### Inhalt des Dokuments

## 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 14.04.15 | 16:15 | MA 313 | Felix Krahmer (TU München) | Sparsity models in compressed sensing (Abstract) | G. Kutyniok |

Di 28.04.15 | 16:15 | MA 313 | Sonja Cox (U Amsterdam) | Numerical simulations for stochastic PDEs (Abstract) | R. Kruse |

Di 5.05.15 | 16:15 | MA 313 | Massimo Fornasier (TU München) | Sparse Mean-Field Optimal Control (Abstract) | G. Kutyniok |

Di 12.05.15 | 16:15 | MA 313 | Elias Jarlebring (KTH Stockholm) | The infinite Arnoldi method and applications (Abstract) | A. Miedlar V. Mehrmann |

Di 19.05.15 | 16:15 | MA 313 | Robert Lipton (Louisiana State U) | Cohesive Dynamics and Fracture (Abstract) | E. Emmrich |

Di 2.06.15 | 16:15 | MA 313 | Zlatko Drmac (U Zagreb) | On robust implementation of vector fitting (Abstract) | V. Mehrmann |

Di 9.06.15 | 14:45 | MA 313 | Arye Nehorai (Washington U in St. Louis) | Computable Performance Analysis of Sparse Recovery (Abstract) | G. Kutyniok |

Di 9.06.15 | 16:15 | MA 313 | Martin Stoll (MPI Magdeburg) | Preconditioners for time-dependent PDE-constrained optimization problems (Abstract) | V. Mehrmann A. Miedlar |

Fr 12.06.15 | 10:15 | MA 313 | Enrique Zuazua (BCAM, Bilbao) | Numerical flow control and optimisation (Abstract) | V. Mehrmann |

Di 16.06.15 | 16:15 | MA 313 | Esther Klann (TU Berlin) | Geometric inverse problems with applications in tomography (Abstract) | -- |

Di 23.06.15 | 16:15 | MA 313 | Bernd Sturmfels (UC Berkeley) | Decomposing Tensors into Frames (Abstract) | G. Kutyniok |

Di 30.06.15 | 16:15 | MA 313 | Luca Grubisic (U Zagreb) | Pollution free method for computing eigenvalues and eigenvectors of Fredholm valued operator functions (Abstract) | V. Mehrmann A. Miedlar |

Di 28.07.15 | 16:15 | MA 313 | Vianey Villamizar (BYU Provo) | Local and Global Absorbing Conditions for Wave Scattering in Unbounded Domains (Abstract) | B. Wagner |

### Felix Krahmer (TU München)

**Sparsity models in compressed sensing**

Dienstag, den 14.04.2015, 16.15 Uhr in MA 313

Abstract

The theory of compressed sensing shows that sparse signals, that is, vectors with only a small number of non-vanishing entries, can be recovered from a number of measurements that is considerably less than the signal dimension. Typically the measurements to allow for such guarantees are chosen at random. Considerable efforts have been spent over the last years to study random measurements with additional structural properties imposed by the applications. However, in contrast to generic (for example Gaussian) measurements, such structured random measurement systems are no longer universal, i.e., without additional modifications, the guarantees only hold for certain sparsity bases. This talk discusses two approaches to obtain recovery guarantees when the sparsity basis under consideration does not allow the application of the standard theory. On the one hand one can adjust the sampling distribution (for example in Fourier imaging applications), and on the other hand, one can apply a suitable temporal transform to achieve sparsity in the standard basis for each time instance (for example in photoacoustic tomography applications).

The talk is based on joint works with Rachel Ward and with Michael Sandbichler, Thomas Berer, Peter Burgholzer, and Markus Haltmeier.

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

### Sonja Cox (U Amsterdam)

**Numerical simulations for stochastic PDEs **

Dienstag, den 28.04.2015, 16.15 Uhr in MA 313

Abstract

Various models arising from finance and natural sciences involve stochastic partial differential equations (SPDEs). As the solutions to an SPDE generally cannot be given explicitly, numerical simulations are used to gain insight in their behaviour. In my talk I will explain the challenges encountered here. In particular, I will explain why generally one cannot expect large convergence rates and why non-linear equations pose difficulties that do not occur with deterministic PDEs. Finally, I will explain my recent results concerning approximations to non-linear S(P)DEs.

