrete level. Structure-preserving finite element methods have been developed for differential forms, such as arise in electromagnetic and flow applications, and for stress fields in solid mechanics. In this talk we will introduce a new family of finite element spaces, devised for discretization of another sort of field important in applications: Riemannian metrics and other symmetric covariant tensors of rank 2. In the lowest order case these new finite elements are intimately related to discrete metrics introduced by Tullio Regge in 1961 for the study of general relativity; hence their name. Special cases of the new elements have connections to classical plate bending elements and to a recent novel approach to elasticity as well. This work is joint with Lizao Li and will appear in his thesis.

Time/place: Monday 08th of May 2017, 11.15-12.00, room 734

Speaker: Douglas N. Arnold (University of Minnesota)
Title: The Regge family of finite elements: structure-preserving elements for metrics

Abstract: Over the past decade there has been a great deal of interest in compatible or structure-preserving discretizations of PDE, that is, discretizations which exactly retain certain key geometric or algebraic properties of the continuous problem at the discrete level. Structure-preserving finite element methods have been developed for differential forms, such as arise in electromagnetic and flow applications, and for stress fields in solid mechanics. In this talk we will introduce a new family of finite element spaces, devised for discretization of another sort of field important in applications: Riemannian metrics and other symmetric covariant tensors of rank 2. In the lowest order case these new finite elements are intimately related to discrete metrics introduced by Tullio Regge in 1961 for the study of general relativity; hence their name. Special cases of the new elements have connections to classical plate bending elements and to a recent novel approach to elasticity as well. This work is joint with Lizao Li and will appear in his thesis.

Time/place: Wednesday 03rd of May 2017, 10.15–11.00, room 656
Speaker: Jørgen Endal (NTNU) Title: On nonlocal equations of porous medium type
Abstract: We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem for nonlocal equations of porous medium type. Here the nonlocal operator can be any symmetric (possibly \(x\)-dependent) degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The nonlinearity is only assumed to be continuous and nondecreasing. The class of equations include nonlocal (generalized) porous medium equations, fast diffusion equations, and Stefan problems. We will also consider numerical schemes of these equations.

Time/place: Wednesday 03rd of May 2017, 10.15–11.00, room 656

Speaker: Jørgen Endal (NTNU)
Title: On nonlocal equations of porous medium type

Abstract: We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem for nonlocal equations of porous medium type. Here the nonlocal operator can be any symmetric (possibly \(x\)-dependent) degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The nonlinearity is only assumed to be continuous and nondecreasing. The class of equations include nonlocal (generalized) porous medium equations, fast diffusion equations, and Stefan problems. We will also consider numerical schemes of these equations.

Time/place: Wednesday 26th of April 2017, 10.15–11.00, room 656
Speaker: Diana Stan (BCAM, Bilbao) Title: Asymptotic behaviour for fractional diffusion-convection equations
Abstract: We are interested in the large time asymptotic behaviour of a convection-diffusion model with linear fractional diffusion. More exactly, we consider the initial value problem \[\begin{cases}u_t (t, x) + (-\Delta)^{\alpha/2} u(t, x) + (f(u))_x = 0 & \text{ for } t > 0 \text{ and } x \in \mathbb{R},\\ u(0, x) = u_0(x) & \text{ for } x \in \mathbb{R}, \end{cases}\] where \(u \colon (0,\infty) \times \mathbb{R} \to \mathbb{R}\), \((-\Delta)^{\alpha/2}\) is the Fractional Laplacian operator of order \(\alpha \in (0,2)\) and \(f(\cdot)\) is a locally Lipschitz function whose prototype is \(f(s) = \|s\|^{q-1} s/q\) with \(q > 1\). The initial data is assumed to be \(u_0 \in L^1(\mathbb{R}) \cap L^\infty(\mathbb{R})\). We prove that, in the subcritical range \(1 < q < \alpha\), the large time asymptotic behaviour of the solution is given by the unique entropy solution of the convective part of the equation. The proof is based on suitable a-priori estimates, among which proving an Oleinik type inequality plays a key role. Previous works in the nonlocal case have been done by N. Alibaud, B. Andreianov, P. Biler, J. Endal, G. Karch, E.R. Jakobsen and many others. For the local case we mention M. Escobedo, J.L. Vázquez, E. Zuazua. This is joint work with Professor Liviu Ignat (IMAR Romania). References * N. Alibaud, Entropy formulation for fractal conservation laws, J. Evol. Equ., vol. 7, no. 1, pp. 145–175, 2007. * M. Escobedo, J. L. Vázquez, and E. Zuazua, Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal., vol. 124, no. 1, pp. 43–65, 1993. * L. Ignat and D. Stan, Asymptotic behaviour for fractional diffusion-convection equations, submitted, arXiv:1703.02908.

