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## Colloquium of the Modeling, Numerics, Differential Equations Group

Responsible Persons: | All Professors of the Modeling • Numerics • Differential Equations Group |
---|---|

Coordination: | Dr. Christian Schröder |

Dates: | Tue 16-18 Uhr in MA 313 and by appointment |

Content: | Talks by visitors and sometimes also our faculty on current resesarch topics |

### Description

The colloquium of the Modeling, Numerics, Differential Equations Group at the institute of mathematics is a conventional colloquium attracting a broad audience consisting of professors and research assistants of all associated work groups, in particular applied functional analysis, numerical linear algebra, and partial differential equations. Graduate students are also attending the colloquium.

For these reasons we look forward to talks aimed at non-specialists that can be be enjoyed by graduate students.

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.

### 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.