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

### Massimo Fornasier (TU München)

**Sparse Mean-Field Optimal Control**

Dienstag, den 5.05.2015, 16.15 Uhr in MA 313

Abstract

Starting with the seminal papers of Reynolds (1987), Vicsek et. al. (1995) Cucker-Smale (2007), there has been a flood of recent works on models of self-alignment and consensus dynamics. Self-organization has been so far the main driving concept. However, the evidence that in practice self-organization does not necessarily occur leads to the natural question of whether it is possible to externally influence the dynamics in order to promote the formation of certain desired patterns. Once this fundamental question is posed, one is also faced with the issue of defining the best way of obtaining the result, seeking for the most “economical” manner to achieve a certain outcome. The first part of this talk precisely addresses the issue of finding the sparsest control strategy for finite dimensional models in order to lead the dynamics optimally towards a given outcome. In the second part of the talk we introduce the rigorous limit process connecting finite dimensional sparse optimal control problems with ODE constraints to an infinite dimensional optimal control problem with a constraint given by a system of ODE for the leaders coupled with a PDE of Vlasov-type, governing the dynamics of the probability distribution of the followers. The technical derivation of the sparse mean-field optimal control is realized by the simultaneous development of the mean-field limit of the equations governing the followers dynamics together with the Gamma-limit of the finite dimensional sparse optimal control problems. Additionally we derive corresponding first order optimality conditions for the infinite dimensional optimal control problem in the form of Hamiltonian ﬂows in the Wasserstein space of probability measures, which correspond to natural limits of the finite dimensional Pontryagin Maximum principles. We conclude the talk by mentioning recent results in sparse optimal control of high-dimensional dynamical systems.

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

### Elias Jarlebring (KTH Stockholm)

**The infinite Arnoldi method and applications**

Dienstag, den 12.05.2015, 16.15 Uhr in MA 313

Abstract

The infinite Arnoldi method [A linear eigenvalue algorithm for the nonlinear eigenvalue problem, Numer. Math., 2012] is a recent approach which is equivalent to the Arnoldi method, but applicable to different types of nonlinear problems. It was originally developed for the nonlinear eigenvalue problem, with an analytic dependence on the spectral parameter, and we have recently shown that the construction can also be used to solve certain types of ODEs in the setting exponential integrators [An infinite Arnoldi exponential integrator for inhomogeneous linear ODEs, Koskela, J., arxiv preprint, 2015]. We present improvements, e.g., structure exploiting restarting, tensor representations, low-rank exploitation, for the infinite Arnoldi method, as well as different applications.

### Robert Lipton (Louisiana State U)

**Cohesive Dynamics and Fracture**

Dienstag, den 19.05.2015, 16.15 Uhr in MA 313

Abstract

Dynamic brittle fracture is a multiscale phenomena operating across a wide range of length and time scales. Apply enough stress or strain to a sample of brittle material and one eventually snaps bonds at the atomistic scale leading to fracture of the macroscopic specimen. At present there is a growing demand for new fracture models capable of predicting complex fracture patterns inside materials used in modern infrastructure. The peridynamic formulation introduced in the work of Silling 2000 is a promising method for modeling free crack propagation. Here we work in the peridynamic formulation and introduce a new type of nonlocal, nonlinear, cohesive continuum model for assessing the deformation state inside a cracking body. In this model short-range forces between material points are initially elastic and then become unstable and soften beyond a critical relative displacement. The dynamics inside the deforming body selects whether a material point lies inside or outside the "process zone" associated with nonlinear behavior corresponding to softening. This is in contrast to a classic cohesive zone fracture model that collapses the process zone onto predetermined surfaces and assumes linear elastic deformation away from these surfaces. Here the natural length scale that controls the size of the process zone is the radius of nonlocal interaction between material points. An explicit inequality is identified showing how the length scale of nonlocal interaction controls the volume of the process zone. The volume of the process zone is shown to vanish as the nonlocal interaction distance is decreased to zero. We apply Gamma convergence arguments coming from the theory of image processing to find that the limiting dynamics has an energy density associated with a process zone confined to a surface. Distinguished limits of cohesive evolutions are identified and are found to have both bounded linear elastic energy and Griffith surface energy. The limit dynamics corresponds to the simultaneous evolution of linear elastic displacement described by the classic wave equation together with a fracture set across which the displacement is discontinuous.

