Talks & Conferences

Selected talks

• Manifolds in Julia – Manifolds.jl and ManifoldsBase.jl JuliaCon 2020 July 27 — 31, 2020 Everywhere (online)

  In many scientific and engineering scenarios, measured data or model variables lie in smooth, nonlinear spaces like the sphere or the group of 3D rotations. Such spaces often have a manifold structure, that is, they locally can be approximately linearized (i.e. they are locally diffeomorphic to a Hilbert space). Equipping such manifolds with a (pseudo-)Riemannian metric tensor, which defines a ruler on the manifold, enables local computation of distances and angles. These features enable construction of efficient algorithms that respect the nonlinear structure, for example for optimization, interpolation, and statistical modeling.

  In this talk, we demonstrate various tools that Manifolds.jl provides for working with manifolds. With these tools, you can for example explicitly locally linearize points using inverse retractions and orthonormal bases. You can also compute intrinsic statistics, such as the Riemannian mean and variance, of data on manifolds. We also extend Distributions.jl to support distributions on arbitrary manifolds, which can be used to generate random points on a manifold or for statistical modeling.

  Manifolds.jl implements a library of manifolds and provides combinators to construct new manifolds from these. Examples are the product manifold of two manifolds, the power manifold, the tangent bundle, and Lie groups. Using a trait-based system, any manifold may be augmented with additional geometric structure, including various metrics, without sacrificing efficiency.

  We also present the light-weight interface package ManifoldsBase.jl, which enables users to easily extend Manifolds.jl with new algorithms and additional manifolds thanks to multiple dispatch. The interface can also be used to develop new, stand-alone projects. It is minimally restrictive and makes it possible to write algorithms that are independent from any concrete implementations of the manifolds. One package that uses this interface is Manopt.jl, which provides optimization algorithms on Riemannian manifolds, with a focus on high-dimensional and non-smooth optimization.

• Nonsmooth, nonconvex Optimization on Riemannian Manifolds Optimization in Oslo Simula Research Laboratory October 18, 2023 Oslo, Norway

  When considering data in nonlinear spaces, like on the sphere, for symmetric positive definite matrices, we arrive naturally at optimization tasks that are defined on a Riemannian manifold. Another approach is to rephrase a constrained optimization problem in Euclidean space as an unconstrained problem in the manifold defined by the constraints. This might even lead to the problem becoming convex, since the kind of convexity considered changes to geodesic convexity on the manifold. In this talk we consider recent advances in optimization on Riemannian manifolds, especially for nonsmooth and/or nonconvex objective functions. We discuss both theoretical aspects as well as numerical algorithms and their implementation in Julia within the two packages Manifolds.jl and Manopt.jl.

• Fenchel Duality Theory and a Primal-Dual Algorithm on Riemannian Manifolds International Conference on Machine Learning July 28 — 29, 2023 Honolulu, Hawaii, USA Invited talk Within the workshop “Duality Principles for Modern Machine Learning”

  In this talk we introduces a new duality theory that generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g., the Fenchel–Young inequality and the characterization of the convex subdifferential using the analogue of the Fenchel–Moreau Theorem. These properties of the Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm, and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions.

  Numerical results in Julia illustrate the performance of the algorithm, which competes with the recently derived Douglas–Rachford algorithm on manifolds of non-positive curvature. Furthermore we show numerically that our novel algorithm even converges on manifolds of positive curvature.

• A Variational Model for Data Fitting on Manifolds by Minimizing the Acceleration of a Bézier Curve Institutsseminar Institut für Numerische Mathematik Technische Universität Dresden November 27, 2018 Dresden, Germany

  Fitting a smooth curve to data points $d_0,\dots,d_n$ lying on a Riemannian manifold $\mathcal M$ and associated with real-valued parameters $t_0,\dots,t_n$ is a common problem in applications like wind field approximation, rigid body motion interpolation, or sphere-valued data analysis. The resulting curve should strike a balance between data proximity and a smoothing regularization constraint.

  In this talk we present a variational model to fit a composite Bézier curve to the set of data points $d_0,\dots,d_n$ on a Riemannian manifold $\mathcal M$. The resulting curve is obtained in such a way that its mean squared acceleration is minimal in addition to remaining close the data points. We approximate the acceleration by discretizing the squared second order derivative along the curve. We derive a closed-form, numerically stable and efficient algorithm to compute the gradient of a Bézier curve on manifolds with respect to its control points. This gradient can be expressed as a concatenation of so called adjoint Jacobi fields. Several examples illustrate the capabilities of this approach both for interpolation and approximation.

Talks

2024

• Nonlinear Fenchel Conjugates Mathicse Seminar Institute of Mathematics EPFL November 29, 2024 Lausanne, Switzerland

  Fenchel conjugates play an important role in nonsmooth optimization. Especially in large scale, nonsmooth problems, like image processing, splitting methods enable fast and efficient optimisation algorithms based on Fenchel duality. Recently, a few different variants of Fenchel conjugates have been introduced on Manifolds. The main challenge is, that the original Fenchel conjugate is defined as a supremum over linear functions, which do not exist on manifolds.

  In this talk we present a generalisation, the so-called nonlinear Fenchel conjugates. We discuss, which properties can still be derived, for example how a biconjugate can be introduced and how the relation between convolution and the Fenchel conjugate are generalized to Lie groups. As an application we present the Primal-Dual-Hybrid-Gradient or Chambolle-Pock algorithm and illustrate its performance in an image denoising task for manifold-valued images.

• Manopt.jl: Optimization on Riemannian Manifolds in Julia Julia for numerical problems in quantum and solid-state physics November 26 — 28, 2024 Lausanne, Switzerland

  In many applications nonlinear data is measured, for example when considering unit vectors, rotations, or (bases of) subspaces of a vector space. Modelling this on a Riemannian manifold allows to both reduce the dimension of the data stored as well as focusing on geometric properties of the measurement space compared to constraining a total space the data is represented in. In optimisation this yields unconstrained optimization algorithms, where we have to take the geometry of the optimization domain into consideration.

  To make algorithms in general more accessible, we present the two Julia packages Manopt.jl and Manifolds.jl. Manopt.jl is a package for optimization on Riemannian manifolds, providing a unified interface to implement solvers as well as a large library of solvers. Since it is build on ManifoldsBase.jl and hence can be used to solve problems in any manifold implemented in Manifolds.jl.

• The Riemannian Difference of Convex Algorithm in Manopt.jl Seminarvortrag Institut für Mathematik Universität Bayreuth July 17, 2024 Bayreuth, Germany

  In many applications nonlinear data is measured, for example when considering unit vectors, rotations, or (bases of) subspaces of a vector space. Modelling this on a Riemannian manifold allows to both reduce the dimension of the data stored as well as focusing on geometric properties of the measurement space compared to constraining a total space the data is represented in. In optimisation this yields unconstrained optimization algorithms, where we have to take the geometry of the optimization domain into consideration.

