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## Kolloquium der Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen

Verantwortliche Dozenten: | Alle Professoren der Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen |
---|---|

Koordination: | Dr. Christian Schröder |

Termine: | Di 16-18 Uhr in MA 313 und nach Vereinbarung |

Inhalt: | Vorträge von Gästen und Mitarbeitern zu aktuellen Forschungsthemen |

## Beschreibung

Das Kolloquium der Arbeitsgruppe "Modellierung, Numerik, Differentialgleichungen" im Institut der Mathematik ist ein Kolloquium klassischer Art. Es wird also von einem breiten Kreis der Professoren und Mitarbeiter aus allen zugehörigen Lehrstühlen, insbesondere Angewandte Funktionalanalysis, Numerische Lineare Algebra, und Partielle Differentialgleichungen, besucht. Auch Studierende nach dem Bachelorabschluss zählen schon zu den Teilnehmern unseres Kolloquiums.

Aus diesen Gründen freuen wir uns insbesondere über Vorträge, die auf einen nicht spezialisierten Hörerkreis zugeschnitten sind und auch von Studierenden nach dem Bachelorabschuss bereits mit Profit gehört werden können.

Datum date | Zeit time | Raum room | Vortragende(r) speaker | Titel title | Einladender invited by |
---|---|---|---|---|---|

Di 2.10.12 | 16:15 | MA 313 | Joseph Traub (Columbia U) | Algorithms and Complexity for Quantum Computing (no Abstract) | H. Yserentant |

Fr 5.10.12 | 14:15 | MA 313 | Helge Holden (Norwegian U of Science and Technology) | The Camassa-Holm equation - a survey (Abstract) | V. Mehrmann |

Di 16.10.12 | 16:15 | MA 313 | Mario Arioli (Rutherford Appleton Lab, UK) | An introduction to Quantum Graphs (Abstract) | V. Mehrmann A. Miedlar |

Di 23.10.12 | 16:15 | MA 313 | Piotr Rybka (U Warsaw) | Motion of Closed Curves by Singular Weighted Mean Curvature (Abstract) | M. Korzec B. Wagner |

Di 30.10.12 | 16:15 | MA 313 | Peter Benner (MPI Magdeburg) | System-Theoretic Model Reduction for Nonlinear Systems (Abstract) | F. Tröltzsch |

Do 1.11.12 | 16:15 | MA 313 | Michel Pierre (ENS Cachan, IRMAR, France) | About two cross-diffusion systems (Abstract) | E. Emmrich |

Di 6.11.12 | no colloquium, room occupied by workshop, guests are welcome | ||||

Fr 9.11.12 | 10:15 | MA 415 | Juliette Chabassier (INRIA Bordeaux, U Pau) | Modeling and numerical simulation of a grand piano (Abstract) | K. Schmidt |

Fr 9.11.12 | 13:15 | MA 313 | Rich Lehoucq (Sandia National Labs, US) | Analysis and approximation of nonlocal diffusion problems with volume constraints (Abstract) | V. Mehrmann |

Di 13.11.12 | 16:15 | MA 313 | Dozenten der AG ModNumDiff | Lehrbesprechung der AG ModNumDiff | D. Puhst |

Di 20.11.12 | 16:15 | MA 313 | Francois Murat (U Paris VI) | Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L^1 (Abstract) | E. Emmrich |

Di 27.11.12 | 16:15 | MA 313 | Nicolas Gillis (UC Louvain, Belgium) | Fast and Robust Algorithms for Separable Nonnegative Matrix Factorization (Abstract) | J. Liesen |

Mi 28.11.12 | 16:15 | MA 313 | Mats Larson (U Umea, Sweden) | Cut finite element methods for fluids and solids: theory, implementation, and applications (Abstract) | V. Mehrmann |

Di 11.12.12 | no colloquium, room occupied by workshop, guests are welcome | ||||

Di 18.12.12 | this date is free | the previously announced talk by Ali Pezeshki had to be cancelled | |||

Di 8.01.13 | 16:15 | MA 313 | Tomas Sauer (U Passau) | Chemistry, Splines, Kronecker (Abstract) | G. Kutyniok |

