# Program

## Thursday, May 4

14:30 – 15:25

Registration

15:25 – 15:30

Opening

15:30 – 15:55

Approximation of Fluid-Structure Interaction in Porous Media

Modeling of coupled mechanical deformation and fluid flow in porous media has become of increasing importance in several branches of natural sciences and technology. Recently, iterative coupling techniques for the numerical simulation of such problems have attracted researchers' interest. In this contribution we consider the quasi-static Biot system of poroelasticity. We present families of higher order space-time finite element approximations of the Biot system of poroelasticity that are based on an optimized iterative coupling of properly defined subproblems of mechanical deformation and fluid flow. For the discretization in time continuous and discontinuous Galerkin methods are studied. Mixed finite element methods are applied for the spatial discretization of the subproblem of fluid flow. The convergence of the iterative coupling scheme is proved on the continuous and discrete level. Efficient solution techniques for solving the arising algebraic systems of equations are addressed. The convergence, stability and performance properties of the approaches are illustrated by numerical experiments. Finally, future extensions of the methods for the construction of monolithic solvers and to the fully dynamic hyperbolic-parabolic Biot-Allard model of poroelasticity are discussed.

This is a joint work with U. Köcher (Helmut Schmidt University) and F. Radu (University of Bergen).

References:

[1] M. Bause, F. A. Radu, U. Köcher,

[2] M. Bause, U. Köcher,

[3] M. Bause, F. A. Radu, U. Köcher,

[4] U. Köcher,

[5] U. Köcher, M. Bause,

This is a joint work with U. Köcher (Helmut Schmidt University) and F. Radu (University of Bergen).

References:

[1] M. Bause, F. A. Radu, U. Köcher,

*Space-time finite element approximation of the Biot poroelasticity system with iterative coupling*, Comp. Meth. Appl. Mech. Eng., in press (2017), doi:10.1016/j.cma.2017.03.017 and arXiv:1611.06335v1, 1--24.[2] M. Bause, U. Köcher,

*Iterative coupling of variational space-time methods for Biot's system of poroelasticity*, in B. Karasözen et al. (eds.),*Numerical Mathematics and Advanced Applications ENUMATH 2015*, Springer, 2016, 143--151.[3] M. Bause, F. A. Radu, U. Köcher,

*Error analysis for discretizations of parabolic problems using continuous finite elements in time and mixed finite elements in space*, Numer. Math., under review (2015), arXiv:1504.04491v2, 1--47.[4] U. Köcher,

*Variational Space-Time Methods for the Elastic Wave Equation and the Diffusion Equation*, Ph.D. thesis, Helmut Schmidt University Hamburg (2015), urn:nbn:de:gbv:705-opus-31129, 1--188.[5] U. Köcher, M. Bause,

*Variational space-time discretisations for the wave equation*, J. Sci. Comput., 61 (2014), 424--453.15:55 – 16:20

Finite Element Approximation of Ultrasonic Wave Propagation under Fluid-Structure Interaction (FSI) for Structural Health Monitoring (SHM) Systems

In this contribution, a concept of coupling fluid-structure interaction (FSI) with an ultrasonic wave propagation is proposed, which is referred to as extended Fluid-Structure Interaction (eXFSI) problem. The eXFSI is a one-directional coupling of typical FSI problem with an ultrasonic wave propagation in fluid-solid and their interaction (WpFSI). The WpFSI is a strongly coupled problem of acoustic and elastic wave equations and automatically adopts the boundary and initial condition from previous time step. To the best of our knowledge, such a model is new in the literature. The FSI is modelled in terms of the arbitrary Lagrangian Eulerian (ALE) technique and couples the isothermal, incompressible Navier-Stokes equations with nonlinear elastodynamics using the Saint-Venant Kirchhoff solid model. The WpFSI problems are solved on the moving mesh which is automatically adopted from the FSI problem at each time step. The ALE approach provides a simple, but powerful procedure to couple solid deformations with fluid flows by a monolithic solution algorithm. In such a setting, the fluid equations are transformed to a fixed reference configuration via the ALE mapping. However, combining fluid dynamics with structural analysis traditionally poses a formidable challenge for even the most advanced numerical techniques due to the disconnected, domain-specific nature of analysis tools. The principal aim of this research is the exploration and development of concepts for the efficient numerical solution of the eXFSI problem. The finite element method is used for the spatial discretization. Temporal discretization is based on finite differences and is formulated as a one step-theta scheme, from which we can consider Crank-Nicolson, shifted Crank-Nicolson and the fractional-step-theta schemes. The nonlinear problem is solved by a Newton-like method. Our application of the eXFSI and WpFSI models is the design of on-live and off-live Structural Health Monitoring (SHM) systems for composite material and lightweight structure for the non-destructive inspection, respectively. Further applications of the models can be found in biomechanics, e.g. hemodynamics, Vibro-mechanics, poroelasticity as well as subsurface and porous media flow. The implementation is accomplished via the software library package DOpElib and deal.II.