  In this talk we consider the task of minimizing the difference of two convex functions defined on a manifold and present the Difference of Convex Algorithm. To make algorithms in general more accessible, we then present the two Julia packages Manifolds.jl and Manopt.jl, that allow to define and use Riemannian manifolds and optimization algorithms employing numerical differential geometry, respectively.

• Quarto Extensions for the Julia Community JuliaCon 2024 July 9 — 13, 2024 Eindhoven, The Netherlands Talk together with P. Altmeyer
• Nonsmooth Optimization on Riemannian Manifolds in Manopt.jl 33rd European Conference on Operational Research (Euro 24) June 30 — July 3, 2024 Copenhagen, Denmark

  In many applications nonlinear data is measured, for example when considering rotations, bases of sub spaces or even sub spaces themselves. While this can be modelled as linear data with constraints, this induce more complicated algorithms. If the constraints on the other hand form a Riemannian manifold, for example the Special Orthogonal group, the Stiefel or the Grassmann manifold, one can employ classical unconstrained algorithms, while staying inherently feasible. In the recent years these algorithms were extended to non-smooth objectives as well as constrained optimization on manifolds.

  In this talk, we present a framework and several algorithms that are available in Manopt.jl, which together with Manifolds.jl forms a family of Julia packages to efficiently solve optimization problems on Manifolds in Julia.

• Nonsmooth Optimization on Riemannian manifolds Kolloquium über Angewandte Mathematik June 11, 2024 Göttingen, Germany

  In many applications nonlinear data is measured, for example when considering unit vectors, rotations, or (bases of) subspaces of a vector space. Modelling this on a Riemannian manifold allows to both reduce the dimension of the data stored as well as focusing on geometric properties of the measurement space compared to constraining a total space the data is represented in. In optimisation this yields unconstrained optimization algorithms, where we have to take the geometry of the optimization domain into consideration.

  In this talk we present recent developments in nonsmooth optimization on manifolds. We generalize the Convex Bundle Method to Riemannian manifolds and investigate its convergence properties. We further consider the task of minimizing the difference of two convex functions defined on a manifold and discuss the Difference of Convex Algorithm.

• The Riemannian Difference of Convex Algorithm in Manopt.jl Optimization Techniques in Quantum Chemistry June 5 — 7, 2024 Aachen, Germany

  In many applications nonlinear data is measured, for example when considering unit vectors, rotations, or (bases of) subspaces of a vector space. Modelling this on a Riemannian manifold allows to both reduce the dimension of the data stored as well as focusing on geometric properties of the measurement space compared to constraining a total space the data is represented in. In optimisation this yields unconstrained optimization algorithms, where we have to take the geometry of the optimization domain into consideration.

  In this talk we consider the task of minimizing the difference of two convex functions defined on a manifold and present the Difference of Convex Algorithm. To make algorithms in general more accessible, we then present the two Julia packages Manifolds.jl and Manopt.jl, that allow to define and use Riemannian manifolds and optimization algorithms employing numerical differential geometry, respectively.

2023

• Optimization on Riemannian Manifolds in Julia Robotics and Automation group research seminar NTNU November 16, 2023 Trondheim, Norway
• Optimierung auf Mannigfaltigkeiten und Open-Source Software Mathematics and Industry November 3, 2023 Berlin, Germany
• Nonsmooth, nonconvex Optimization on Riemannian Manifolds Optimization in Oslo Simula Research Laboratory October 18, 2023 Oslo, Norway

  When considering data in nonlinear spaces, like on the sphere, for symmetric positive definite matrices, we arrive naturally at optimization tasks that are defined on a Riemannian manifold. Another approach is to rephrase a constrained optimization problem in Euclidean space as an unconstrained problem in the manifold defined by the constraints. This might even lead to the problem becoming convex, since the kind of convexity considered changes to geodesic convexity on the manifold. In this talk we consider recent advances in optimization on Riemannian manifolds, especially for nonsmooth and/or nonconvex objective functions. We discuss both theoretical aspects as well as numerical algorithms and their implementation in Julia within the two packages Manifolds.jl and Manopt.jl.

• The Difference of Convex Algorithm on Riemannian Manifolds European Conference on Computational Optimization September 25 — 27, 2023 Heidelberg, Germany
• Nonsmooth, nonconvex Optimization on Riemannian Manifolds From Modeling and Analysis to Approximation and Fast Algorithms September 2 — 6, 2023 Hasenwinkel, Germany
• The Difference of Convex Algorithm on Riemannian Manifolds 10th International Congress on Industrial and Applied Mathematics August 20 — 25, 2023 Tokio, Japan & virtual

  In this talk we propose a difference of convex algorithm (DCA) on Riemannian manifolds to solve optimisation problems involving a difference of two functions.

  We establish both its relation to recently introduced Fenchel duality on manifolds, and its well-posedness. On Hadamard manifolds, we prove that every cluster point of the sequence generated by the algorithm is a cluster point.

  Finally, we illustrate that several optimisation problems can be written as difference of convex (DC) functions on manifolds, and that some Euclidean problems that are differences of non-convex problems become DC problems when rephrased on a manifold. Numerical examples illustrate that such a rephrasing even for DC problems is beneficial numerically.

• Fenchel Duality Theory and a Primal-Dual Algorithm on Riemannian Manifolds International Conference on Machine Learning July 28 — 29, 2023 Honolulu, Hawaii, USA Invited talk Within the workshop “Duality Principles for Modern Machine Learning”

  In this talk we introduces a new duality theory that generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g., the Fenchel–Young inequality and the characterization of the convex subdifferential using the analogue of the Fenchel–Moreau Theorem. These properties of the Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm, and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions.

  Numerical results in Julia illustrate the performance of the algorithm, which competes with the recently derived Douglas–Rachford algorithm on manifolds of non-positive curvature. Furthermore we show numerically that our novel algorithm even converges on manifolds of positive curvature.

• Nonsmooth Optimization on Riemannian Manifolds in Manopt.jl Analysis and PDE Seminar Department of Mathematics University of Bergen May 9, 2023 Bergen, Norway
• An Introduction to Optimization on Riemannian Manifolds Constrained Optimization Methods for Models in Data Fusion March 27, 2023 Oslo, Norway
• The Difference of Convex Algorithm on Riemannian Manifolds Manifold and Geometric Integration Colloquia 2023 February 27 — March 3, 2023 Ilsetra, Norway

2022

• Nonsmooth Optimization on Riemannian Manifolds in Manopt.jl Optimization Research Seminar Scoop Heidelberg University December 8, 2022 Heidelberg, Germany
• Fenchel Duality Theory and a Primal-Dual Algorithm on Riemannian Manifolds Workshop in Geometry and Geometric integration September 7 — 9, 2022 Bergen, Norway

  In this talk we introduces a new duality theory that generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g., the Fenchel–Young inequality and the characterization of the convex subdifferential using the analogue of the Fenchel–Moreau Theorem. These properties of the Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm, and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions.

