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## Kolloquium der Arbeitsgruppe Modellierung • Numerik • Differentialgleichungen

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

Koordination: | Dr. Janusz
Ginster |

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

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

Kontakt: | kolloquium-mnd@math.tu-berlin.de
[1] |

## Terminplanung

Datum date | Zeit time | Raum room | Vortragende(r) speaker | Einladende(r) invited by |
---|---|---|---|---|

15.10.2019 | 16:15 | MA 313 | ||

22.10.2019 | 16:15 | MA 313 | Stefan Heinrich (Uni
Kaiserslautern) | G.
Bärwolff |

29.10.2019 | 16:15 | MA 313 | ||

05.11.2019 | 16:15 | MA 313 | Federico Poloni
(University of Pisa) | V.
Mehrmann |

12.11.2019 | 16:15 | MA 313 | ||

19.11.2019 | 16:15 | MA 313 | ||

26.11.2019 | 16:15 | MA 313 | Thomas
Strohmer (UC Davis) | G. Kutyniok |

03.12.2019 | 16:15 | MA 313 | ||

10.12.2019 | 16:15 | MA 313 | Matthias Chung
(Virginia Tech) | V.
Mehrmann |

17.12.2019 | 16:15 | MA 313 | Roland Duduchava
(University of Georgia and Ivane Javakhishvili Tbilisi State
University) | R. Schneider |

07.01.2020 | 16:15 | MA 313 | Daniel B. Szyld (Temple University) | V. Mehrmann |

14.01.2020 | 16:15 | MA 313 | ||

21.01.2020 | 16:15 | MA 313 | ||

28.01.2020 | 16:15 | MA 313 | Evelyn Buckwar (Johannes
Kepler University Linz) | G. Bärwolff |

04.02.2020 | 16:15 | MA 313 | Ruili Zhang (TU Berlin) | V.
Mehrmann |

11.02.2020 | 16:15 | MA 313 | Julianne Chung
(Virginia Tech) | G.
Kutyniok |

# Abstracts zu den Vorträgen

**Stefan Heinrich (Universität
Kaiserslautern): ** * Stochastic integration in various
function classes – algorithms and complexity: *

Abstract [2]

**Frederico Poloni (University of
Pisa): ** *Inverses of quasidefinite matrices in
block-factored form, with an application to control theory (joint work
with P. Benner) : *

We describe an algorithm to
compute the explicit inverse of a dense quasi-definite matrix, i.e., a
symmetric matrix of the form [-B*B^T, A;A^T, C^T*C], with the (1,1)
block negative semidefinite and the (2,2) block positive semidefinite.
The algorithm is a variant of Gauss-Jordan elimination that works on
the low-rank factors B and C directly without ever forming those
blocks. The individual elimination steps amount to a transformation
called principal pivot transform; it was shown in [Poloni, Strabic
2016] how to perform it by working only on A, B, C, and we rely on
that procedure here.

We discuss the stability of the resulting
method, and show how the algorithm (and in particular the produced
low-rank factors) can be of use in control theory, in the context of
the matrix sign iteration, a method used to solve algebraic Riccati
equations.

**Thomas Strohmer (UC
Davis): *** Taming non-convex optimization landscapes
in data analysis *

Nonconvex optimization problems are the
bottleneck in many applications in science and technology. In my talk
I will report on two recent breakthroughs in solving some important
nonconvex optimization problems. The first example concerns blind
deconvolution, a topic that pervades many areas of science and
technology, including geophysics, medical imaging, and communications.
Blind deconvolution is obviously ill-posed and its optimization
landscape is full of undesirable local minima. I will first describe
how I once failed to catch a murderer (dubbed the "graveyard
murderer" by the media), because I failed in solving a blind
deconvolution problem. I will then present a host of new algorithms to
solve such nonconvex optimization problems. The proposed methods come
with theoretical guarantees, are numerically efficient, robust, and
require little or no parameter tuning, thus making them useful for
massive data sets. The second example concerns the classical topics of
data clustering and graph cuts. Organizing data into meaningful groups
is one of the most fundamental tasks in data analysis and machine
learning. While spectral clustering has become one of the most popular
clustering techniques, a rigorous and meaningful theoretical
justification has still been elusive so far. I will discuss a convex
relaxation approach, which gives rise to a rigorous theoretical
analysis of graph cuts. I derive deterministic bounds of finding
optimal graph cuts via a natural and intuitive spectral proximity
condition. Moreover, our theory provides theoretical guarantees for
spectral clustering and for community detection.

**Matthias Chung (Virginia Tech): ** *Sampled
Limited Memory Optimization Methods for Least Squares Problems with
Massive Data *

Emerging fields such as data analytics,
machine learning, and uncertainty quantification heavily rely on
efficient computational methods for solving inverse problems. With
growing model complexities and ever increasing data volumes, state of
the art inference method exceeded their limits of applicability and
novel methods are urgently needed. Hence, new inference method need to
focus on the scalability to large dimension and to address eventual
model complexities.

In this talk, we discuss massive least
squares problems where the size of the forward model matrix exceeds
the storage capabilities of computer memory or the data is simply not
available all at once. We introduce sampled limited memory
optimization method for least squares problems, where an approximation
of the global curvature of the underlying least squares problem is
used to speed up the initial convergence and to improve the accuracy
of iterates. Our proposed methods can be applied to ill-posed inverse
problem, where we establish sampled regularization parameter selection
methods. Numerical experiments on very large superresolution and
tomographic reconstruction examples demonstrate the efficiency of
these sampled limited memory row-action methods. This is joint work
with Julianne Chung, Tanner Slagel, and Luis Tenorio.

