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## Kolloquium der Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen

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

Koordination: | Dr. Christian Schröder, Dr. Hans-Christian Kreusler |

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 15.04.14 | 16:15 | MA 313 | Vu Hoang Linh (Vietnam National U Hanoi) | Spectral analysis of differential-algebraic equations (Abstract) | V. Mehrmann |

Di 22.04.14 | 16:15 | MA 313 | Bernhard Bodmann (U Houston) | Near-optimal robust transmissions with random fusion frames (Abstract) | J. Vybiral |

Di 29.04.14 | 16:15 | MA 313 | Alexander Linke (WIAS Berlin) | On the Role of the Helmholtz Decomposition in Mixed Methods for Incompressible Flows and a New Variational Crime (Abstract) | G. Bärwolff |

Di 6.05.14 | 16:15 | MA 313 | Raphael Kruse (TU Berlin) | Consistency and stability of numerical methods for stochastic differential equations (Abstract) | - |

Di 13.05.14 | 16:15 | MA 313 | Hongguo Xu (U of Kansas, Lawrence) | A classical Poisson-Nernst-Planck model for ionic flow (Abstract) | V. Mehrmann |

Mi 14.05.14 | 16:15 | MA 313 | Ira Livshits (Ball State U, USA) | Developments toward scalable multigrid solver for the indefinite Helmholtz equations (Abstract) | R. Nabben |

Di 20.05.14 | 16:15 | MA 313 | Elias Wegert (TU Freiberg) | Exploring Complex Functions Using Phase Plots (Abstract) | J. Liesen |

Di 27.05.14 | 16:15 | MA 313 | Martin Lotz (U Manchester) | Phase Transitions in Convex Optimization (Abstract) | G. Kutyniok |

Di 03.06.14 | 16:15 | MA 313 | Michael Hinze (U Hamburg) | Reconstruction of matrix parameters from noisy measurements (Abstract) | G. Bärwolff |

Di 10.06.14 | 16:15 | MA 313 | Hermann Matthies (TU Braunschweig) | Factorisations, Tensor Methods, and Inverse Problems (Abstract) | R. Schneider |

Di 17.06.14 | 16:15 | MA 313 | Lucy Weggler (HU Berlin) | Advances in High Order Boundary Element Methods (Abstract) | K. Schmidt R. Schneider |

Di 24.06.14 | 16:15 | MA 313 | Martin Ehler (U Wien) | Signal reconstruction from magnitude measurements (Abstract) | B. Bodmann |

Di 01.07.14 | 16:15 | MA 313 | Kees Vuik (TU Delft) | Fast and robust iterative solvers for the Helmholtz equation (Abstract) | R. Nabben |

Di 08.07.14 | 16:15 | MA 313 | Yoel Shkolnisky (Tel Aviv U) | Viewing Directions Estimation in cryo-EM Using Synchronization (Abstract) | G. Kutyniok |

Di 15.07.14 | 16:15 | MA 313 | K.C. Sivakumar (IIT Madras) | Moore-Penrose inverse positivity of interval matrices (Abstract) | V. Mehrmann |

Di 22.07.14 | 16:15 | MA 313 | Raymond Chan (Chinese U Hong Kong) | Point-spread function reconstruction in ground-based astronomy (Abstract) | G. Kutyniok |

### Vu Hoang Linh (Vietnam National U Hanoi)

**Spectral analysis of differential-algebraic equations**

Dienstag, den 15.04.2014, 16.15 Uhr in MA 313

Abstract:

The spectral theory of differential equations started with the historical works of Lyapunov and Perron. This is an important part of the qualitative theory of differential equations and it gives a powerful tool for analyzing the stability and the asymptotic behavior of solutions. For ordinary differential equations, while the theoretical part of the spectral analysis (for both the finite and the infinite dimensional cases) was fairly well established by late 70's, the numerical aspects have attracted researchers' attention since 80's. In particular, in the last fifteen years, a significant progress has happened with the numerical methods for approximating spectral intervals of differential and differential-algebraic equations. In this talk, we first summarize our recent contributions to the theory and to the numerical methods for the spectral analysis of differential-algebraic equations. Then, we discuss also works in progress and several open problems that are worth being investigated in the next coming years.