Time/place: Wednesday 26th of April 2017, 10.15–11.00, room 656

Speaker: Diana Stan (BCAM, Bilbao)
Title: Asymptotic behaviour for fractional diffusion-convection equations

Abstract: We are interested in the large time asymptotic behaviour of a convection-diffusion model with linear fractional diffusion. More exactly, we consider the initial value problem \[\begin{cases}u_t (t, x) + (-\Delta)^{\alpha/2} u(t, x) + (f(u))_x = 0 & \text{ for } t > 0 \text{ and } x \in \mathbb{R},\\ u(0, x) = u_0(x) & \text{ for } x \in \mathbb{R}, \end{cases}\] where \(u \colon (0,\infty) \times \mathbb{R} \to \mathbb{R}\), \((-\Delta)^{\alpha/2}\) is the Fractional Laplacian operator of order \(\alpha \in (0,2)\) and \(f(\cdot)\) is a locally Lipschitz function whose prototype is \(f(s) = |s|^{q-1} s/q\) with \(q > 1\). The initial data is assumed to be \(u_0 \in L^1(\mathbb{R}) \cap L^\infty(\mathbb{R})\).
We prove that, in the subcritical range \(1 < q < \alpha\), the large time asymptotic behaviour of the solution is given by the unique entropy solution of the convective part of the equation. The proof is based on suitable a-priori estimates, among which proving an Oleinik type inequality plays a key role.
Previous works in the nonlocal case have been done by N. Alibaud, B. Andreianov, P. Biler, J. Endal, G. Karch, E.R. Jakobsen and many others. For the local case we mention M. Escobedo, J.L. Vázquez, E. Zuazua.
This is joint work with Professor Liviu Ignat (IMAR Romania).
References
* N. Alibaud, Entropy formulation for fractal conservation laws, J. Evol. Equ., vol. 7, no. 1, pp. 145–175, 2007.
* M. Escobedo, J. L. Vázquez, and E. Zuazua, Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal., vol. 124, no. 1, pp. 43–65, 1993.
* L. Ignat and D. Stan, Asymptotic behaviour for fractional diffusion-convection equations, submitted, arXiv:1703.02908.

Time/place: Thursday 23rd of March 2017, 09.15–10.00, room 734
Speaker: Poul G. Hjorth (DTU Denmark) Title: Applied Mathematics Applied
Abstract: 50 years ago a new type of interaction between industry and academia was initiated by applied mathematicians at University of Oxford: the so-called Study Groups with Industry. These problem-solving workshops were later introduced at other UK universities, at universities in Europe, and on other continents as well. I will describe the format, the practice, and give a few examples of the wide variety of problems in industrial mathematics that I have come across at Study Groups in the UK, and in Denmark.

Time/place: Thursday 23rd of March 2017, 09.15–10.00, room 734

Speaker: Poul G. Hjorth (DTU Denmark)
Title: Applied Mathematics Applied

Abstract: 50 years ago a new type of interaction between industry and academia was initiated by applied mathematicians at University of Oxford: the so-called Study Groups with Industry. These problem-solving workshops were later introduced at other UK universities, at universities in Europe, and on other continents as well. I will describe the format, the practice, and give a few examples of the wide variety of problems in industrial mathematics that I have come across at Study Groups in the UK, and in Denmark.