  Numerical results in Julia illustrate the performance of the algorithm, which competes with the recently derived Douglas–Rachford algorithm on manifolds of nonpositive curvature. Furthermore we show numerically that our novel algorithm even converges on manifolds of positive curvature.

• Manifolds.jl Workshop in Geometry and Geometric integration September 7 — 9, 2022 Bergen, Norway
• The RIemannian Chambolle-Pock algorithm Jahrestagung der Gesellschaft für angewandte Mathematik und Mechanik August 15 — 19, 2022 Aachen, Germany

  Based on a Fenchel dual notion on Riemannian manifolds we investigate the saddle point problem related to a nonsmooth convex optimization problem. We derive a primal-dual hybrid gradient algorithm, also known as Chambolle-Pock algorithm in imaging, to compute the saddle point using either an exact or a linearized approach for the involved nonlinear operator. We investigate a sufficient condition for convergence of the linearized algorithm on Hadamard manifolds. Numerical examples illustrate, that on Hadamard manifolds we are on par with state of the art algorithms and on general manifolds we outperform existing approaches.

• Manopt.jl – Optimization on Riemannian Manifolds JuliaCon 2022 July 27 — 29, 2022 online & everywhere

  In many applications and optimization tasks, non-linear data appears naturally. For example, when data on the sphere is measured, diffusion data can be captured as a signal or even multivariate data of symmetric positive definite matrices and orientations like they appear for electron backscattered diffraction (EBSD) data. Another example are fixed rank matrices, appearing in matrix completion. Working on these data, for example doing data interpolation and approximation, denoising, inpainting, or performing matrix completion, can usually be phrased as an optimization problem

  Manopt.jl (manoptjl.org) provides a set of optimization algorithms for optimization problems given on a Riemannian manifold. Build upon on a generic optimization framework, together with the interface ManifoldsBase.jl for Riemannian manifolds, classical and recently developed methods are provided in an efficient implementation. Algorithms include the derivative-free Particle Swarm and Nelder–Mead algorithms, as well as classical gradient, conjugate gradient and stochastic gradient descent. Furthermore, quasi-Newton methods like a Riemannian L-BFGS and nonsmooth optimization algorithms like a Cyclic Proximal Point Algorithm, a (parallel) Douglas-Rachford algorithm and a Chambolle-Pock algorithm are provided, together with several basic cost functions, gradients and proximal maps as well as debug and record capabilities.

• Manopt.jl – Optimization on Riemannian Manifolds JuliaCon 2022 July 27 — 29, 2022 online & everywhere This is an extended version of the Lightning Talk given at the JuliaCon 2022.

  In many applications and optimization tasks, non-linear data appears naturally. For example, when data on the sphere is measured, diffusion data can be captured as a signal or even multivariate data of symmetric positive definite matrices and orientations like they appear for electron backscattered diffraction (EBSD) data. Another example are fixed rank matrices, appearing in matrix completion. Working on these data, for example doing data interpolation and approximation, denoising, inpainting, or performing matrix completion, can usually be phrased as an optimization problem

  Manopt.jl (manoptjl.org) provides a set of optimization algorithms for optimization problems given on a Riemannian manifold. Build upon on a generic optimization framework, together with the interface ManifoldsBase.jl for Riemannian manifolds, classical and recently developed methods are provided in an efficient implementation. Algorithms include the derivative-free Particle Swarm and Nelder–Mead algorithms, as well as classical gradient, conjugate gradient and stochastic gradient descent. Furthermore, quasi-Newton methods like a Riemannian L-BFGS and nonsmooth optimization algorithms like a Cyclic Proximal Point Algorithm, a (parallel) Douglas-Rachford algorithm and a Chambolle-Pock algorithm are provided, together with several basic cost functions, gradients and proximal maps as well as debug and record capabilities.

• Manopt.jl – Numerical Optimization on Riemannian Manifolds International Conference on Continuous Optimization 2022 July 25 — 28, 2022 Bethlehem, PA, USA
• Modelling Fire Incidents Response Times in Ålesund European Study Group with Industry (ESGI) 156 June 13 — 17, 2022 Ålesund, Norway
• Fenchel Duality Theory and a Primal-Dual Algorithm on Riemannian Manifolds Seminar IMADA Syddansk Universitet June 2, 2022 Odense, Denmark

  In this talk we introduces a new duality theory that generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g., the Fenchel–Young inequality and the characterization of the convex subdifferential using the analogue of the Fenchel–Moreau Theorem. These properties of the Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm, and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions.

  Numerical results in Julia illustrate the performance of the algorithm, which competes with the recently derived Douglas–Rachford algorithm on manifolds of nonpositive curvature. Furthermore we show numerically that our novel algorithm even converges on manifolds of positive curvature.

• Splitting Methods for Non-smooth Optimization on Manifolds The DNA Seminar Institutt for matematiske fag Norges teknisk-naturvitenskapelige universitet April 4, 2022 Trondheim, Norway
• The Riemannian Chambolle–Pock Algorithm Manifold and Geometric Integration Colloquia 2022 February 28 — March 4, 2022 Ilsetra, Norway

2021

• Reproduzierbarkeit von numerischen Experimenten Seminar der GMDS-Nachwuchsförderung November 12, 2021 online
• A Primal-Dual Algorithm for Convex Nonsmooth Optimization on Riemannian Manifolds Informs Annual Meeting 2021 October 24 — 27, 2021 Anaheim, CA, USA – and online Talk within the Session “Optimization on Manifolds”.

  In this talk we introduce a new duality theory that generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. The Fenchel conjugate is employed to derive a Riemannian primal-dual optimization algorithm. This generalizes the primal-dual algorithm by Chambolle and Pock to Riemannian manifolds. We prove that this algorithm converges to a critical point for the case of Hadamard manifolds under appropriate assumptions. Numerical examples illustrate the performance of the algorithm, which competes with the recently derived Douglas-Rachford algorithm on manifolds of non-positive curvature. Furthermore we show numerically that our novel algorithm even converges on manifolds of positive curvature.

• Fenchel Duality Theory on Riemannian Manifolds IFIP TC7 Conference on System Modelling and Optimization August 30 — September 3, 2021 online (originally Quito, Ecuador) Talk within the Minisymposium “Manifolds and Optimization”.

  In this talk we introduces a new duality theory that generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g., the Fenchel–Young inequality and the characterization of the convex sub differential using the analogue of the Fenchel–Moreau Theorem.

• A Primal-Dual Algorithm for Convex Nonsmooth Optimization on Riemannian Manifolds IFIP TC7 Conference on System Modelling and Optimization August 30 — September 3, 2021 online (originally Quito, Ecuador) Talk within the Minisymposium “Non-Smooth First-order Methods, Convex, and Non-convex”.