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

### Bernhard Bodmann (U Houston)

**Near-optimal robust transmissions with random fusion frames**

Dienstag, den 22.04.2014, 16.15 Uhr in MA 313

Abstract:

This talk describes recent results on fusion frames which are motivated by communication theory. Fusion frames are families of weighted orthogonal projection operators which sum to an approximate identity. Fusion frames are a natural tool in packet-based communication systems when a vector to be transmitted is encoded in terms of lower-dimensional components obtained by projecting the vector onto subspaces. The possible non-orthogonality between the subspaces can be used to correct errors that occur in the transmission. This talk discusses the problem of fusion frame design for the recovery from partial data loss when the content of a relatively small number of subspace components is lost in the transmission. Optimal fusion frames are in general hard to find, but randomized constructions give near-optimal fusion frames in a straightforward way. Essential techniques for frame design in this context come from the literature on compressed sensing.

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

### Alexander Linke (WIAS Berlin)

**On the Role of the Helmholtz Decomposition in Mixed Methods for Incompressible Flows and a New Variational Crime **Dienstag, den 29.04.2014, 16.15 Uhr in MA 313

Abstract:

Unfortunately, nearly all mixed discretizations for the incompressible Navier-Stokes equations do not preserve exactly the fundamental identity of vector calculus `gradient fields are irrotational’. In consequence, mixed methods suffer from a non-physical interaction of divergence-free and irrotational forces in the momentum balance, resulting in the well-known numerical instability of `poor mass conservation'. The origin of this problem is the lack of L2-orthogonality between discretely divergence-free velocities and irrotational vector fields.

Therefore, a new variational crime for the nonconforming Crouzeix-Raviart element is proposed, where divergence-free, lowest-order Raviart-Thomas velocity reconstructions reestablish L2-orthogonality. This approach allows to construct an efficient flow discretization for unstructured and even anisotropic 2D and 3D simplex meshes that fulfills `gradient fields are irrotational'. In the Stokes case, optimal a-priori error estimates for the velocity gradients and the pressure are derived. Moreover, the discrete velocity is independent of the continuous pressure. Several detailed linear and nonlinear numerical examples illustrate the theoretical findings.

### Raphael Kruse (TU Berlin)

**Consistency and stability of numerical methods for stochastic differential equations**

Dienstag, den 6.05.2014, 16.15 Uhr in MA 313

Abstract:

In this talk we introduce a stability concept for the numerical analysis of one-step methods, which are used for the temporal discretization of stochastic differential equations. An important ingredient in this concept is the so called stochastic Spijker norm, which allows for two-sided estimates of the strong error of convergence. We apply these two-sided estimates to prove the maximum order convergence for the Euler-Maruyama method. If time permits we discuss several generalizations of the concept to multistep methods and stochastic partial differential equations.

### Hongguo Xu (U of Kansas, Lawrence)

**A classical Poisson-Nernst-Planck model for ionic flow**

Dienstag, den 13.05.2014, 16.15 Uhr in MA 313

Abstract:

Ion channels are small holes embedded in cell membranes. They open and close to control the flow of ions. In this way, cells function properly. Poisson-Nernst-Planck (PNP) model is a mathematical model commonly used to study ionic flow. In this talk, a classical PNP model is considered. We show that the system has a unique solution. The proof is constructive. With singular perturbation analysis the problem is reduced to an inverse problem of a first order linear system with boundary conditions. Then with the help of matrix theory, it is further reduced to a root problem of a meromorphic function, which is solved by Cauchy Argument Principle. We show how matrix eigenvalue theory is used to solve the problem.

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

### Ira Livshits (Ball State U, USA)

**Developments toward scalable multigrid solver for the indefinite Helmholtz equations**

Mittwoch, den 14.05.2014, 16.15 Uhr in MA 313

Abstract:

In this talk, new developments toward a fully scalable and fast multigrid solvers for the indefinite Helmholtz equations are discussed. The foundation of the new approaches is the wave-ray algorithm developed by Brandt and Livshits for the Helmholtz operator with constant wave numbers. Until now, however, its application was limited to constant and mildly varying coefficients, while most real-life applications involve operators with variable, including discontinuous, wave numbers. We propose different strategies that allow to adapt the wave-approach to be applicable to a variety of wave numbers, while maintaining its scalability and fast convergence.

### Elias Wegert (TU Freiberg)

**Exploring Complex Functions Using Phase Plots**

Dienstag, den 20.05.2014, 16.15 Uhr in MA 313

Abstract:

During the last years it became quite popular to visualize complex (analytic) functions as images. So-called ``phase plots'' depict a function f directly on a domain by color-coding the argument of f. The picture shows a phase plot of the Riemann zeta function. Phase plots are like fingerprints: though part of the information (the modulus) is neglected, meromorphic functions are (almost) uniquely determined by their phase plot. In the first part of the lecture we learn how basic properties of a function can be read off from these images. In the second part we investigate the phase plots of some special functions and illustrate several known results (theorems of Jentzsch and Szegö, universality of the Riemann zeta function). Finally, we demonstrate that phase plots and related ``phase diagrams'' are useful tools for exploring complex functions which may help to deepen our understanding and pose new quetsions.