Time/place: Wednesday 22nd of March 2017, 10.15–11.00, room 656
Speaker: Kathrin Flaßkamp (Bremen) Title: Variational integrators for optimal control and estimation of mechanical systems
Abstract: Control, optimization, and estimation tasks for the underlying mechanical system commonly arise in many fields of application, e.g. robotics, biomechanics, or automotive engineering. Those problems require numerical solution techniques. This talk focusses on structure-preserving solution methods based on variational integrator (VI) discretization, which has gained growing interest in recent years. In particular, transcription methods for direct optimal control methods have been developed in literature. Rarely, VI have been used in indirect methods. However, for linear quadratic optimal control problems, modified discrete-time Riccati equations can be derived. This approach then preserves not only the Hamiltonian structure of the mechanics but also the Hamiltonian structure of the state-adjoint equations characterizing the optimal solution. As this is the dual problem, we also consider the optimal state estimation (Kalman filtering) and we show various application examples throughout the talk.

Time/place: Wednesday 22nd of March 2017, 10.15–11.00, room 656

Speaker: Kathrin Flaßkamp (Bremen)
Title: Variational integrators for optimal control and estimation of mechanical systems

Abstract: Control, optimization, and estimation tasks for the underlying mechanical system commonly arise in many fields of application, e.g. robotics, biomechanics, or automotive engineering. Those problems require numerical solution techniques. This talk focusses on structure-preserving solution methods based on variational integrator (VI) discretization, which has gained growing interest in recent years. In particular, transcription methods for direct optimal control methods have been developed in literature. Rarely, VI have been used in indirect methods. However, for linear quadratic optimal control problems, modified discrete-time Riccati equations can be derived. This approach then preserves not only the Hamiltonian structure of the mechanics but also the Hamiltonian structure of the state-adjoint equations characterizing the optimal solution. As this is the dual problem, we also consider the optimal state estimation (Kalman filtering) and we show various application examples throughout the talk.

Time/place: Wednesday 15th of March, 10.15–11.00 2017, room 656
Speaker: Raj Dhara (University of Warsaw) Title: Existence and regularity theory in weighted Sobolev spaces and applications
Abstract: My emphasis in this talk will be on functional analytical tools to the solvability and uniqueness of solutions to the nonhomogeneous boundary value problems, dealing with degenerate PDEs of elliptic type. My aim is to consider possibly general classes of weights. In particular, I consider the \(B_{p}\)-class of weights, introduced by Kufner and Opic, which is a much more general class than the commonly studied Muckenhoupt \(A_{p}\)-class.

Time/place: Wednesday 15th of March, 10.15–11.00 2017, room 656

Speaker: Raj Dhara (University of Warsaw)
Title: Existence and regularity theory in weighted Sobolev spaces and applications

Abstract: My emphasis in this talk will be on functional analytical tools to the solvability and uniqueness of solutions to the nonhomogeneous boundary value problems, dealing with degenerate PDEs of elliptic type. My aim is to consider possibly general classes of weights. In particular, I consider the \(B_{p}\)-class of weights, introduced by Kufner and Opic, which is a much more general class than the commonly studied Muckenhoupt \(A_{p}\)-class.

Time/place: Tuesday 28th of February, 11.15–12.00 2017, room 734
Speaker: David Lannes (Bordeaux) Title: On the dynamics of floating structures
Abstract: The goal of this talk is to derive some equations describing the interaction of a floating solid structure and the surface of a perfect fluid. This is a double free boundary problem since in addition to the water waves problem (determining the free boundary of the fluid region), one has to find the evolution of the contact line between the solid and the surface of the water. The so-called floating body problem has been studied so far as a three-dimensional problem. Our first goal is to reduce it to a two-dimensional problem that takes the form of a coupled compressible-incompressible system. We will also show that the hydrodynamic forces acting on the solid can be partly put under the form of an added mass-inertia matrix, which turns out to be affected by the dispersive terms of the equations.