References:

[1] C. Goll, T. Wick and W. Wollner.

[2] C. Goll, T. Wick, and W. Wollner.

[3] W. Bangerth, T. Heister and G. Kanschat.

[4] W. Bangerth, R. Hartmann and G. Kanschat.

[5] B.S.M. Ebna Hai, and M. Bause.

[6] B.S.M. Ebna Hai, and M. Bause.

[7] B.S.M. Ebna Hai, M. Bause and P. Kuberry.

[8] B.S.M. Ebna Hai and M. Bause.

[9] B.S.M. Ebna Hai and M. Bause.

[10] B.S.M. Ebna Hai and M. Bause.

[11] B.S.M. Ebna Hai and M. Bause.

[12] B.S.M. Ebna Hai and M. Bause.

[13] B.S.M. Ebna Hai and M. Bause.

[14] B.S.M. Ebna Hai.

[15] B.S.M. Ebna Hai and M. Bause.

References:

[1] C. Goll, T. Wick and W. Wollner.

*DOpElib: The Differential Equation and Optimization Environment.*web: http://www.dopelib.net.[2] C. Goll, T. Wick, and W. Wollner.

*DOpElib: The Differential Equation and Optimization Environment: A Goal Oriented Software Library for Solving PDEs and Optimization Problems with PDEs.*Preprint, University of Heidelberg, 2012[3] W. Bangerth, T. Heister and G. Kanschat.

*deal.II: Differential Equations Analysis Library.*web: http://www.dealii.org.[4] W. Bangerth, R. Hartmann and G. Kanschat.

*deal.II - a general-purpose object-oriented finite element library.*ACM Transactions on Mathematical Software, Vol. 33, Issue 4, pp 24/1-27, 2007.}[5] B.S.M. Ebna Hai, and M. Bause.

*Finite Element Approximation of Fluid-Structure Interaction (FSI) Problem with Coupled Wave Propagation.*In proceedings of: the 88$^{th}$ GAMM Annual Meeting of the International Association of Applied Mathematics and Mechanics, Weimar, Germany, March 6-10, 2017.[6] B.S.M. Ebna Hai, and M. Bause.

*Coupling Fluid-Structure Interaction with an Ultrasonic Wave Propagation.*The 19$^{th}$ International Conference on Finite Elements in Flow Problems, Rome, Italy, April 5-7, 2017.[7] B.S.M. Ebna Hai, M. Bause and P. Kuberry.

*Finite Element Approximation of the eXtended Fluid-Structure Interaction (eXFSI) Problem.*In proceedings of: the ASME Fluids Engineering Division Summer Meeting, Vol. 1A, Washington, D.C., USA, July 10--14, 2016.[8] B.S.M. Ebna Hai and M. Bause.

*Finite Element Model-based Structural Health Monitoring (SHM) Systems for Composite Material under Fluid-Structure Interaction (FSI) Effect.*In proceedings of: the $7^{th}$ European Workshop on Structural Health Monitoring, Nantes, France, July 08--11, 2014 and The e-Journal of Nondestructive Testing & Ultrasonics, NDT.net issue, Vol. 20, Issue 2 (Feb 2015).[9] B.S.M. Ebna Hai and M. Bause.

*Adaptive Multigrid Methods for An Integrated Structural Health Monitoring (SHM) Systems for Composite Material with Fluid-Structure Interaction (FSI) Effect.*The SIAM Conference on Computational Science and Engineering, Salt Lake City, Utah, USA, March 14--18, 2015.[10] B.S.M. Ebna Hai and M. Bause.