### Zlatko Drmac (U Zagreb)

**On robust implementation of vector fitting**

Dienstag, den 2.06.2015, 16.15 Uhr in MA 313

Abstract

Vector Fitting (VF) is a popular method of constructing rational approximants that provides a least squares fit to frequency response measurements. In an earlier work, we provided an analysis of VF for scalar-valued rational functions and established a connection with optimal H2 approximation. We build on this work and extend the previous framework to include the construction of effective rational approximations to matrix-valued functions, a problem which presents significant challenges that do not appear in the scalar case. Transfer functions associated with multi-input/multi-output (MIMO) dynamical systems typify the class of functions that we consider here.

Others have also considered extensions of VF to matrix-valued functions and related numerical implementations are readily available. However to our knowledge, a detailed analysis of numerical issues that arise does not yet exist. We offer such an analysis including critical implementation details here.

One important issue that arises for VF on matrix-valued functions that has remained largely unaddressed is the control of the McMillan degree of the resulting rational approximant; the McMillan degree can grow very high in the case of large input/output dimensions. We introduce two new mechanisms for controlling the McMillan degree of the final approximant, one based on alternating least-squares minimization and one based on ancillary system-theoretic reduction methods.

Motivated in part by our earlier work on the scalar VF problem as well as by recent innovations for computing optimal H2 approximation, we establish a connection with optimal H2 approximation, and are able to improve significantly the fidelity of VF through numerical quadrature, with virtually no increase in cost or complexity. We provide several numerical examples to support the theoretical discussion and proposed algorithms.

### Arye Nehorai (Washington U in St. Louis)

**Computable Performance Analysis of Sparse Recovery**

Dienstag, den 9.06.2015, 14.45 Uhr in MA 313

Abstract

The last decade has witnessed burgeoning developments in the reconstruction of signals based on exploiting their low-dimensional structures, particularly their sparsity, block-sparsity, and low-rankness. The reconstruction performance of these signals is heavily dependent on the structure of the operating matrix used in sensing. The quality of these matrices in the context of signal recovery is usually quantified by the restricted isometry constant and its variants. However, the restricted isometry constant and its variants are extremely difficult to compute.

We present a framework for analytically computing the performance of the recovery of signals with sparsity structures. We define a family of incoherence measures to quantify the goodness of arbitrary sensing matrices. Our primary contribution is the design of efficient algorithms, based on linear programming and second order cone programming, to compute these incoherence measures. As a by-product, we implement efficient algorithms to verify sufficient conditions for exact signal recovery in the noise-free case. The utility of the proposed incoherence measures lies in their relationship to the performance of reconstruction methods. We derive closed-form expressions of bounds on the recovery errors of convex relaxation algorithms in terms of these measures.

Preceding this talk there will be coffee, tea, and biscuits at 14:30 in room MA 315 - everybody's welcome.

### Martin Stoll (MPI Magdeburg)

**Preconditioners for time-dependent PDE-constrained optimization problems**

Dienstag, den 9.06.2015, 16.15 Uhr in MA 313

Abstract

With the advance of both computers and algorithms many previously studied partial differential equation models have become constraints in an optimization problem. The goal within this framework is to determine the configuration of the governing equations that best fits a desired state or observed measurements. Our aim for this talk is to introduce space-time discretizations that when treated in an all-at-once approach, where we solve for all unknowns in one step, lead to very large and sparse linear systems. These systems will be beyond the reach of any direct method and we therefore discuss preconditioning strategies that can be embedded into iterative solvers. We illustrate their performance on a number of different problems. One disadvantage of this approach is the large storage demand for the space-time vectors. In certain cases this drawback can be overcome by using low-rank techniques, i.e., lifting the curse of dimensionality. Our approach combines recent developments in preconditioning with state-of-the-art solvers for matrix equations and we show how this can also lead to efficient solvers for optimization problems subject to PDEs with uncertain coefficients.

### Enrique Zuazua (BCAM, Bilbao)

**Numerical flow control and optimisation**

Freitag, den 12.06.2015, 10.15 Uhr in MA 313

Abstract

We address the classical optimal control problem of inverse design, aiming to identify the initial source leading to a desired final configuration, in the context of hyperbolic or viscous conservation laws. This issue arises frequently in areas such as aeronautics or water management. We mainly focus on the one-dimensional case, considering Burgers like equations.