Time/place: Tuesday 28th of February, 11.15–12.00 2017, room 734

Speaker: David Lannes (Bordeaux)
Title: On the dynamics of floating structures

Abstract: The goal of this talk is to derive some equations describing the interaction of a floating solid structure and the surface of a perfect fluid. This is a double free boundary problem since in addition to the water waves problem (determining the free boundary of the fluid region), one has to find the evolution of the contact line between the solid and the surface of the water. The so-called floating body problem has been studied so far as a three-dimensional problem. Our first goal is to reduce it to a two-dimensional problem that takes the form of a coupled compressible-incompressible system. We will also show that the hydrodynamic forces acting on the solid can be partly put under the form of an added mass-inertia matrix, which turns out to be affected by the dispersive terms of the equations.

Time/place: Tuesday 28th of February, 14.15–15.00 2017, room 734
Speaker: Caren Tischendorf (HU-Berlin) Title: Asymptotic Stability of Autonomous (P)DAEs with Network Structure
Abstract: We analyze autonomous differential algebraic equations (DAEs) with the following particular structure: \(y_1 = \frac{d}{dt}f_1(x_1) + g_1(x_1)\), \(x_2 = \frac{d}{dt}f_2(y_2)+g_2(y_2)\), \(x_1 = A_1^Tz\), \(x_2 = A_2^Tz\), \(0=A_1y_1 + A_2y_2\), where \((A_1,A_2)\) is an incidence matrix of full row rank. First we show that a transient modeling of the flow of different kind of networks (circuits, water networks, gas networks, blood circuits) leads to DAEs with this structure. The functions \(f_1\), \(f_2\), \(g_1\), \(g_2\) represent the element models and their spatial discretizations (in case of PDE element models). The matrices \(A_1\), \(A_2\) describe the network topology. Furthermore we show the equivalence to a Port-Hamiltonian system in the special case that \(A_1\) is nonsingular and \(f_1\), \(g_1\) are globally invertible. Finally we present criteria on the element model functions \(f_1\), \(f_2\), \(g_1\), \(g_2\) for asymptotic stability of DAEs with this structure. It includes a characterization of the eigenvalue structure for the related generalized eigenvalue problem of the form \(y_1 = \lambda F_1x_1 + G_1x_1\), \(x_2 = \lambda F_2y_2 + G_2y_2\), \(x_1 = A_1^Tz\), \(x_2 = A_2^Tz\), \(0 = A_1y_1 + A_2y_2\).

Time/place: Tuesday 28th of February, 14.15–15.00 2017, room 734

Speaker: Caren Tischendorf (HU-Berlin)
Title: Asymptotic Stability of Autonomous (P)DAEs with Network Structure

Abstract: We analyze autonomous differential algebraic equations (DAEs) with the following particular structure:
\(y_1 = \frac{d}{dt}f_1(x_1) + g_1(x_1)\),
\(x_2 = \frac{d}{dt}f_2(y_2)+g_2(y_2)\),
\(x_1 = A_1^Tz\),
\(x_2 = A_2^Tz\),
\(0=A_1y_1 + A_2y_2\),
where \((A_1,A_2)\) is an incidence matrix of full row rank. First we show that a transient modeling of the flow of different kind of networks (circuits, water networks, gas networks, blood circuits) leads to DAEs with this structure. The functions \(f_1\), \(f_2\), \(g_1\), \(g_2\) represent the element models and their spatial discretizations (in case of PDE element models). The matrices \(A_1\), \(A_2\) describe the network topology. Furthermore we show the equivalence to a Port-Hamiltonian system in the special case that \(A_1\) is nonsingular and \(f_1\), \(g_1\) are globally invertible.
Finally we present criteria on the element model functions \(f_1\), \(f_2\), \(g_1\), \(g_2\) for asymptotic stability of DAEs with this structure. It includes a characterization of the eigenvalue structure for the related generalized eigenvalue problem of the form
\(y_1 = \lambda F_1x_1 + G_1x_1\),
\(x_2 = \lambda F_2y_2 + G_2y_2\),
\(x_1 = A_1^Tz\),
\(x_2 = A_2^Tz\),
\(0 = A_1y_1 + A_2y_2\).