*Adaptive Multigrid Methods for eXtended Fluid-Structure Interaction (eXFSI) Problem: Part I - Mathematical Modelling.*In proceedings of: the ASME International Mechanical Engineering Congress & Exposition, Vol. 7B, Houston, Texas, USA, Nov 13--19, 2015.[11] B.S.M. Ebna Hai and M. Bause.

*Finite Element Approximation of Wave Propagation in Composite Material With Asymptotic Homogenization.*In proceedings of: the ASME Turbo Expo: Turbine Technical Conference and Exposition, Vol. 7A, Duesseldorf, Germany, June 16--20, 2014.[12] B.S.M. Ebna Hai and M. Bause.

*Adaptive Finite Elements Simulation Methods and Applications for Monolithic Fluid-Structure Interaction (FSI) Problem.*In proceedings of: the ASME 4$^{th}$ Joint US-European Fluids Engineering Division Summer Meeting, Vol. 1B, Chicago, Illinois, USA, August 3--7, 2014.[13] B.S.M. Ebna Hai and M. Bause.

*Adaptive Multigrid Methods for Fluid-Structure Interaction (FSI) Optimization in an Aircraft and design of integrated Structural Health Monitoring (SHM) Systems.*In proceedings of: the $2^{nd}$ ECCOMAS Young Investigators Conference, Bordeaux, France, Sept 02--06, 2013.[14] B.S.M. Ebna Hai.

*Numerical Approximation of Fluid Structure Interaction (FSI) Problem.*In proceedings of: the ASME Fluids Engineering Division Summer Meeting, Vol. 1A, Incline Village, Nevada, USA, July 7--11, 2013.[15] B.S.M. Ebna Hai and M. Bause.

*Finite Element Approximation of Fluid Structure Interaction (FSI) Optimization in Arbitrary Lagrangian-Eulerian Coordinates.*In proceedings of: the ASME International Mechanical Engineering Congress & Exposition, Vol. 7B, San Diego, California, USA, November 15--21, 2013.16:20 – 16:45

High order approximation with the Virtual Element Method

The Virtual Element Method (VEM) is a very recent generalization of the Finite Element Method. VEM utilizes polygonal/polyhedral meshes in lieu of the classical triangular/tetrahedral and quadrilateral/hexaedral meshes. This automatically includes nonconvex elements, hanging nodes (enabling natural handling of interface problems with nonmatching grids), easy construction of adaptive meshes and efficient approximations of geometric data features.

In this talk we review the basic construction of the method and discuss an extension of VEM to approximations of high and increasing order of accuracy.

In this talk we review the basic construction of the method and discuss an extension of VEM to approximations of high and increasing order of accuracy.

16:45 – 17:10

Coffee Break

17:10 – 17:35

Modeling, Analysis, and Homogenization of Multiscale Models for Bainitic Phase Transformations in Steel

Steel is known to exhibit a complex material behavior where the macroscopic properties are highly dependent on the underlying microstructures or phases. Bainite is a particular type of microstructure that can form in undercooled Austenite steel, a transformation that is usually accompanied by mechanical stresses. In this talk, we present a microscale, sharp-interface model describing the bainitic phase transformation in steel. For a simplified linear, fully coupled thermoelasticity problem with given phase-interface movement, we investigate well-posedness and a priori estimates. Based on these results, we then rigorously derive a homogenized, macroscopic model which is coupled with distributed time-dependent microstructures that account for the underlying microscopical structure.

17:35 – 18:00

Some theoretical notes on the variable-fidelity surrogate modeling technique Cokriging

Cokriging is a variable-fidelity extension of the response surface method Kriging. It requires the modeling of the spatial cross- and auto-correlation of the input sample data. For Kriging this is done via positive definite correlation kernels but it is not obvious that this approach remains valid for Cokriging.

In this talk, we discuss theoretical aspects of the positive definiteness of the Cokriging matrix. Moreover, we present results on the likelihood-optimal choice of the Cokriging hyper-parameters.

In this talk, we discuss theoretical aspects of the positive definiteness of the Cokriging matrix. Moreover, we present results on the likelihood-optimal choice of the Cokriging hyper-parameters.