  Based on a Fenchel dual notion on Riemannian manifolds we investigate the saddle point problem related to a nonsmooth convex optimization problem. We derive a primal-dual hybrid gradient algorithm to compute the saddle point using either an exact or a linearized approach for the involved nonlinear operator. We investigate a sufficient condition for convergence of the linearized algorithm on Hadamard manifolds. Numerical examples illustrate, that on Hadamard manifolds we are on par with state of the art algorithms and on general manifolds we outperform existing approaches.

2020

• The Riemannian Chambolle–Pock Algorithm and Optimization on Manifolds in Julia Mini-Workshop: Computational Optimization on Manifolds November 15 — 21, 2020 Oberwolfach - everywhere (online)
• Manifolds in Julia – Manifolds.jl and ManifoldsBase.jl JuliaCon 2020 July 27 — 31, 2020 Everywhere (online)

  In many scientific and engineering scenarios, measured data or model variables lie in smooth, nonlinear spaces like the sphere or the group of 3D rotations. Such spaces often have a manifold structure, that is, they locally can be approximately linearized (i.e. they are locally diffeomorphic to a Hilbert space). Equipping such manifolds with a (pseudo-)Riemannian metric tensor, which defines a ruler on the manifold, enables local computation of distances and angles. These features enable construction of efficient algorithms that respect the nonlinear structure, for example for optimization, interpolation, and statistical modeling.

  In this talk, we demonstrate various tools that Manifolds.jl provides for working with manifolds. With these tools, you can for example explicitly locally linearize points using inverse retractions and orthonormal bases. You can also compute intrinsic statistics, such as the Riemannian mean and variance, of data on manifolds. We also extend Distributions.jl to support distributions on arbitrary manifolds, which can be used to generate random points on a manifold or for statistical modeling.

  Manifolds.jl implements a library of manifolds and provides combinators to construct new manifolds from these. Examples are the product manifold of two manifolds, the power manifold, the tangent bundle, and Lie groups. Using a trait-based system, any manifold may be augmented with additional geometric structure, including various metrics, without sacrificing efficiency.

  We also present the light-weight interface package ManifoldsBase.jl, which enables users to easily extend Manifolds.jl with new algorithms and additional manifolds thanks to multiple dispatch. The interface can also be used to develop new, stand-alone projects. It is minimally restrictive and makes it possible to write algorithms that are independent from any concrete implementations of the manifolds. One package that uses this interface is Manopt.jl, which provides optimization algorithms on Riemannian manifolds, with a focus on high-dimensional and non-smooth optimization.

• Fenchel Duality Theory and a Primal-Dual Algorithm on Riemannian Manifolds Optimization and Numerical Analysis Seminar School of Mathematics University of Birmingham February 26, 2020 Birmingham, United Kingdom

2019

• Fenchel Duality Theory and a Primal-Dual Algorithm on Riemannian Manifolds Oberseminar Angewandte Algebra und Analysis Institut für Analysis und Algebra Technische Universität Braunschweig December 10, 2019 Braunschweig, Germany
• A Primal-Dual Algorithm for Convex Nonsmooth Optimization on Riemannian Manifolds European Numerical Mathematics and Advanced Applications Conference 2019 September 30 — October 4, 2019 Egmond aan Zee, The Netherlands

  Based on a Fenchel dual notion on Riemannian manifolds we investigate the saddle point problem related to a nonsmooth convex optimization problem. We derive a primal-dual hybrid gradient algorithm to compute the saddle point using either an exact or a linearized approach for the involved nonlinear operator. We investigate a sufficient condition for convergence of the linearized algorithm on Hadamard manifolds. Numerical examples illustrate, that on Hadamard manifolds we are on par with state of the art algorithms and on general manifolds we outperform existing approaches.

• Fenchel Duality Theory and a Primal-Dual Algorithm on Riemannian Manifolds Workshop: Modeling, Interpolation, and Approximation for Waves and Signals August 20 — 22, 2019 Lübeck Invited Plenary Talk
• A Primal-Dual Algorithm for Convex Nonsmooth Optimization on Riemannian Manifolds Workshop on Optimization on Manifolds August 12, 2019 Chemnitz

  Based on a Fenchel dual notion on Riemannian manifolds we investigate the saddle point problem related to a nonsmooth convex optimization problem. We derive a primal-dual hybrid gradient algorithm to compute the saddle point using either an exact or a linearized approach for the involved nonlinear operator. We investigate a sufficient condition for convergence of the linearized algorithm on Hadamard manifolds. Numerical examples illustrate, that on Hadamard manifolds we are on par with state of the art algorithms and on general manifolds we outperform existing approaches.

• A Primal-Dual Algorithm for Convex Nonsmooth Optimization on Riemannian Manifolds International Conference on Continuous Optimization August 5 — 8, 2019 Berlin

  Based on a Fenchel dual notion on Riemannian manifolds we investigate the saddle point problem related to a nonsmooth convex optimization problem. We derive a primal-dual hybrid gradient algorithm to compute the saddle point using either an exact or a linearized approach for the involved nonlinear operator. We investigate a sufficient condition for convergence of the linearized algorithm on Hadamard manifolds. Numerical examples illustrate, that on Hadamard manifolds we are on par with state of the art algorithms and on general manifolds we outperform existing approaches.

• A Variational Model for Data Fitting on Manifolds by Minimizing the Acceleration of a Bézier Curve 9th International Congress on Industrial and Applied Mathematics July 15 — 19, 2019 Valencia, Spain

  We present a variational model to fit a composite Bézier curve to data points on a Riemannian manifold. The resulting curve has a minimal mean squared acceleration while also remaining close the data. We discretize the acceleration of the curve and derive an efficient algorithm to compute the gradient w.r.t. its control points, expressed as a concatenation of adjoint Jacobi fields. Several examples illustrate the capabilities of this approach both for interpolation and approximation.

• A Variational Model for Data Fitting on Manifolds by Minimizing the Acceleration of a Bézier Curve 30^th European Conference on Operational Research June 23 — 26, 2019 Dublin, Ireland

  Fitting a smooth curve to a set of n data points lying on a Riemannian manifold and associated with real-valued time points is a common problem in applications like wind field approximation, rigid body motion interpolation, or sphere-valued data analysis. The resulting curve should strike a balance between data proximity and a smoothing regularization constraint.

  In this talk we present a variational model to fit a composite Bézier curve to the set of data points on a Riemannian manifold. The resulting curve is obtained in such a way that its mean squared acceleration is minimal in addition to remaining close the data points. We approximate the acceleration by discretizing the squared second order derivative along the curve. We derive a closed-form, numerically stable and efficient algorithm to compute the gradient of a Bézier curve on manifolds with respect to its control points. This gradient can be expressed as a concatenation of so called adjoint Jacobi fields. Several examples illustrate the capabilities of this approach both for interpolation and approximation.