Time/place: Monday 5th, Tuesday 6th of February, 13.15–14.00 in 734 2017
Speaker: John Maddocks (EPFL/Lousanne) Title: Unconstrained Hamiltonian formulations of constrained Lagrangian dynamics, Parts I and II
Abstract: Part I: I will describe how second-order constrained systems of Lagrangian DAE can be formulated as unconstrained first-order Hamiltonian systems where the constraints are by construction integrals of the Hamiltonian system. This apparently magical transformation can be viewed as the dual procedure to Dirac's theory of constraints. It relies on the fact that in Lagrangian systems subject to holonomic constraints, there is no single most natural choice of the associated conjugate variable in the Legendre transform. The theory will be illustrated by two simple ODE examples: a) the point-mass spherical pendulum in a form that you have probably never seen before, and b) a Hamiltonian formulation of rigid body dynamics using quaternions. Part II: (which I will aim to make understandable independent of Part I) will be applications of the theory of Part I to the analysis and computation of the equations for the statics (ODE in the rigid body group SE(3)) and dynamics (1+1 PDE in SE(3)) of elastic rods. One of the main conclusions of the presentation is that there are non-standard classes of interesting Hamiltonian problems with practical applications that could very well benefit from the construction of specialised geometrical integration schemes.

Time/place: Monday 5th, Tuesday 6th of February, 13.15–14.00 in 734 2017

Speaker: John Maddocks (EPFL/Lousanne)
Title: Unconstrained Hamiltonian formulations of constrained Lagrangian dynamics, Parts I and II

Abstract: Part I: I will describe how second-order constrained systems of Lagrangian DAE can be formulated as unconstrained first-order Hamiltonian systems where the constraints are by construction integrals of the Hamiltonian system. This apparently magical transformation can be viewed as the dual procedure to Dirac's theory of constraints. It relies on the fact that in Lagrangian systems subject to holonomic constraints, there is no single most natural choice of the associated conjugate variable in the Legendre transform. The theory will be illustrated by two simple ODE examples: a) the point-mass spherical pendulum in a form that you have probably never seen before, and b) a Hamiltonian formulation of rigid body dynamics using quaternions. Part II: (which I will aim to make understandable independent of Part I) will be applications of the theory of Part I to the analysis and computation of the equations for the statics (ODE in the rigid body group SE(3)) and dynamics (1+1 PDE in SE(3)) of elastic rods. One of the main conclusions of the presentation is that there are non-standard classes of interesting Hamiltonian problems with practical applications that could very well benefit from the construction of specialised geometrical integration schemes.

Time/place: Monday 23th of January, 11.15–12.00 in 734 2017
Speaker: Chongchun Zeng (Georgia Institute of Technology Title: Instability, index theorems, and exponential dichotomy of Hamiltonian PDEs
Abstract: In this talk, we start with a general linear Hamiltonian system \(u_t = JL u\) in a Hilbert space \(X\) – the energy space. The main assumption is that the energy functional \(\frac 12 \langle Lu, u\rangle\) has only finitely many negative dimensions – \(n^-(L) < \infty\). Our first result is an index theorem related to the linear instability of \(e^{tJL}\), which gives some relationship between \(n^-(L)\) and the dimensions of spaces of generalized eigenvectors of eigenvalues of \(JL\). Under some additional non-degeneracy assumption, for each eigenvalue \(\lambda \in i R\) of \(JL\) we also construct special 'good' choice of generalized eigenvectors which both realize the corresponding Jordan canonical form corresponding to \(\lambda\) and work well with \(L\). Our second result is the linear exponential trichotomy of the group \(e^{tJL}\). This includes the nonexistence of exponential growth in the finite co-dimensional invariant center subspace and the optimal bounds on the algebraic growth rate there. Thirdly we consider the structural stability of this type of systems under perturbations if time permits. Finally we discuss applications to examples of nonlinear Hamiltonian PDEs such as BBM, GP, and 2-D Euler equations, including the construction of some local invariant manifolds near some coherent states (standing wave, steady state, traveling waves etc.). This is a joint work with Zhiwu Lin.