Di 15.01.13 | 16:15 | MA 313 | Kees Vuik (TU Delft, NL) | An efficient and robust Krylov method for Discontinuous Galerkin problems (Abstract) | R. Nabben |

Di 29.01.13 | 16:15 | MA 313 | Martin Burger (U Münster) | Mathematical Challenges in Neuronal Polarization (Abstract) | G. Kutyniok |

Di 5.02.13 | 16:15 | MA 313 | Julio D. Rossi (U Alicante) | A game theory approach to the p-Laplacian and its limit, the infinity Laplacian (Abstract) | E. Emmrich |

Di 26.02.13 | 16:15 | MA 313 | Andrea Bertozzi (UCLA) | Mathematics of Crime (Abstract, Flyer) | B. Wagner |

Di 5.03.13 | no talk, room is occupied | ||||

Di 26.03.13 | no talk, room is occupied |

### Helge Holden (Norwegian U of Science and Technology)

**The Camassa-Holm equation - a survey**

Freitag, den 5.10.2012, 14.15 Uhr in MA 313

Abstract:

The Camassa-Holm equation u_t-u_{xxt}+ u_x+3u u_x-2u_x u_{xx}-u u_{xxx}=0 has received considerable attention the last 20 years due to its many intriguing mathematical properties. In particular, the Cauchy problem possesses two distinct classes of solutions due to the wave breaking of the solution. We review the current understanding of this problem, with emphasis on the Lipschitz stability of the solution of the Cauchy problem. Extensions to a two-component generalization of the Camassa-Holm equation will also be discussed. The talk is based on joint work with X. Raynaud (University of Oslo) and K. Grunert (Norwegian University of Science and Technology).

Preceding this talk there will be coffee, tea, and biscuits at 13:45 in room MA 315 - everybody's welcome.

### Mario Arioli (Rutherford Appleton Lab, UK)

**An introduction to Quantum Graphs**

Dienstag, den 16.10.2012, 16.15 Uhr in MA 313

Abstract:

We will present an elementary introduction to metric graphs and to the solution and modelling of differential problems on them. A metric graph with a global differential problem deﬁned on its vertices and edges is called a Quantum Graph.

We will describe several elementary properties that make the problem of solving differential equations on metric graphs different from the standard, and we will illustrate several potential applications related to complex network theory.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

### Piotr Rybka (U Warsaw)

**Motion of Closed Curves by Singular Weighted Mean Curvature**

Dienstag, den 23.10.2012, 16.15 Uhr in MA 313

Abstract:

We study evolution of simple closed curves, driven by the driven singular weighted mean curvature (wmc) with forcing. The weighted mean curvature is so singular that the closed curves with constant curvature (i.e spheres) are rectangles.

We construct variational solutions of the flow when the initial data are from a class of perturbed constant curvature curves. We follow the evolution of facets.

We expose the parabolic nature of the problem in question, in particular we formulate a suitable version viscosity solutions. We also discuss the question of uniqueness of solutions.

### Peter Benner (MPI Magdeburg)

**System-Theoretic Model Reduction for Nonlinear Systems**

Dienstag, den 30.10.2012, 16.15 Uhr in MA 313

Abstract:

We discuss Krylov-subspace based model reduction techniques for nonlinear control systems. Since reduction procedures of existent approaches like TPWL and POD methods require simulation of the original system and are therefore dependent on the chosen input function, models that are subject to variable excitations might not be sufficiently approximated. We will overcome this problem by generalizing Krylov-subspace methods known from linear systems to a more general class of bilinear and quadratic-bilinear systems, respectively. As has recently been shown, a lot of nonlinear dynamics can be represented by the latter systems. We will explain advantages and disadvantages of the different approaches and illustrate their behavior for several benchmark examples from the literature.

This is joint work with Tobias Breiten (MPI Magdeburg).

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

### Michel Pierre (ENS Cachan, IRMAR, France)

**About two cross-diffusion systems **

Donnerstag, den 1.11.2012, 16.15 Uhr in MA 313

Abstract:

Cross-diffusion occurs for instance when interaction between species takes place through motion and not only through reaction. This leads to more complex models whose mathematical structure is only partially understood yet. In particular, much needs to be done for the question of global existence and regularity of solutions.