• Optimization on Manifolds for Models using Second Order Differences Kolloquium Institut für Mathematik Universität zu Lübeck June 6, 2019 Lübeck, Germany

  In many real life scenarios, measured data appears as values on a Riemannian manifold. For example in interferometric Synthetic Aperture Radar (InSAR) data is given as a phase, in electron backscattered diffraction (EBSD) as data items being from a quotient of the orientation group SO(3), and in diffusion tensor magnetic resonance imaging (DT-MRI) the measured data are symmetric positive definite matrices. These data items are often measured on an equispaced grid like usual signals and image but they also suffer from the same measurement errors like presence of noise or incompleteness. Hence there is a need to perform data processing tasks like denoising, inpainting or interpolation on these manifold-valued data.

  In this talk we present variational models for these tasks involving discrete second order differences for manifold-valued data. To compute a minimizer of such a the model, we obtain a high-dimensional, possibly non-smooth, optimization problem defined on a Riemannian manifold. We present algorithms to efficiently solve these problems and illustrate their performance.

• Optimization on Manifolds for Models using Second Order Differences Kolloquium über Angewandte Mathematik Institut für Numerische und Angewandte Mathematik Georg-August-Universität Göttingen May 28, 2019 Göttingen, Germany

  In many real life scenarios, measured data appears as values on a Riemannian manifold. For example in interferometric Synthetic Aperture Radar (InSAR) data is given as a phase, in electron backscattered diffraction (EBSD) as data items being from a quotient of the orientation group SO(3), and in diffusion tensor magnetic resonance imaging (DT-MRI) the measured data are symmetric positive definite matrices. These data items are often measured on an equispaced grid like usual signals and image but they also suffer from the same measurement errors like presence of noise or incompleteness. Hence there is a need to perform data processing tasks like denoising, inpainting or interpolation on these manifold-valued data.

  In this talk we present variational models for these tasks involving discrete second order differences for manifold-valued data. To compute a minimizer of such a the model, we obtain a high-dimensional, possibly non-smooth, optimization problem defined on a Riemannian manifold. We present algorithms to efficiently solve these problems and illustrate their performance.

• Nonsmooth Optimization on Riemannian Manifolds and Manifold-Valued Data Processing Geometry and Learning from Data in 3D and Beyond I: Geometric Processing April 1 — 5, 2019 IPAM, UCLA, Los Angeles, USA

  In many real life scenarios measured data appears as values from a Riemannian manifold. For example in interferometric Synthetic Aperture Radar (InSAR) data is given as a phase, in electron backscattered diffraction (EBSD) as data items being from a quotient of the orientation group SO(3), and in diffusion tensor magnetic resonance imaging (DT-MRI) the measured data are symmetric positive definite matrices. These data items are often measured on a equispaced grid like usual signals and image but they also suffer from the same measurement errors like presence of noise or incompleteness. Hence there is a need to perform data processing tasks like denoising, inpainting or even interpolation on these manifold-valued data.

  In this talk we present variational methods and nonsmooth optimization algorithms for processing manifold-valued data. For the first, we present first and second order difference based priors and their application to the tasks from image processing. To compute minimizers of the variational methods, we present algorithms to efficiently solve nonsmooth optimization tasks on manifolds.

• Intrinsic Formulation of KKT Conditions and Constraint Qualifications on Smooth Manifolds Jahrestagung der Gesellschaft für angewandte Mathematik und Mechanik February 18 — 22, 2019 Vienna, Austria

  We formulate Karush-Kuhn-Tucker (KKT) conditions for equality and inequality con- strained optimization problems on smooth manifolds. Under the Guignard constraint qualification, local minimizers are shown to admit Lagrange multipliers. We also investigate other constraint qualifications and provide results parallel to those in Euclidean space. Illustrating numerical examples will be presented.

2018

• A Variational Model for Data Fitting on Manifolds by Minimizing the Acceleration of a Bézier Curve Kolloquium Friedrich-Alexander-Universität Erlangen-Nürnberg December 11, 2018 Erlangen, Germany

  Fitting a smooth curve to data points $d_0,\dots,d_n$ lying on a Riemannian manifold $\mathcal M$ and associated with real-valued parameters $t_0,\dots,t_n$ is a common problem in applications like wind field approximation, rigid body motion interpolation, or sphere-valued data analysis. The resulting curve should strike a balance between data proximity and a smoothing regularization constraint.

  In this talk we present a variational model to fit a composite Bézier curve to the set of data points $d_0,\dots,d_n$ on a Riemannian manifold $\mathcal M$. The resulting curve is obtained in such a way that its mean squared acceleration is minimal in addition to remaining close the data points. We approximate the acceleration by discretizing the squared second order derivative along the curve. We derive a closed-form, numerically stable and efficient algorithm to compute the gradient of a Bézier curve on manifolds with respect to its control points. This gradient can be expressed as a concatenation of so called adjoint Jacobi fields. Several examples illustrate the capabilities of this approach both for interpolation and approximation.

• A Variational Model for Data Fitting on Manifolds by Minimizing the Acceleration of a Bézier Curve Institutsseminar Institut für Numerische Mathematik Technische Universität Dresden November 27, 2018 Dresden, Germany

  Fitting a smooth curve to data points $d_0,\dots,d_n$ lying on a Riemannian manifold $\mathcal M$ and associated with real-valued parameters $t_0,\dots,t_n$ is a common problem in applications like wind field approximation, rigid body motion interpolation, or sphere-valued data analysis. The resulting curve should strike a balance between data proximity and a smoothing regularization constraint.

  In this talk we present a variational model to fit a composite Bézier curve to the set of data points $d_0,\dots,d_n$ on a Riemannian manifold $\mathcal M$. The resulting curve is obtained in such a way that its mean squared acceleration is minimal in addition to remaining close the data points. We approximate the acceleration by discretizing the squared second order derivative along the curve. We derive a closed-form, numerically stable and efficient algorithm to compute the gradient of a Bézier curve on manifolds with respect to its control points. This gradient can be expressed as a concatenation of so called adjoint Jacobi fields. Several examples illustrate the capabilities of this approach both for interpolation and approximation.

• A Variational Model for Data Fitting on Manifolds by Minimizing the Acceleration of a Bézier Curve Mecklenburg Workshop — Approximation Methods and Fast Algorithms September 10 — 14, 2018 Hasenwinkel

  Fitting a smooth curve to data points $d_0,\dots,d_n$ lying on a Riemannian manifold $\mathcal M$ and associated with real-valued parameters $t_0,\dots,t_n$ is a common problem in applications like wind field approximation, rigid body motion interpolation, or sphere-valued data analysis. The resulting curve should strike a balance between data proximity and a smoothing regularization constraint.

  In this talk we present a variational model to fit a composite Bézier curve to the set of data points $d_0,\dots,d_n$ on a Riemannian manifold $\mathcal M$. The resulting curve is obtained in such a way that its mean squared acceleration is minimal in addition to remaining close the data points. We approximate the acceleration by discretizing the squared second order derivative along the curve. We derive a closed-form, numerically stable and efficient algorithm to compute the gradient of a Bézier curve on manifolds with respect to its control points. This gradient can be expressed as a concatenation of so called adjoint Jacobi fields. Several examples illustrate the capabilities of this approach both for interpolation and approximation.