Time/place: Monday 23th of January, 11.15–12.00 in 734 2017

Speaker: Chongchun Zeng (Georgia Institute of Technology
Title: Instability, index theorems, and exponential dichotomy of Hamiltonian PDEs

Abstract: In this talk, we start with a general linear Hamiltonian system \(u_t = JL u\) in a Hilbert space \(X\) – the energy space. The main assumption is that the energy functional \(\frac 12 \langle Lu, u\rangle\) has only finitely many negative dimensions – \(n^-(L) < \infty\). Our first result is an index theorem related to the linear instability of \(e^{tJL}\), which gives some relationship between \(n^-(L)\) and the dimensions of spaces of generalized eigenvectors of eigenvalues of \(JL\). Under some additional non-degeneracy assumption, for each eigenvalue \(\lambda \in i R\) of \(JL\) we also construct special 'good' choice of generalized eigenvectors which both realize the corresponding Jordan canonical form corresponding to \(\lambda\) and work well with \(L\). Our second result is the linear exponential trichotomy of the group \(e^{tJL}\). This includes the nonexistence of exponential growth in the finite co-dimensional invariant center subspace and the optimal bounds on the algebraic growth rate there. Thirdly we consider the structural stability of this type of systems under perturbations if time permits. Finally we discuss applications to examples of nonlinear Hamiltonian PDEs such as BBM, GP, and 2-D Euler equations, including the construction of some local invariant manifolds near some coherent states (standing wave, steady state, traveling waves etc.). This is a joint work with Zhiwu Lin.

Time/place: Thursday 12th of January, 13.15–14.00 in 734 2017
Speaker: John Carter (Seattle/Bergen) Title: Frequency downshifting in a viscous fluid
Abstract: Frequency downshift, i.e. a shift in the spectral peak to a lower frequency, in a train of nearly monochromatic gravity waves was first reported by Lake et al. (1977). Even though it is generally agreed upon that frequency downshifting (FD) is related to the Benjamin-Feir instability and many physical phenomena (including wave breaking and wind) have been proposed as mechanisms for FD, its precise cause remains an open question. Dias et al. (2008) added a viscous correction to the Euler equations and derived the dissipative NLS equation (DNLS). In this talk, we introduce a higher-order generalization of the DNLS equation, which we call the viscous Dysthe equation. We outline the derivation of this new equation and present many of its properties. We establish that it predicts FD in both the spectral mean and spectral peak senses. Finally, we demonstrate that predictions obtained from the viscous Dysthe equation accurately model data from experiments in which frequency downshift occurred.

Time/place: Thursday 12th of January, 13.15–14.00 in 734 2017

Speaker: John Carter (Seattle/Bergen)
Title: Frequency downshifting in a viscous fluid

Abstract: Frequency downshift, i.e. a shift in the spectral peak to a lower frequency, in a train of nearly monochromatic gravity waves was first reported by Lake et al. (1977). Even though it is generally agreed upon that frequency downshifting (FD) is related to the Benjamin-Feir instability and many physical phenomena (including wave breaking and wind) have been proposed as mechanisms for FD, its precise cause remains an open question. Dias et al. (2008) added a viscous correction to the Euler equations and derived the dissipative NLS equation (DNLS). In this talk, we introduce a higher-order generalization of the DNLS equation, which we call the viscous Dysthe equation. We outline the derivation of this new equation and present many of its properties. We establish that it predicts FD in both the spectral mean and spectral peak senses. Finally, we demonstrate that predictions obtained from the viscous Dysthe equation accurately model data from experiments in which frequency downshift occurred.