18:00 – 18:25

Parallel-in-time methods for climate simulation

While processor core clock speeds are not increasing any more, the available number of cores is growing rapidly in nowadays’ high performance hardware. Since the size of a spatial domain that is reasonable to distribute onto a single core is limited from below, an increase in the number of cores is generally used to increase the spatial resolution, which in almost every case is theoretically desirable. However, the temporal resolution is usually coupled to the spatial one by a CFL-type condition. Thus, any finer spatial discretization (theoretical possible by the usage of more processor cores) results in the necessity of a higher number of (smaller) time-steps and thus increase in the overall time to solution. Since the traditional time-stepping algorithms in all models are sequential, this becomes the main bottleneck in a long-time run. Parallel-in-time integration basically means that the whole time interval is splitted into time slices, and the time integration on these slices is performed in parallel on different (groups of) processors. The main tasks in this approach are (i) the generation of appropriate initial states for the time slices, i.e., on a coarse time-grid and (ii) the design of an algorithm that ensures the matching of the solutions on the time slices. As a result, a parallel-in-time algorithm has to iterate the descibed process, e.g. by a Newton-type procedure that minimizes the misfits at the matching points of the subintervals corresponding to the coarse time-grid. The method obtained in this way can be interpreted and embedded in the framework of deferred correction (e.g. Skeel 1982), multiple-shooting (e.g. Osborne 1969) and space-time-multigrid methods (Hackbusch 1984, 1985). Introduced by Nievergelt (1964) for simple, but already non-linear ordinary differential equations, it was further analyzed for non-linear, time-dependent partial differential equations, e.g. by Lions et al (2001). There the method was called „parareal algorithm“. The relation or interpretation as multiple shooting or multigrid is described, for example, by Gander and Vandervalle (2007) and Falgout et al (2013). For the model configuration to be used in long-term interglacial climate simulation runs (about 120'000 years), an appropriate model hierarchy has to be chosen. The basic idea of the parallel-in-time method remains applicable if coarse and fine propagator (i.e. model) differ not only in the resolution (and some parametrizations), but are different models. We present first examples for simple climate model configurations and the planned setting for more realistic, high resolution models.

19:15

Dinner at the Restaurant Al Limone (open end)

## Friday, May 5

09:00 – 09:25

Elementary proofs that the determinant of a symplectic matrix is 1

It seems to be of recurring interest in the literature to give alternative proofs for the fact that the determinant of a symplectic matrix is one. We state four short and elementary proofs for symplectic matrices over general fields. Two of them seem to be new.

09:25 – 09:50

On a new kind of Ansatz Spaces for the Linearization of Matrix Polynomials

Polynomial eigenvalue problems arise in the analysis and numerical solution of higher order systems of ordinary differential equations. One approach to solve a polynomial eigenvalue problem corresponding to a matrix polynomial $P(\lambda)$ is to consider a strong linearization for $P(\lambda)$. That is, the matrix polynomial is converted into a larger matrix pencil with the same eigenstructure as $P(\lambda)$ (e.g. the Frobenius companion form). Then the generalized eigenvalue problem for this strong linearization can be solved by standard methods. For any matrix polynomial $P(\lambda)$ there exist infinitely many strong linearizations. In the previous decade a systematic approach to generate strong linearizations has been developed. In particular, in [3] the ''ansatz spaces'' $\mathbb{L}_1(P), \mathbb{L}_2(P)$ and $\mathbb{DL}(P)$ of matrix pencils for a matrix polynomial $P(\lambda)$ were introduced. Its elements satisfy a certain ''ansatz equation'' and may be regarded as generalizations of the Frobenius companion form. The ansatz space approach yields a a large class of linearizations (i.e. almost every pencil in $\mathbb{L}_1(P)$ or $\mathbb{L}_2(P)$ is a strong linearization for $P(\lambda)$) from which one can select a linearization guaranteed to be as well-conditioned as the original problem.

In the light of recent developments ([1]) the question arose, whether there are alternative ''ansatz equations'' for defining other ansatz spaces of matrix pencils with similar properties. In fact, the linearizations in such an ansatz space should be easy to construct from $P(\lambda)$, flexible with respect to structure-preservation (e.g. symmetric linearizations for symmetric $P(\lambda)$) and well analyzable with respect to conditioning and perturbation analysis. In this talk we show how alternative ansatz equations with these properties may be derived on the basis of $\mathbb{L}_1(P)$ and how this leads to the large-dimensional ''Block Kronecker ansatz spaces'' [2]. We comprehensively characterize this family of ansatz spaces and show that they serve as an abundant source of (structured) strong linearizations.