• Variational Methods for Manifold-valued image Processing Seminar “Theory and Algorithms in Data Science” Alan Turing Institute September 3, 2018 Invited talk and a week of research stay Mihai Cucuringu, and Hemant Tyagi

  In many real life scenarios measured data appears as values from a Riemannian manifold. For example in interferometric Synthetic Aperture Radar (InSAR) data is given as a phase, in electron backscattered diffraction (EBSD) as data items being from a quotient of the orientation group SO(3), and in diffusion tensor magnetic resonance imaging (DT-MRI) the measured data are symmetric positive definite matrices. These data items are often measured on a pixel grid like usual images and they also suffer from the same measurement errors like noisy or incompleteness.Hence there is a need to perform image processing tasks like denoising, inpainting or segmentation on these manifold-valued images. Recently the ROF-Model for TV regularization has been transferred to the setting of manifold-valued images to perform such image processing tasks.

  In this talk we present methods and algorithms to compute the TV regularization for manifold-valued images efficiently using variational methods. We present generalizations to second order models, and to the case, where the data is not given on a pixel grid, for example when dealing with nonlocal methods.

• A parallel Douglas-Rachford Algorithm for Data on Hadamard Manifolds International Symposium on Mathematical Programming 2018 July 1 — 6, 2018 Bordeaux, France Invited talk within the Session “Riemannian Geometry in Optimization for Learning” organized by N. Boumal

  In many applications like DT-MRI or a data set of multivariate Gaussian distributions. Since these measurements suffer from errors or even loss of data, we aim to transfer image processing methods to manifold-valued data.

  In this talk we present a parallel Douglas-Rachford algorithm for manifold-valued data and proof its convergence to a minimizer for Hadamard manifolds of constant curvature. We illustrate the algorithm using the total variation regularization or ROF model and demonstrate the performance within the Manifold-Valued Image Restoration Toolbox (MVIRT).

• Nonlocal Inpainting of Manifold-valued Data on Finite Weighted Graphs SIAM Conference on Imaging Science 2018 June 5 — 8, 2018 Bologna, Italy Invited talk within the Session Minisymposium #31 “Variational Approaches for Regularizing Nonlinear Geometric Data”

  When dealing with manifold-valued data one faces the same challenging processing tasks as, e.g., in classical imaging. In this talk we consider image inpainting for manifold-valued data in which missing information have to be filled in suitably. We present a generalization of the graph infinity-Laplacian to manifold-valued data based on the min-max characterization of the local discrete Lipschitz constant. We derive a numerical scheme to solve the obtained manifold-valued infinity-Laplace equation and inpaint missing data.

• The Graph-Infinity Laplacian for Manifold-valued Data 89^thAnnual Meeting of the International Association of Applied Mathematics and Mechanics (Gesellschaft für angewandte Mathematik und Mechanik, GAMM) March 19 — 23, 2018 München, Germany

  Due to recent technological advances in the development of modern sensors new imaging modalities have emerged for which the acquired data values are given on a Riemannian manifold, e.g., in phase values in InSAR imaging or when dealing with diffusion tensors in DT-MRI. Furthermore, to process not only rectangular domains and regular grids, graphs can be employed to model both data on arbitrary (sampled) manifolds as well as nonlocal vicinity. In this talk we extend a framework for processing discrete manifold-valued data to the task of inpainting missing values, such that the inpainted area nonlocally incorporates data into the reconstruction. We discuss a generalization of the graph infinity-Laplacian to manifold-valued data based on the min-max characterization of the local discrete Lipschitz constant. Finally, we present a numerical scheme to solve the obtained manifold-valued infinity-Laplace equation.

2017

• Nonlocal Inpainting of Manifold-valued Data on Finite Weighted Graphs 3^rd Conference on Geometric Science of Information November 7 — 9, 2017 Paris, France Contributed talk within the session “Optimization on Manifolds” organized by P.-A. Absil, and Rudolphe Sepulchre

  Recently, there has been a strong ambition to translate models and algorithms from traditional image processing to non-Euclidean domains, e.g., to manifold-valued data. While the task of denoising has been extensively studied in the last years, there was rarely an attempt to perform image inpainting on manifold-valued data. In this paper we present a nonlocal inpainting method for manifold-valued data given on a finite weighted graph. We introduce a new graph infinity-Laplace operator based on the idea of discrete minimizing Lipschitz extensions, which we use to formulate the inpainting problem as PDE on the graph. Furthermore, we derive an explicit numerical solving scheme, which we evaluate on two classes of synthetic manifold-valued images.

• MVIRT – A Toolbox for Manifold-valued Image Restoration IEEE International Conference on Image Processing (ICIP) 2017 September 17 — 20, 2017 Bejing, China Invited talk within the Special Session “Trends in Statistical Analysis of Manifold-Valued Data – Theory and Applications to Imaging”, supported by a DAAD conference travel grant

  In many real life application measured data takes its values on Riemannian manifolds. For the special case of the Euclidean space this setting includes the classical grayscale and color images. Like these classical images, manifold-valued data might suffer from measurement errors in form of noise or missing data. In this talk we present the manifold-valued image restoration toolbox (MVIRT) that provides implementations of classical image processing tasks. Based on recent developments in variational methods for manifold-valued image processing methods, like total variation regularization, the toolbox provides easy access to work with these algorithms. The toolbox is implemented in Matlab, open source, and easily extendible, e.g. with own manifolds, noise models or further algorithms. This paper introduces the main mathematical methods as well as numerical examples.

• A Graph Framework for Manifold-valued Data Kolloquium angewandte Mathematik Institute of Computational and Applied Mathematics Westfälische Wilhelms-Universität Münster June 26, 2017 Invited talk and research stay within the research group of Benedikt Wirth

  In many real-world applications measured data are not in a Euclidean vector space but rather are given on a Riemannian manifold. This is the case, e.g., when dealing with Interferometric Synthetic Aperture Radar (InSAR) data consisting of phase values or data obtained in Diffusion Tensor Magnetic Resonance Imaging (DT-MRI).

  In this talk we present a framework for processing discrete manifold-valued data, for which the underlying (sampling) topology is modeled by a graph. We introduce the notion of a manifold-valued differences on a graph and based on this deduce a family of manifold-valued graph operators. In particular, we introduce the graph p-Laplacian and graph infinity-Laplacian for manifold-valued data. We discuss a numerical scheme to compute a solution to the corresponding parabolic PDEs and apply this algorithm to different manifold-valued data, illustrating the diversity and flexibility of the proposed framework in denoising and inpainting applications.