We will consider two specific systems. One is of conservative form and is the relaxed version of a general model in which solutions are spatially "regularized" to take into account that the interaction between species occurs not only pointwise, but in the neighborhood of each point. We prove that the problem is globally well-posed.

The second nonlinear cross-diffusion system is obtained as the limit of infinitely fast reactions in more classical reaction-diffusion systems. Global existence of weak solutions are thus derived, but it is not clear to compare them with the known local strong solutions.

Preceding this talk there will be coffee, tea, and biscuits at 15:45 in room MA 315 - everybody's welcome.

### Juliette Chabassier (INRIA Bordeaux, U Pau)

**Modeling and numerical simulation of a grand piano**

Freitag, den 9.11.2012, 10.15 Uhr in MA 415

Abstract:

The piano is an instrument of remarkably complexity. Not less than 12'000 elements are necessary to build the Steinay D-model, the largest Steinway grand piano ! Our goal is to model the acoustical and vibratory behavior of the whole instrument. We only consider the main parts : hammer, strings, soundboard, and sound radiation in the air. This allows us to design a mathematical and numerical model for the piano. Experimental studies have shown that the nonlinear behavior of the strings had a considerable influence on the percussive tone quality. We suggest a nonlinear model for the strings , also taking the stiffness into account. This yields a first nonlinear PDE system. Moreover, the strings-hammer coupling is nonlinear. The strings' extremity are attached to the bridge, so that their energy is transmitted to the soundboard. Finally, the soundboard radiated into the surrounding air, modifying the pressure field, and our ears detect a sound.

Putting this coupled system into a discrete form is a challenge, especially as several elements are nonlinear. The global stability of the numerical scheme is achieved through an energy technique. We design numerical schemes that decay a total discrete energy, ensuring the reciprocal circulation of energy between sub-systems. Space discretisation is done with high order finite elements. Very different time discretisation methods are used on each sub-system (innovative scheme for the strings, analytic method for the soundboard, finite differences for the sound radiation). All these methods are coupled efficienlty by means of Schur complements and the use of Lagrange multipliers.

We will present numerical results which show that this comprehensive model allows to explain phenomena that were previously observed experimentally, but never simulated until today.

### Rich Lehoucq (Sandia National Labs, US)

**Analysis and approximation of nonlocal diffusion problems with volume constraints**

Freitag, den 9.11.2012, 13.15 Uhr in MA 313

Abstract:

A recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on a bounded domain. The nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion, including a notion of nonlocal flux. In particular, it is shown that fractional Laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion. As an application, we compute the exit-time for a symmetric finite-range jump process in direct analogy to the classical diffusion equation with a homogeneous Dirichlet boundary condition, the nonlocal diffusion equation is augmented with a homogeneous volume constraint. The volume-constrained master equation provides an efficient alternative over simulation for computing an important statistic of the process. Several numerical examples are given.

### Francois Murat (U Paris VI)

**Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L ^{1}**

Dienstag, den 20.11.2012, 16.15 Uhr in MA 313

Abstract:

In this lecture I will report on joint work with J. Casado-Díaz, T. Chacón Rebollo, V. Girault and M. Gómez Marmol which has been published in Numerische Mathematik, vol. 105, (2007), pp. 337-374. We consider, in dimension d≥2, the standard P1 finite elements approximation of the second order linear elliptic equation in divergence form with coefficients in L^{∞}(Ω) which generalizes Laplace's equation, i.e.

- div A Du = f.

We assume that the family of triangulations is regular and that it satisfies an hypothesis close to the classical hypothesis which implies the discrete maximum principle. When the right-hand side belongs to L^{1}(Ω), we prove that the unique solution of the discrete problem converges in W^{1,q}_{0}(Ω) (for every q with 1 ≤ q < d/(d-1) to the unique renormalized solution of the problem. We obtain a weaker result when the right-hand side is a bounded Radon measure. In the case where the dimension is d=2 or d=3 and where the coefficients are smooth, we give an error estimate in W^{1,q}_{0}(Ω) when the right-hand side belongs to L^{r}(Ω) for some r> 1.