References:

[1] Froilán M. Dopico, Piers W. Lawrence, Javier Pérez, and Paul van Dooren.

[2] Heike Faßbender and Philip Saltenberger.

[3] D. Steven Mackey, Niloufer Mackey, Christian Mehl, and Volker Mehrmann.

In the light of recent developments ([1]) the question arose, whether there are alternative ''ansatz equations'' for defining other ansatz spaces of matrix pencils with similar properties. In fact, the linearizations in such an ansatz space should be easy to construct from $P(\lambda)$, flexible with respect to structure-preservation (e.g. symmetric linearizations for symmetric $P(\lambda)$) and well analyzable with respect to conditioning and perturbation analysis. In this talk we show how alternative ansatz equations with these properties may be derived on the basis of $\mathbb{L}_1(P)$ and how this leads to the large-dimensional ''Block Kronecker ansatz spaces'' [2]. We comprehensively characterize this family of ansatz spaces and show that they serve as an abundant source of (structured) strong linearizations.

References:

[1] Froilán M. Dopico, Piers W. Lawrence, Javier Pérez, and Paul van Dooren.

*Block Kronecker Linearizations of Matrix Polynomials and their Backward Errors*. (MIMS Eprint 2016.34), 2016.[2] Heike Faßbender and Philip Saltenberger.

*Block Kronecker Ansatz Spaces for Matrix Polynomials.*To appear in*Linear Algebra and its Applications*, 2017. See also*On a new kind of Ansatz Spaces for Matrix Polynomials*, Preprint 2016, arXiv:1610.05988v2.[3] D. Steven Mackey, Niloufer Mackey, Christian Mehl, and Volker Mehrmann.

*Vector spaces of linearizations for matrix polynomials*. SIAM*Journal on Matrix Analysis and Applications*, 28(4):971–1004, 2006.09:50 – 10:15

Stability Aspects of Persistent Homology and Frame Theory

Some interactions between frame theory and persistent homology are discussed including alternative ways to build invariants in signal processing. Frame theory extends ideas of time-frequency analysis as the short term Fourier transform and wavelet theory. Persistent homology uses concepts from algebraic topology to design data analysis algorithms. We explain how stability properties play a role when combining these tools.

10:15 – 10:40

Coffee Break

10:40 – 11:05

Wavelet and Shearlet-Based Image Representations for Visual Servoing

We present a novel approach to visual servoing in which the guiding feature vector is given by the coefficients of wavelet- and shearlet-based transforms of the currently observed image.

A visual servoing scheme consists of a closed-loop control approach in which visual information feedback is used to control the motion of a robotic system. Vision-based control schemes find numerous applications in industrial manipulation tasks, medical robotics and, more recently, drone navigation. The goal of a vision-based control law is to make a positioning task succeed by minimizing the difference $\mathbf{e} = \mathbf{s} - \mathbf{s^*}$ between a set of desired visual features $\mathbf{s^*} \in\mathbb{R}^N$ and a set of current features $\mathbf{s}\in\mathbb{R}^N$. This can be achieved by continuously adjusting the position of a robotic system according to the Levenberg-Marquardt method \begin{equation*} \mathbf{v} = -\lambda \left(({\mathbf{L}_{\mathbf{s}}}^\intercal\mathbf{L}_{\mathbf{s}} + \mu \operatorname{diag}({\mathbf{L}_{\mathbf{s}}}^\intercal\mathbf{L}_{\mathbf{s}})\right)^{-1}{\mathbf{L}_{\mathbf{s}}}^\intercal \mathbf{e}, \end{equation*} where the movement of a robot with $k$ degrees of freed is expressed by the velocity vector $\mathbf{v} \in \mathbb{R}^k$ and $\mathbf{L}_{\mathbf{s}}\in\mathbb{R}^{N\times k}$ denotes a so-called interaction matrix that linearizes the time-variation of the observed features $\mathbf{s}$ in terms of $\mathbf{v}$.