  This is joint work with Daniel Tenbrinck (WWU Münster)

• Infimal Convolution Coupling of First and Second Order Differences on Manifold-Valued Images 6^th International Conference on Scale Space and Variational Methods in Computer Vision June 4 — 8, 2017 Kolding, Denmark
• Variational Models for Manifold-valued Image Processing Seminars in Mathematical Engineering ICTEAM Université catolique de Louvain March 21, 2017 Louvain-la-Neuve, Belgium Invited talk and a week of research stay P.-A. Absil
• A Graph Framework for Manifold-valued Data 87^thAnnual Meeting of the International Association of Applied Mathematics and Mechanics (Gesellschaft für angewandte Mathematik und Mechanik, GAMM) March 6 — 10, 2017 Weimar & Ilmenau, Germany

2016

• A Second Order Non-smooth Variational Model for Restoring Manifold-valued Image Mecklenburg Workshop – Approximation Methods and Data Analysis September 5 — 9, 2016 Hasenwinkel, Germany
• Second Order TV-Type Regularization Methods for Manifold-Valued Images SIAM Conference of Imaging Sciences 2016 May 23 — 26, 2016 Albuquerque, NM, USA Invited talk within the Minisymposium I on “Inversion on Non-linear Image Formation Models”, supported by a DAAD conference travel grant.

  In many real world situations, measured data is noisy and nonlinear, i.e., the data is given as values in a certain manifold. Examples are Interferometric Synthetic Aperture Radar (InSAR) images or the hue channel of HSV, where the entries are phase-valued, directions in $\mathbb R^n$, which are data given on $\mathbb S^{n-1}$, and diffusion tensor magnetic resonance imaging (DT-MRI), where the obtained pixel of the image are symmetric positive definite matrices.<br/><br> In this talk we extend the recently introduced total variation model on manifolds by a second order TV type model. We first introduce second order differences on manifolds in a sound way using the induced metric on Riemannian manifolds. By avoiding a definition involving tangent bundles, this definition allows for a minimization employing the inexact cyclic proximal point algorithm, where the proximal maps can be computed using Jacobian fields. The algorithm is then applied to several examples on the aforementioned manifolds to illustrate the efficiency of the algorithm.

  This is joint work with M. Bačák, J. Persch, G. Steidl, and A. Weinmann.

• A Second Order Non-smooth Variational Model for Restoring Manifold-valued Image Seminar AG Imaging at the Imaging Workgroup Institute of Computational and Applied Mathematics Westfälische Wilhelms-Universität Münster May 11, 2016 Münster, Germany Invited talk and week research stay D. Tenbrinck

  In many real world situations, measured data is noisy and nonlinear, i.e., the data is given as values in a certain manifold or on a submanifold of the euclidean space the measurements are taken in. Examples are InSAR images and the hue channel of HSV, where the entries are phase-valued, directions in $\mathbb R n$, which are data given on sphere $\mathbb S^{n−1}$, and diffusion tensor magnetic resonance imaging, where the obtained data items are symmetric positive definite $3\times 3$ matrices.

  In this talk we extend the recently introduced total variation model on manifolds by a second order TV type model. We first introduce second order differences on manifolds in a sound way using the induced metric on Riemannian manifolds. By avoiding a definition involving tangent bundles, this definition allows for a minimization employing the inexact cyclic proximal point algorithm, where the proximal maps can be computed using Jacobian fields. The algorithm is then applied to several examples on the aforementioned manifolds to illustrate the efficiency of the algorithm.

• A Second Order Non-smooth Variational Model for Restoring Manifold-valued images 37^th Northern German Colloquium on Applied Analysis and Numerical Mathematics April 22 — 23, 2016 Lübeck, Germany
• A Second Order Non-smooth Variational Model for Restoring Manifold-valued images 86^thAnnual Meeting of the International Association of Applied Mathematics and Mechanics (Gesellschaft für angewandte Mathematik und Mechanik, GAMM) March 7 — 11, 2016 Braunschweig, Germany Invited talk within the Young Researchers Minisymposium YR 3 “Dedicated regularization for variational image processing”

2015

• A Second Order Non-smooth Variational Model for Restoring Manifold-valued images Variational Methods for Dynamic Inverse Problems and imaging September 28 — 30, 2015 Münster, Germany
• A Second Order Non-smooth Variational Model for Restoring Manifold-valued images 25th Rhein-Main Arbeitskreis – Mathematics of Computation July 10, 2015 Darmstadt, Germany Invited talk

  In many real world situations, measured data is noisy and nonlinear, i.e., the data is given as values in a certain manifold. Examples are InSAR images and the hue channel of HSV, where the entries are phase-valued, directions in $R^n$, which are data given on $S^{n-1}$, and diffusion tensor magnetic resonance imaging, where the obtained data items are symmetric positive definite matrices.

• The anisotropic Strang-Fix conditions Mecklenburger Workshop – Approximation Methods and Function Spaces March 16 — 20, 2015 Hasenwinkel, Germany
• Second Order Differences for Combined Cyclic and Vector Space Data and their Application to Denoising and Inpainting Cambridge Image Analysis seminars March 4, 2015 Cambridge, United Kingdom Invited talk and a week research stay J. Lellmann
• Inpainting and Denoising of Cyclic Data Using First and Second Order Differences Research Seminar Institute of Computational Biology, Helmholtzzentrum München February 18, 2015 Munich, Germany Invited talk and research stay at ICB
• Inpainting of Cyclic Data using First and Second Order Differences 10th International Conference on Energy Minimizing Methods in Computer Vision and Pattern Recognition January 15, 2015 Hong Kong

  Cyclic data arise in various image and signal processing applications such as interferometric synthetic aperture radar, electroencephalogram data analysis, and color image restoration in HSV or LCh spaces. In this paper we introduce a variational inpainting model for cyclic data which utilizes our definition of absolute cyclic second order differences. Based on analytical expressions for the proximal mappings of these differences we propose a cyclic proximal point algorithm (CPPA) for minimizing the corresponding functional. We choose appropriate cycles to implement this algorithm in an efficient way. We further introduce a simple strategy to initialize the unknown inpainting region. Numerical results both for synthetic and real-world data demonstrate the performance of our algorithm.

2014

• Second Order Differences of Cyclic Data and Applications in Variational Denoising Deutsch-französischer Workshop Mathematische Bildverarbeitung October 6 — 8, 2014 Fraunhofer ITWM, Kaiserslautern, Germany
• Second Order Differences of Cyclic Data and Applications in Variational Denoising Mathematical Image Processing and Phase Retrieval September 1 — 3, 2014 Göttingen, Germany
• Second Order Differences of Cyclic Data and Application to Variational Denoising Reserach Seminar Institute of Computational Biology, Helmholtzzentrum München May 14, 2014 München, Germany Invited talk and research stay at ICB
• Multivariate anisotrope Interpolation auf dem Torus Mecklenburger Workshop – Approximationsmethoden und schnelle Algorithmen March 17 — 20, 2014 Hasenwinkel, Germany
• Multivariate Anisotropic periodic Wavelets 85^thAnnual Meeting of the International Association of Applied Mathematics and Mechanics (Gesellschaft für angewandte Mathematik und Mechanik, GAMM) March 10 — 14, 2014 Erlangen, Germany