### Martin Lotz (U Manchester)

**Phase Transitions in Convex Optimization**

Dienstag, den 27.05.2014, 16.15 Uhr in MA 313

Abstract:

Convex regularization has become a popular approach to solve large scale inverse or data separation problems. A prominent example is the problem of identifying a sparse signal from linear samples my minimizing the l_1 norm under linear constraints. Recent empirical research indicates that many convex regularization problems on random data exhibit a phase transition phenomenon: the probability of successfully recovering a signal changes abruptly from zero to one as the number of constraints increases past a certain threshold. We present a rigorous analysis that explains why phase transitions are ubiquitous in convex optimization. It also describes tools for making reliable predictions about the quantitative aspects of the transition, including the location and the width of the transition region. These techniques apply to regularized linear inverse problems, to demixing problems, and to cone programs with random affine constraints. These applications depend on a new summary parameter, the statistical dimension of cones, that canonically extends the dimension of a linear subspace to the class of convex cones.

Joint work with Dennis Amelunxen, Mike McCoy and Joel Tropp.

### Michael Hinze (U Hamburg)

**Reconstruction of matrix parameters from noisy measurements**

Dienstag, den 3.06.2014, 16.15 Uhr in MA 313

Abstract:

We consider identification of the diffusion matrix in elliptic PDEs from measurements. We prove existence of solutions using the concept of H-convergence. We discretize the problem using variational discretization and prove Hd-convergence of the discrete solutions by adapting the concept of Hd-convergence introduced by Eymard and Gallouet for finite-volume discretizations to finite element approximations. Furthermore, we prove strong convergence of the discrete coefficients in L2, and of the associated discrete states in the norm of the oberservation space. Finally, assuming a projected source condition we prove error estimates for the discrete coefficients.

Joint work with Klaus Deckelnick (Magdeburg).

### Hermann G. Matthies (TU Braunschweig)

**Factorisations, Tensor Methods, and Inverse Problems**

Dienstag, den 10.06.2014, 16.15 Uhr in MA 313

Abstract:

Parametric problems typically involve models of mappings from the parameter space into some vector space. Such maps can be associated to certain linear operators, and an analysis of the factorisations of these operators is shown to be intimately connected with different representations of the parametric map. This in turn leads to tensor representation and approximations. These tools are then used both in the forward problem as well as in the identification of the parameters from observation.

Parameter identification involves the observation of a function of the state of some system – usually described by a PDE – which depends on some unknown parameters. The mapping from parameter to observable is commonly not invertible, which causes the problem to be ill-posed. In a probabilistic setting the knowledge prior to the observation is encoded in a probability destribution which is updated according to Bayes ́s rule through the observation, equivalent to a conditional (conditioned on the observation) expectation. The conditional expectation is equivalent to a minimisation of a functional, hence an optimisation problem. To perform the update one has to solve the forward problem, propagating the parameter distribution to the forecast observable. The difference with the real observation leads to the update. It is common to change the underlying measure in the update, but here we update – which is the minimiser – directly the random variables describing the parameters, thus changing the parameter distribution only implicitly. This is achieved by a functional approximation of the random variables involved. The solution of the forward problem can be addressed more efficiently through the use of tensor approximations. We show that the same is true for the inverse problem. Both the computation of the update map – a “filter” – and the update itself can be sped up considerably through the use of tensor approximation methods. The computations will be demonstrated on some examples from continuum mechanics, for linear as well as highly non-linear systems like elasto-plasticity.

### Lucy Weggler (HU Berlin)

**Advances in High Order Boundary Element Methods**

Dienstag, den 17.06.2014, 16.15 Uhr in MA 313

Abstract:

The use of high order approximation methods is very effective in achieving high accuracy numerical simulations while keeping the number of unknowns moderate, in particular for piecewise smooth solutions of partial differential equations. The numerical and theoretical studies for this kind of methods began in the late 70s - early 80s. Since that time high order discretization schemes are getting increasingly popular in many practical applications such as fluid dynamics, structural mechanics, electromagnetics, acoustics, etc. Nowadays, one can say that the high order methods are an established field of research in the finite element community [1, 2]. In what concerns the boundary element community, most recent publications show that theory and numerics of the high order boundary element methods are getting more and more interesting [3, 4, 5, 6].