**Roland Duduchava (University of Georgia and Ivane
Javakhishvili Tbilisi State University): **

Abstract [3]

**Daniel B. Szyld (Temple
University): ***One- and Two-level Asynchronous Optimized
Schwarz Methods for the solution of PDEs*

Abstract
[4]

**Evelyn Buckwar (Johannes Kepler University
Linz): ** *Splitting methods in Approximate Bayesian
Computation for partially observed diffusion processes : *

Approximate Bayesian Computation (ABC) has become one of the
major tools of likelihood-free statistical inference in complex
mathematical models. Simultaneously, stochastic differential equations
(SDEs) have developed as an established tool for modelling time
dependent, real world phenomena with underlying random effects. When
applying ABC to stochastic models, two major difficulties arise.
First, the derivation of effective summary statistics and proper
distances is particularly challenging, since simulations from the
stochastic process under the same parameter configuration result in
different trajectories. Second, exact simulation schemes to generate
trajectories from the stochastic model are rarely available, requiring
the derivation of suitable numerical methods for the synthetic data
generation. In this talk we consider SDEs having an invariant density
and apply measure-preserving splitting schemes for the synthetic data
generation. We illustrate the results of the parameter estimation with
the corresponding ABC algorithm with simulated data.

**Ruili Zhang (TU Berlin): ** * Classical
Instabilities of Conservative Systems are the Results of Parity-Time
Symmetry Breaking : *

We show that the governing
equations of the classical two-fluid interaction and the
incompressible fluid system are PT-symmetric, and the well-known
Kelvin-Helmholtz instability is the result of spontaneous PT-symmetry
breaking. Specifically, it is shown that the boundaries between the
stable and unstable regions are locations for Krein collisions between
eigenmodes with different Krein signatures. In terms of physics, this
rigorously implies that the system is destabilized when a
positive-action mode resonates with a negative-action mode, and that
this is the only mechanism by which the system can be destabilized. It
is anticipated that this physical mechanism of destabilization is
valid for other collective instabilities in conservative systems in
plasma physics, accelerator physics, and fluid dynamics systems.

**Julianne Chung (Virginia Tech): **
*Advancements in Hybrid Iterative Methods for Inverse Problems
: *

In many physical systems, measurements can only be
obtained on the exterior of an object (e.g., the human body or the
earth's crust), and the goal is to estimate the internal structures.
In other systems, signals measured from machines (e.g., cameras) are
distorted, and the aim is to recover the original input signal. These
are natural examples of inverse problems that arise in fields such as
medical imaging, astronomy, geophysics, and molecular biology.

Hybrid iterative methods are increasingly being used to solve large,
ill-posed inverse problems, due to their desirable properties of (1)
avoiding semi-convergence, whereby later reconstructions are no longer
dominated by noise, and (2) enabling adaptive and automatic
regularization parameter selection. In this talk, we describe some
recent advancements in hybrid iterative methods for computing
solutions to large-scale inverse problems. First, we consider a hybrid
approach based on the generalized Golub-Kahan bidiagonalization for
computing Tikhonov regularized solutions to problems where explicit
computation of the square root and inverse of the covariance kernel
for the prior covariance matrix is not feasible. This is useful for
large-scale problems where covariance kernels are defined on irregular
grids or are only available via matrix-vector multiplication. Second,
we describe flexible hybrid methods for solving l_p regularized
inverse problems, where we approximate the p-norm penalization term as
a sequence of 2-norm penalization terms using adaptive regularization
matrices, and we exploit flexible preconditioning techniques to
efficiently incorporate the weight updates. We introduce a flexible
Golub-Kahan approach within a Krylov-Tikhonov hybrid framework, such
that our approaches extend to general (non-square) l_p regularized
problems. Numerical examples from dynamic photoacoustic tomography,
space-time deblurring, and passive seismic tomography demonstrate the
range of applicability and effectiveness of these approaches.

# Rückblick

- Kolloquium ModNumDiff Sommer 2019 [5]
- Kolloquium ModNumDiff Winter 2018/2019 [6]
- Kolloquium ModNumDiff Sommer 2018 [7]
- Kolloquium ModNumDiff Winter 2017/2018 [8]
- Kolloquium ModNumDiff Sommer 2017 [9]
- Kolloquium ModNumDiff Winter 2016/17 [10]
- Kolloquium ModNumDiff Sommer 2016 [11]
- Kolloquium ModNumDiff Winter 2015/16 [12]
- Kolloquium ModNumDiff Sommer 2015 [13]
- Kolloquium ModNumDiff Winter 2014/15 [14]
- Kolloquium ModNumDiff Sommer 2014 [15]
- Kolloquium ModNumDiff Winter 2013/14 [16]
- Kolloquium ModNumDiff Sommer 2013 [17]
- Kolloquium ModNumDiff Winter 2012/13 [18]
- Kolloquium ModNumDiff Sommer 2012 [19]
- Kolloquium ModNumDiff Winter 2011/12 [20]
- Kolloquium ModNumDiff Sommer 2011 [21]

parameter/en/font2/maxhilfe/id/209161/?no_cache=1&a

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6WxLLBr2AQA%3D%3D&ask_name=KOLLOQUIUM-MND

-kolloquium/Abstracts/heinrich-berlin19.pdf

-kolloquium/Abstracts/Duduchava2019.pdf

-kolloquium/Abstracts/Abstract_Szyld_2020.pdf

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