We show how the matrix $\mathbf{L}_{\mathbf{s}}$ can be explicitly computed at a given point in time for shearlet- and wavelet-based feature vectors $\mathbf{s}$ and present experimental evidence regarding the reliability of wavelet- and shearlet-based visual servoing schemes in nominal and unfavorable conditions (partial occlusions, unstable illumination).

This is joint work with Lesley-Ann Duflot and Alexandra Krupa from INRIA Rennes-Bretagne Atlantique and Brahim Tamadazte and Nicolas Andreff from FEMTO-ST Institute Besan\c{c}on.

A visual servoing scheme consists of a closed-loop control approach in which visual information feedback is used to control the motion of a robotic system. Vision-based control schemes find numerous applications in industrial manipulation tasks, medical robotics and, more recently, drone navigation. The goal of a vision-based control law is to make a positioning task succeed by minimizing the difference $\mathbf{e} = \mathbf{s} - \mathbf{s^*}$ between a set of desired visual features $\mathbf{s^*} \in\mathbb{R}^N$ and a set of current features $\mathbf{s}\in\mathbb{R}^N$. This can be achieved by continuously adjusting the position of a robotic system according to the Levenberg-Marquardt method \begin{equation*} \mathbf{v} = -\lambda \left(({\mathbf{L}_{\mathbf{s}}}^\intercal\mathbf{L}_{\mathbf{s}} + \mu \operatorname{diag}({\mathbf{L}_{\mathbf{s}}}^\intercal\mathbf{L}_{\mathbf{s}})\right)^{-1}{\mathbf{L}_{\mathbf{s}}}^\intercal \mathbf{e}, \end{equation*} where the movement of a robot with $k$ degrees of freed is expressed by the velocity vector $\mathbf{v} \in \mathbb{R}^k$ and $\mathbf{L}_{\mathbf{s}}\in\mathbb{R}^{N\times k}$ denotes a so-called interaction matrix that linearizes the time-variation of the observed features $\mathbf{s}$ in terms of $\mathbf{v}$.

We show how the matrix $\mathbf{L}_{\mathbf{s}}$ can be explicitly computed at a given point in time for shearlet- and wavelet-based feature vectors $\mathbf{s}$ and present experimental evidence regarding the reliability of wavelet- and shearlet-based visual servoing schemes in nominal and unfavorable conditions (partial occlusions, unstable illumination).

This is joint work with Lesley-Ann Duflot and Alexandra Krupa from INRIA Rennes-Bretagne Atlantique and Brahim Tamadazte and Nicolas Andreff from FEMTO-ST Institute Besan\c{c}on.

11:05 – 11:30

A mixed stochastic - numeric algorithm for transported interacting particles

A coupled system of population balance and convection-diffusion equations is solved numerically, employing stochastic and finite element techniques in combination. While the evolution of the particle population is modeled as a Markov jump process and solved with a stochastic simulation algorithm, transport of temperature and species concentration are subject to a finite element approximation. We want to briefly introduce both the stochastic and the deterministic approach and discuss some difficulties to overcome when combining them. A proof of concept simulation of a flow crystallizer in 2D is presented.

11:30 – 11:55

How Do Electrons Move in Space? Flux Discretizations for Non-Boltzmann Statistics

When modelling semiconductor devices via the van Roosbroeck system one often uses statistical functions to describe the correspondence between carrier densities and chemical potentials. For 3D bulk semiconductors the most general choice is given by the Fermi-Dirac integral of order 1/2. However, how to numerically solve the van Roosbrock in this general (non-Boltzmann) case is still an open problem. We will present and compare several flux discretization schemes which generalize the well-known Scharfetter-Gummel scheme. Our main goal is to discretely preserve important properties from the continuous system such as existence and uniqueness of the solution, consistency with the thermodynamical equilibrium as well as unconditional stability. We also show how these new numerical schemes can be efficiently implemented for 2D and 3D applications.

11:55 – 13:10

Lunch Break

13:10 – 13:35

Optimal control of multiphase flows

We consider the optimal control of a two-phase fluid that is described by the thermodynamically consistent diffuse interface model proposed in 2012 by Abels/Garcke/Grün. As key ingredient we present an energy stable simulation scheme proposed by the authors in 2016. It allows us to simulate two-phase fluids in an energy stable way and provides enough regularity to apply classic theory from optimal control. We prove existence of solutions to a semi-discrete in time optimal control problem, and present a convergence analysis for its finite element discretization. We illustrate the performance of our approach with some numerical examples.