This talk is concerned with the development and application of high order boundary elements as presented in [7]. After shortly recalling the theoretical results on the high order convergence rates for Galerkin solutions, the key ideas behind the high order boundary element implementation are discussed. At the one hand, this is the abstract relation between energy spaces and trace spaces that appear in variational formulations of elliptic and Maxwell boundary value problems. On the other hand, this is a general access to the definition of curved element shapes that go along well with the high order basis functions needed to discretize the variational formulations resulting from a boundary integral equation. In the second part of this talk we consider the problem of electromagnetic scattering at the perfect electric conductor. Numerical results for the high order boundary element methods are presented. Our tests bring awareness of the necessity to enable a high order description of the geometry. This, in turn, gives rise to ongoing work on theoretical and practical tasks that come into play here, i.e., the approximation theory of finite-dimensional spaces of tangential vector fields on curved manifolds or isogeometric analysis in general.

References

[1] Demkowicz, L., Computing with hp-adaptive finite elements. Vol. 1, Chapman & Hall/CRC Applied Math. and Nonlin. Science Series, 2007.

[2] L. Demkowicz, J. Kurtz, D. Pardo, M. Paszynski, W. Rachowicz and A. Zdunek, Computing with hp-adaptive finite elements. Vol. 2, Chapmen & Hall/CRC Applied Math. and Nonlin. Science Series, 2008.

[3] E.P. Stephan, M. Maischak and F. Leydecker, An hp-adaptive fem/bem coupling method for electromagnetic problems, Comput. Mech, Vol. 39, 2007.

[4] M. Maischak and E.P. Stephan, Adaptive hp-version of boundary element methods for elastic contact problems, Comput. Mech, Vol. 39, 2007.

[5] S. Sauter and Ch. Schwab, Boundary element methods, Vol. 39, Comp. Math., Springer, 2011 (translated from the German original published in 2004).

[6] A. Bespalov, N. Heuer and R. Hiptmair, Convergence of the Natural hp-BEM for the Electric Field Integral Equation on Polyhedral Surfaces., J. Numer. Anal., Vol 48, 2010.

[7] L. Weggler, High Order Boundary Element Methods, Dissertation, Saarland University, 2011.

### Martin Ehler (U Wien)

**Signal reconstruction from magnitude measurements**

Dienstag, den 24.06.2014, 16.15 Uhr in MA 313

Abstract:

Many optical physics experiments lead to phaseless measurements, so that a signal must be reconstructed from magnitude measurements alone. By using more mathematical terms, this can be formulated as a ``simple’’ linear algebra problem. We have or choose a few linear subspaces and aim to determine a vector from the norms of its orthogonal projection onto these subspaces. It turns out that this type of phase retrieval problem is much more complicated than it appears at first sight and is currently under massive investigation in some mathematical communities. Questions and problems are centered around how to choose subspaces, how many subspaces are necessary in relation to the ambient dimension, and how to design an efficient algorithm to reconstruct a vector from its magnitude subspace components. We shall answer few open problems in this field, and the mathematical ingredients involve (at a basic level) harmonic analysis, optimization, and algebraic geometry.

### Kees Vuik (TU Delft)

**Fast and robust iterative solvers for the Helmholtz equation**

Dienstag, den 1.07.2014, 16.15 Uhr in MA 313

Abstract:

Many wave phenomena are well described by the wave equation. When the considered wave has a fixed frequency the wave equation is mostly re-written in the frequency domain which results in the Helmholtz equation. It also possible to approximate the time domain solution with a summation of solutions for several frequencies. Applications are the propagation of sound, sonar, seismics, and many more. We take as an example the search for oil and gas using seismics. It is well known that an increase of the frequency leads to a higher resolution, so more details of the underground become visible.

The Helmholtz equation in its most simple form is a combination of the symmetric positive Poisson operator and a negative constant, the so-called wave number, multiplied with the identity operator. In order to find the solution in a complicated domain a discretization has to be done. There are two characteristic properties of the discretized system:

- the product of the wave number and the step size should be smaller than a given constant

- if the wavenumber increases the operator has more and more negative eigenvalues.

If damping is involved the operator also has a part with an imaginary value. The resulting matrix is however not Hermitian.