### Nicolas Gillis (UC Louvain, Belgium)

**Fast and Robust Algorithms for Separable Nonnegative Matrix Factorization**

Dienstag, den 27.11.2012, 16.15 Uhr in MA 313

Abstract:

Nonnegative Matrix Factorization (NMF) is a linear dimensionality reduction technique for nonnegative data. It consists in approximating a nonnegative data matrix with the product of two low-rank nonnegative matrices. NMF has become a very popular technique in data mining and machine learning because it automatically extracts meaningful features through a sparse and part-based representation. Although NMF is NP-hard in general, it has been shown very recently that it is possible to compute an optimal solution under the assumption that the input nonnegative data matrix is separable (i.e., there exists a cone spanned by a small subset of the columns containing all columns). Current approaches solving the separable NMF problem are either computationally expensive or not robust to noise. In this talk, we first introduce NMF and illustrate its usefulness with some application examples (namely, image processing, text mining and hyperspectral data analysis). Then, we present a new family of fast and robust recursive algorithms for separable NMF problems.

This is joint work with Stephen Vavasis.

### Mats Larson (U Umea, Sweden)

**Cut finite element methods for fluids and solids: theory, implementation, and applications**

Mittwoch, den 28.11.2012, 16.15 Uhr in MA 313

Abstract:

Multi-domain and multi-physics problems with moving interfaces or changing geometric domains can be severely limited by the use of conforming meshes when complex geometries in three dimensions are involved. An alternative to conforming meshes is fixed-grid methods based where the geometry as well as the solution is represented on a background mesh. In this talk we formulate so called Cut Finite Element Methods which weakly impose conditions on boundaries and interfaces that cut through the underlying mesh, i.e., does not conform to the mesh. We discuss implementation, theory and finally present some applications including fluids and fluis-structure interaction.

References

[1] A. Massing, M. G. Larson, A. Logg, and M. E. Rognes, “A stabilized Nitsche fictitious domain method for the Stokes problem”, submitted for publication, 2012 (available as arXiv preprint arXiv:1206.1933)

[2] A. Massing, M. G. Larson, A. Logg, and M. E. Rognes, “A stabilized Nitsche overlapping mesh method for the Stokes problem”, submitted for publication, 2012 (available as arXiv preprint arXiv:1205.6317)

[3] A. Massing, M. G. Larson, and A. Logg, “Efficient implementation of finite element methods on non-matching and overlapping meshes in 3D ”, SIAM J. Sci. Comput., accepted, 2012 (available as arXiv preprint arXiv:1210.7076)

[4] A. Massing, M. G. Larson and A. Logg, "Towards an Implementation of Nitsche’s Method on Overlapping Meshes in 3D", AIP Conference Proceedings 1281, 2010

### Tomas Sauer (U Passau)

**Chemistry, Splines, Kronecker**

Dienstag, den 8.01.2013, 16.15 Uhr in MA 313

Abstract:

High dimensional problems arise natural in the numerical treatement of chemical processes where any reasonable relationship with reality requires the consideration of various input parameters. One way to access interpolation problems emerging from catalyzer simulation is to use tensor product functions where the resulting linear problems provide a Kronecker product structure that makes them accessible in "factorized form" even for quite large scale problems. Then, however, the problem is to provide numerical methods that work entirely in the factorized which becomes interesting in more than two variables and an extension of concepts like the SVD for more general multilinear forms. The talk will describe some results in this direction and the numerical application that motivated them.

This is joint work with F. Lamping (Giessen) and J.M. Pena (Zaragoza).

### Kees Vuik (TU Delft)

**An efficient and robust Krylov method for Discontinuous Galerkin problems**

Dienstag, den 15.01.2013, 16.15 Uhr in MA 313

Abstract:

Discontinuous Galerkin (DG) methods can be interpreted as finite volume methods that use (discontinuous) higher-order polynomials rather than piecewise constants. As such, it combines the best of both classical finite element methods and finite volume methods. However, a challenge for DG methods is that the resulting linear system is often ill-conditioned and relatively large compared to e.g. classical finite element methods. Especially for problems with large contrasts in the coefficients, this can lead to long computational times.