2013

• Multivariate Anisotropic Wavelets on the Torus Workshop on Advances in Image Processing September 30 — October 2, 2013 Anweiler am Trifels, Germany
• Multivariate anisotrope Wavelets auf dem Torus Mecklenburger Workshop — Approximationsmethoden und schnelle Algorithmen June 20 — 23, 2013 Hasenwinkel, Germany
• Multivariate anisotrope Wavelets auf dem Torus 32. Norddeutsches Kolloquium über angewandte Analysis und Numerische Mathematik May 3 — 4, 2013 Clausthal, Germany
• Die multivariate anisotropen Wavelet-Transformation auf dem Torus Kolloquium of the Image Processing and Data Analysis Group Fachbereich Mathematik Technische Universität Kaiserslautern April 8, 2013 Kaiserslautern, Germany

2012

• The multivariate anisotropic Wavelet Transform on the Torus Workshop on Mathematics for Life Sciences September 3 — 14, 2012 Kiev, Ukraine

2011

• Die multivariate periodische Wavelet-Transformation 32. Norddeutsches Kolloquium über angewandte Analysis und Numerische Mathematik May 27 — 28, 2011 Institut für Mathematik, Universität Osnabrück, Osnabrück, Germany

  Für periodische Wavelets im eindimensionalen Fall ist seit Mitte der 1990er Jahre eine umfassende Theorie bekannt. Dieser Vortrag stellt die multivariate Verallgemeinerung vor. Ausgehend von anisotropen gleichmäßigen Mustern auf dem $d$-dimensionalen Torus wird eine diskrete Fourier-Transformation definiert. Dies führt zu diskreten Frequenzmengen – die erzeugende Gruppe –, die bestimmte Richtungen bevorzugen und trotzdem schnelle Algorithmen ermöglichen. Dazu ist es notwendig, eine bestimmte Anordnung der Musterpunkte, sowie der erzeugenden Gruppe, zu finden. Die darauf aufbauende Wavelet-Zerlegung wird für den Spezialfall einer dyadischen Zerlegung präsentiert. Sie zerlegt einen Raum von Translaten in zwei zueinander orthogonale Unterräume mit jeweils halber Dimension des Ursprungsraumes. Auch hier ist eine schnelle Implementierung möglich, wenn man für die Elemente der erzeugenden Gruppe eine entsprechende Anordnung wählt.

• Multivariate translationsinvariante Räume 21. Rhein-Ruhr Workshop February 4 — 5, 2011 Königswinter, Germany
• Multivariate Periodic Function Spaces 21. Rhein-Ruhr Workshop February 4 — 5, 2011 Königswinter, Germany

2009

• Drawing Hypergraphs using NURBS curves Presentation of my diploma thesis TU Bergakademie Freiberg September 16, 2009 Freiberg, Germany

Organizer at Conferences

• TMF workshop “Foundations of Numerical Differential Geometry”, May 7-8, 2024, Trondheim, Norway, together with Elena Celledoni, André Massing Brynjulf Owren, and Alexander Schmeding.

• Minisymposium “MS2 Optimization of Shapes and on Manifolds”, within the GAMM 2022, August 17, 2022, Aachen, Germany, together with Stephan Schmidt

• Session “Software for Optimization on Manifolds”, within the ICCOPT 2022, July 26, 2022, Bethlehem, PA, USA, together with Jan Lellmann.

• Minisymposium: “Manifolds and Optimization”, September 2nd, 2021, within the IFIP TC7 Conference on System Modelling and Optimization, Quito and online, September 2nd, 2021, together with Orizon P. Ferreira.

• Workshop on Optimization on Manifolds, August 9, 2019, Technische Universität Chemnitz, Germany, together with Glaydston de Carvalho Bento, Roland Herzog, José Vidal-Núñez.

• Young Researchers Minisymposium “YR3 Local and nonlocal methods for processing manifold and point cloud data” at the Jahrestagung der Gesellschaft für angewandte Mathematik und Mechanik 2017, March 6—10, Weimar, Germany together with Daniel Tenbrinck.

• Workshop on “manifold-valued image processing”, December 1st, 2016, Technische Universität Kaiserslautern, Germany, together with Gabriele Steidl.

Further Conferences

Furthermore I was lucky enough to join the following conferences.

• JuliaCon 2021 July 28 — 30, 2021 online & everywhere
• 7^th International Conference on Scale Space and Variational Methods in Computer Vision June 30 — July 4, 2019 Hofgeismar
• Chemnitz Symposium on Inverse Problems 2018 September 27 — 28, 2018 Chemnitz
• Chemnitz Finite Element Symposium 2018 September 24 — 26, 2018 Chemnitz
• Mathematics and Image Analysis 2018 January 15 — 17, 2018 Berlin, Germany
• Rhein-Main-Arbeitskreis Mathematics of Computation July 7, 2017 Mannheim, Germany
• Workshop: Shape, Images and Optimization February 28 — March 3, 2017 Münster, Germany
• Project Meeting of the Marie Curie Inital Training Network Scheme: INTREPID Forensics February 5, 2016 Leicester, United Kingdom
• Summer School on Applied Analysis 2015 September 21 — 25, 2015 Chemnitz, Germany
• 5^th International Conference on Scale Space and Variational Methods in Computer Vision May 30 — June 4, 2015 Bordeaux, France
• Workshop “Optimisation Géométrique sur les Variétés” November 21, 2014 Paris, France
• Colloquium Commemorating Bernd Fischer October 10, 2014 Lübeck, Gemany
• Summer School on Applied Analysis 2014 September 22 — 26, 2014 Chemnitz, Germany
• Mathematik in Medizin und Lebenswissenschaften May 1, 2014 Lübeck, Germany Meeting on occasion of the 20^th birthday of the Institute of Mathematics at the University of Lübeck.
• Rhein-Main-Arbeitskreis Mathematics of Computation February 7, 2014 Marburg, Germany
• Summer School on Applied Analysis 2013 September 23 — 27, 2013 Chemnitz, Germany
• Summer School on Applied Analysis 2012 September 24 — 28, 2012 Chemnitz, Germany
• 33. Norddeutsches Kolloquium über Angewandte Analysis und Numerische Mathematik May 4 — 5, 2012 Rostock, Germany
• 22. Rhein-Rhuhr-Workshop February 3 — 4, 2012 Bestwig, Germany
• 28th GAMM-Seminar Leipzig on Analysis and Numerical Methods in Higher Dimension January 16 — 18, 2012 Leipzig, Germany
• Summer School on Applied Analysis 2011 September 26 — 30, 2011 Chemnitz, Germany
• 1^st Workshop: A Computational Approach to Harmonic Analysis August 22 — 26, 2011 Marburg, Germany
• From Abstract to Computational Harmonic Analysis June 13 — 19, 2011 Strobl, Austria
• Summer School on Applied Analysis 2010 October 4 — 8, 2010 Chemnitz, Germany
• 20. Rhein-Rhuhr-Workshop February 12 — 13, 2010 Burg Gemen, Germany