In the years around 2000 no good iterative solvers are known. The standard approaches: Krylov and multi-grid break down if the wavenumber increases. Around 2005 a new preconditioner based on the shifted Laplace preconditioner was proposed, which leads to a class of fast and robust Helmholtz solvers. It appears that the amount of work increases linearly with the wavenumber. At this moment, this is the method of choice to solve the Helmholtz equation. Various papers have appeared to analyze the good convergence behavior. Recently a multi level Krylov solver has been proposed that seems to be scalable, which means that the number of iterations is also independent of the wavenumber. An analysis of this method is given and recent results for industrial problems are given.

References:

Y.A. Erlangga and C.W. Oosterlee and C. Vuik, A Novel Multigrid Based Preconditioner For Heterogeneous Helmholtz Problems, SIAM J. Sci. Comput.,27, pp. 1471-1492, 2006, ta.twi.tudelft.nl/nw/users/vuik/papers/Erl06OV.pdf

Y.A. Erlangga and R. Nabben, On a multilevel Krylov method for the Helmholtz equation preconditioned by shifted Laplacian, Electronic Transactions on Numerical Analysis, 31, pp. 403-424, 2008

A.H. Sheikh and D. Lahaye and C. Vuik, On the convergence of shifted Laplace preconditioner combined with multilevel deflation Numerical Linear Algebra with Applications, 20, pp. 645-662, 2013, ta.twi.tudelft.nl/nw/users/vuik/papers/She13LV.pdf

### Yoel Shkolnisky (Tel Aviv U)

**Viewing Directions Estimation in cryo-EM Using Synchronization**

Dienstag, den 8.07.2014, 16.15 Uhr in MA 313

Abstract:

A central stage in recovering the structure of large proteins (3D density maps) from their 2D cryo-electron microscopy (cryo-EM) images, is to determine a three-dimensional model of the protein given many of its 2D projection images. The direction from which each image was taken is unknown, and the images are small and extremely noisy. The goal is to determine the direction from which each image was taken, and then to combine the images into a three-dimensional model of the molecule.

We present an algorithm for determining the viewing directions of all cryo-EM images at once, which is robust to extreme levels of noise. The algorithm is based on formulating the problem as a synchronization problem, that is, we estimate the relative spatial configuration of pairs of images, and then estimate a global assignment of orientations that satisfies all pairwise relations. Information about the spatial relation of pairs of images is extracted from common lines between triplets of images. These noisy pairwise relations are combined into a single consistent orientations assignment, by constructing a matrix whose entries encode the pairwise relations. This matrix is shown to have rank 3, and its non-trivial eigenspace is shown to reveal the projection orientation of each image. In particular, we show that the non-trivial eigenvectors encode the rotation matrix that corresponds to each image.

No prior knowledge is required.

This is a joint work with Amit Singer from Princeton University.

### K.C. Sivakumar (IIT Madras)

**Moore-Penrose inverse positivity of interval matrices**

Dienstag, den 15.07.2014, 16.15 Uhr in MA 313

Abstract:

For real m-by-n matrices A, B with A <= B, define J(A,B) as the set of real m-by-n matrices C such that A <= C <= B, where the order is the component wise order. A rather well known result among researchers in interval mathematics is that if A and B are both invertible and that their inverses are nonnegative, then every C in J(A,B) is invertible and its inverse is nonnegative. In this talk, I would like to present newly published results (jointly with Rajesh Kannan) on a generalization where the nonnegativity of the usual inverse is replaced by the nonnegativity of the Moore-Penrose inverse.

I will try and keep the talk accessible to graduate students.

### Raymond Chan (Chinese U Hong Kong)

**Point-spread function reconstruction in ground-based astronomy**

Dienstag, den 22.07.2014, 16.15 Uhr in MA 313

Abstract:

Ground-based astronomy refers to acquiring images of objects in outer space via ground-based telescopes. Because of atmospheric turbulence, images so acquired are blurry. One way to estimate the unknown blur or point spread function (PSF) is by using natural or artificial guide stars. Once the PSF is known, the images can be deblurred using well-known deblurring methods.

Another way to estimate the PSF is to make use the aberration of wavefronts received at the telescope, i.e., the phase, to derive the PSF. However, the phase is not readily available; instead only its low-resolution gradients can be collected by wavefront sensors. In this talk, we will discuss how to use regularization methods to reconstruct high-resolution phase gradients and then use them to recover the phase and then the PSF in high accuracy.

Our model can be solved efficiently by alternating direction method of multiplier whose convergence has been well established. Numerical results will be given to illustrate that our new model is efficient and give more accurate estimation for the PSF.