For this reason, we have studied the Conjugate Gradient (CG) method for linear systems resulting from Symmetric Interior Penalty (discontinuous) Galerkin (SIPG) discretizations for diffusion problems with strong variations in the coefficients. In particular, we have investigated the impact of choosing the SIPG penalty parameter diffusion-dependent, instead of the usual strategy to use a constant value. Furthermore, we have studied the potential of casting the scalable spectral two-level preconditioner introduced by Dobrev et al. [1] into the deflation framework, using the analysis of Tang et al. [2]. In this talk, we demonstrate numerically the impact of both strategies on the CG and SIPG convergence for several diffusion problems with strong variations in the coefficients.

We have found that a diffusion-dependent penalty parameter yields more accurate SIPG approximations and significantly faster CG convergence. Furthermore, the proposed two-level deflation technique yields fast and scalable CG convergence, just like the original preconditioning variant. Nevertheless, the deflation variant is still faster as it uses only one rather than two smoothing steps. A combination of the two strategies above can increase the efficiency up to 100 times [3]. Future research includes the theoretical support for these findings.

This is joint work with P. van Slingerland.

[1] V. A. Dobrev, R. D. Lazarov, P. S. Vassilevski, L. T. Zikatanov, Two-level preconditioning of discontinuous {G}alerkin approximations of second-order elliptic equations, Numer. Linear Algebra Appl., vol 13(9), pp 753--770, 2006.

[2] J. M. Tang, R. Nabben, C. Vuik, Y. A. Erlangga, Comparison of two-level preconditioners derived from deflation, domain decomposition and multigrid methods, J. Sci. Comput., vol 39(3), pp 340--370, 2009.

[3] P. van Slingerland and C. Vuik, Spectral two-level deflation for DG: a preconditioner for CG that does not need symmetry, Report 11-12, Institute of Applied Mathematics, Delft University of Technology, Delft, 2011.

### Martin Burger (U Münster)

**Mathematical Challenges in Neuronal Polarization**

Dienstag, den 29.01.2013, 16.15 Uhr in MA 313

Abstract:

Understanding neuronal polarization, in particular the robust establishment of a single axon and multiple dendrites, is a question of obvious importance in neurobiology. Several recent results have established connections between cell polarization and the asymmetric activation of certain GTPases and PI3kinase, which is the basis for our investigation.

In this talk we will highlight to mathematical challenges related to the above question. The first is related to image reconstruction and analysis, used in order to quantify transport of GTPases along axons. A major challenge is the limited resolution of microscopes in z-direction and time, which we tackle by combined motion reconstruction and inpainting models related to optimal transport. The second challenge is to understand how robust symmetry breaking can occur by activation of GTPases. For this sake we develop reaction-diffusion models and also investigate the role of transport. In a simplified model still incorporating the main structure we give a rigorous proof of symmetry breaking properties as found in experiments.

### Julio D. Rossi (U Alicante)

**A game theory approach to the p-Laplacian and its limit, the infinity Laplacian**

Dienstag, den 5.02.2013, 16.15 Uhr in MA 313

Abstract:

We will review some recent results concerning the p_laplacian, the inifnity Laplacian, Tug-of-War games and their relation to some well known PDEs. In particular, we will show that solutions to certain nonlinear PDEs can be obtained as limits of values of Tug-of-War games when the parameter that controls the length of the possible movements goes to zero. Since the equations under study are nonlinear and not in divergence form we will make extensive use of the concept of viscosity solutions.

### Andrea Bertozzi (UCLA)

**Mathematics of Crime** (Flyer)

Dienstag, den 26.02.2013, 16.15 Uhr in MA 313

Abstract:

There is an extensive applied mathematics literature developed for the biological and physical sciences. Our understanding of social science from a mathematical standpoint is less developed, but also presents some very interesting problems, especially for young researchers. This lecture uses crime as a case study for using mathematical techniques in a social science application covering a variety of methods. From an application standpoint we will look at residential burglaries and gang crimes. We will consider both "bottom up'' and "top down'' approaches to understanding the mathematics of crime, and how the two approaches could converge to a unifying theory.