We emphasize the relevance of the accuracy of the numerical scheme employed for forward resolution and to which extent schemes that are too-diffusive may affect the efficiency of the computational inversion process. To illustrate this fact we discuss the application to sonic-boom minimisation in supersonic aircrafts, a topic in which, naturally, one is led to analyse the control problem in long time horizons. We shall also discuss the links with the classical turnpike property.

Finally we shall empasize the fact that, due to the possible presence of shocks, the inverse design problem lacks of uniqueness and we shall discuss some possible paths to recover some of the most significant possible ones. We will end up pointing towards some perspectives of future research.

This lecture will be accessible to a large public with broad interests in Fluid Mechanics, PDE, Numerics, Control and Optimisation. Unnecessary technicalities will be avoided.

### Esther Klann (TU Berlin)

**Geometric inverse problems with applications in tomography**

Dienstag, den 16.06.2015, 16.15 Uhr in MA 313

Abstract

In inverse problems one wants to find unknown parameters or structures by indirect measurements. Typical inverse problems occur in medical imaging, e.g., tomography. The task is to find the inner structure of an object, e.g., a human torso, from line integrals of some form of radiation that traveled through the object. In geometric inverse problems one wants to recover this inner structure also in terms of geometric information, i.e., the number of different interior objects (bone, liver, lung, etc.), their location and their shape. We present a Mumford-Shah type functional for the simultaneous reconstruction and segmentation of an object from tomography data. The minimization of the functional is realized using shape sensitivity analysis and carried out in the level-set framework. We show reconstructions and/or segmentations for several tomographic applications (SPECT/CT, limited angle tomography, region of interest CT). We also give a brief theoretical analysis regarding existence and convergence of minimizers of the Mumford-Shah type functional.

### Bernd Sturmfels (UC Berkeley)

**Decomposing Tensors into Frames**

Dienstag, den 23.06.2015, 16.15 Uhr in MA 313

Abstract

A symmetric tensor of small rank decomposes into a configuration of only few vectors. We study the variety of tensors for which this configuration is a unit norm tight frame. This is joint work with Luke Oeding and Elina Robeva, aimed at bridging the gap between frame theory and applied algebraic geometry.

### Luca Grubisic (U Zagreb)

**Pollution free method for computing eigenvalues and eigenvectors of Fredholm valued operator functions **

Dienstag, den 30.06.2015, 16.15 Uhr in MA 313

Abstract

In this talk we present methods to compute eigenvalues of a nonlinear eigenvalue problem based on numerical integration of the Cauchy integral of the generalized resolvent. With the help of numerical integration we construct a bounded operator with two distinct well separated spectral components. A cluster of eigenvalues around one and a cluster of eigenvalues around zero. Few sweeps of subspace iteration, applied directly to the bounded operator, yield improved approximate eigenvectors. Approximate eigenvalues can then be recovered by a postprocesing procedure (cf. work by WJ Beyn, and E. Polizzi). We then use perturbation theory to assess the quality of the approximation and define the convergence threshold. Let us note that the perturbation theoretic approach allows for estimating eigenvalues in gaps of essential spectrum. Numerical examples will be presented to illustrate finer points of the algorithm.

### Vianey Villamizar (BYU Provo)

**Local and Global Absorbing Conditions for Wave Scattering in Unbounded Domains**

Dienstag, den 28.07.2015, 16.15 Uhr in MA 313

Abstract

One of the main difficulties encountered by the application of numerical methods based on finite difference or finite elements to wave scattering problems is the efficient handling of the unboundedness of the physical domain. A common approach is to introduce an artificial boundary B and then define an appropriate boundary condition on B. This boundary condition should be such that the far-field behavior of the scattered field is maintained by avoiding spurious reflection from the artificial boundary. Some of the most commonly used boundary conditions on B include local absorbing boundary conditions based on annihilating differential operators, perfectly matched layers, and global absorbing conditions. I will discuss their implementation and computationally efficiency for single and multiple acoustic scattering including my work on Dirichlet-to-Neumann (DtN) non-reflecting boundary condition and local absorbing conditions based on exact representation of the scattered waves in the far-field.