Michael Hinze (joint with Harald Garcke (UniRE) and Christian Kahle (TUM)

Michael Hinze (joint with Harald Garcke (UniRE) and Christian Kahle (TUM)

13:35 – 14:00

Optimal Transport–Based Restoration and Reconstruction of Q-Ball Data

In medical applications, the diffusivity of water in tissues that exhibit fibrous microstructures, such as muscle fibres or axons in cerebral white matter, contains valuable information about the fiber architecture in the living organism. Diffusion-weighted (DW) magnetic resonance imaging (MRI) is well-established as a way of measuring the main diffusion directions. A widely used reconstruction scheme for DW-MRI data is Q-ball imaging where the quantity of interest is the marginal probability of diffusion in a given direction, the orientation distribution function (ODF).

In joint work with J. Lellmann (Vogt, Lellmann: An Optimal Transport-Based Restoration Method for Q-Ball Imaging, accepted for SSVM 2017), we propose a variational approach for edge-preserving total variation (TV)-based regularization of Q-ball data. While total variation is among the most popular regularizers for variational problems, its application to ODFs is not straightforward. We propose to write the difference quotients in the TV seminorm in terms of the Wasserstein statistical distance from optimal transport and combine this regularizer with a matching Wasserstein data fidelity term. Using the Kantorovich-Rubinstein duality, the variational model can be formulated as a convex optimization problem that can be solved using a primal-dual algorithm.

Current work is focussing on how to apply this to the reconstruction of Q-ball data from so called high angular resolution diffusion imaging (HARDI). Furthermore, the mathematically precise formulation of the function spaces for the continuous model as well as the existence of minimizers of the variational formulation are interesting theoretical questions for future work.

In joint work with J. Lellmann (Vogt, Lellmann: An Optimal Transport-Based Restoration Method for Q-Ball Imaging, accepted for SSVM 2017), we propose a variational approach for edge-preserving total variation (TV)-based regularization of Q-ball data. While total variation is among the most popular regularizers for variational problems, its application to ODFs is not straightforward. We propose to write the difference quotients in the TV seminorm in terms of the Wasserstein statistical distance from optimal transport and combine this regularizer with a matching Wasserstein data fidelity term. Using the Kantorovich-Rubinstein duality, the variational model can be formulated as a convex optimization problem that can be solved using a primal-dual algorithm.

Current work is focussing on how to apply this to the reconstruction of Q-ball data from so called high angular resolution diffusion imaging (HARDI). Furthermore, the mathematically precise formulation of the function spaces for the continuous model as well as the existence of minimizers of the variational formulation are interesting theoretical questions for future work.

14:00 – 14:25

Fishing Strategies as a Nonconvex Optimal Control Problem

The action of a fishermen fleet onto the biomass of fish can be described by a parabolic diffusion-reaction equation with a bilinear control. An optimal fishing strategy leads to a non-convex optimal control problem with nonlinear state equation. In this work, we concentrate on the solvability of the state equation, existence of global minima and the derivation of the first and second order optimality system. Numerical solutions of first numerical test problems show typical features as so-called No-Take-Zones (NTZ) and maximal fishing quota (total allowable catches, TACs) as parts of an optimal fishing strategy.

14:25 – 14:50

Coffee Break

14:50 – 15:15

Prony's method and applications

Prony's method is a prototypical eigenvalue analysis based method for the reconstruction of a finitely supported complex measure on the unit circle from its moments up to a certain degree. In this talk, we give a generalization of this method to the multivariate case and show that there is sharp transition of identifiability with respect to the ratio of the separation distance of the parameter and the order of moments. We present some algorithmic variants based on matrix pencil methods and conclude with some applications in super-resolution imaging, the flat extension of Toeplitz matrices, and the canonical tensor decomposition.

15:15 – 15:40

The linearized KdV-equation on star graphs

We study the linearized Korteweg-de-Vries equation on metric star graphs and identify coupling conditions, under which the corresponding evolution equation yields a reasonable dynamics.

This is joint work with Delio Mugnolo (Hagen) and Diego Noja (Milano).

This is joint work with Delio Mugnolo (Hagen) and Diego Noja (Milano).

15:40 – 15:50

Closing