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## Absolventen-Seminar • Numerische Mathematik

Verantwortliche Dozenten: | Prof. Dr. Christian Mehl, Prof. Dr. Volker Mehrmann |
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Koordination: | Ines Ahrens |

Termine: | Do 10:00-12:00 in MA 376 |

Inhalt: | Vorträge von Bachelor- und Masterstudenten, Doktoranden, Postdocs und manchmal auch Gästen zu aktuellen Forschungsthemen |

# Rückblick

- Absolventen Seminar SS 19
- Absolventen Seminar WS 18/19
- Absolventen Seminar SS 18
- Absolventen Seminar WS 17/18
- Absolventen Seminar SS 17
- Absolventen Seminar WS 16/17
- Absolventen Seminar SS 16
- Absolventen Seminar WS 15/16
- Absolventen Seminar SS 15
- Absolventen Seminar WS 14/15
- Absolventen Seminar SS 14
- Absolventen Seminar WS 13/14
- Absolventen Seminar SS 13
- Absolventen Seminar WS 12/13
- Absolventen Seminar SS 12
- Absolventen Seminar WS 11/12

### Philipp Krah (TU Berlin)

Donnerstag, 23. Januar 2020

**Wavelet Adaptive Proper Orthogonal Decomposition with Application to Insects Flight**

In this talk I will present some new results of the wavelet adaptive proper orthogonal decomposition (wPOD). Given numerical or experimental data, U = {u(x, μ_1 ), . . . , u(x, μ_N )} with u(·, μ) : Ω → R^K, the wPOD combines a sparse representation of u in the spatial domain Ω ⊂ R^d (d ∈ {2, 3}), using wavelet adaptation methods, with the proper orthogonal decomposition, a well known model order reduction technique to approximate the μ-dependence with a few basis functions φ_n(x). The framework is well suited for experimental methods like Particle Image Velocimetry or Direct Numerical Simulations which are highly resolved in space and sampled with a moderate number of parameters μ.

In my talk, I will introduce the snapshot POD, which is the foundation of my algorithm and explain the wavelet adaptation techniques. Furthermore, I will explain the error estimation and present numerical results for the flow around a cylinder (d=2) and the flight of a bumblebee (d=3).

### Karim Cherifi (MPI Magdeburg)

Donnerstag, 16. Januar 2020

**Data driven Port Hamiltonian realizations **

Port Hamiltonian systems have gained a lot of attention in recent years due their interesting properties in modelling and control. They are particularly interesting for systems interconnection since they preserve their port Hamiltonian structure. However, to be able to benefit from this structure, one has to model the system in Port Hamiltonian framework. In this talk, we focus on the construction of minimal realizations of linear time-invariant (LTI) port-Hamiltonian (pH) systems.The goal is to be able to compute the port Hamiltonian minimal realization directly from time domain input/output data of the system.

### Benjamin Unger (TU Berlin)

Donnerstag, 09. Januar 2020

**Nonlinear Galerkin Model Reduction for Systems with Multiple Transport Velocities**

We propose a new model reduction framework for problems that exhibit transport phenomena. As in the moving finite element method (MFEM), our method employs time-dependent transformation operators and, especially, generalizes MFEM to arbitrary basis functions. The new framework is suitable to obtain a low-dimensional approximation with small errors even in situations when the Kolmogorov n-widths do not decay exponentially. In such cases, classical model order reduction techniques require much higher dimensions for a similar approximation quality. In this talk, we discuss the existence of optimal modes and the construction of the reduced order model. If time permits, we discuss a-posteriori error estimation and the close connection to the symmetry reduction framework. The talk describes joint work with F. Black und P. Schulze (both TU Berlin).

### Attila Karsai (TU Berlin)

Donnerstag, 19. Dezember 2019

**Computation of the Distance to Instability for Large Scale Systems**

Although dissipative Hamiltonian systems often are asymptotically stable in theory, in practice truncation and model errors can introduce perturbations such that this property is lost while the DH structure is kept. Without asymptotic stability, arbitrarily small perturbations can make these systems unstable. To cope with this problem, the stability analysis must focus on robust stability. In this talk, an overview of the computation of the distance to instability of dissipative Hamiltonian systems with focus on large scale systems is given. Further, techniques to speed up the computation are presented.

### Benjamin Unger (TU Berlin)

Donnerstag, 19. Dezember 2019

**Semi-explicit Discretization Schemes for Weakly-Coupled Elliptic-Parabolic Problems**

We study the time-discretization of an elliptic-parabolic problem that is weakly coupled. This setting includes poroleasticity, thermoelasticity, as well as multiple-network models used in medical applications. We propose a semi-explicit Euler scheme in time combined with a finite element discretization in space, which decouples the system such that each time step requires the solution of two small and well-structured linear systems rather than the solution of one large system. The decoupling improves the computational efficiency without decreasing the convergence rates. Our convergence proof is based on an interpretation of the scheme as an implicit method applied to a constrained partial differential equation with delay term. Here, the delay time equals the used step size. This connection also allows a deeper understanding of the weak coupling condition, which we accomplish to quantify explicitly.

### Christoph Zimmer (TU Berlin)

Donnerstag, 12. Dezember 2019

**Exponential Integrators for Semi-Linear Parabolic Problems with Linear Constraints **

Exponential integrators provide a powerful tool for the time integration of spatial discretized partial differential equations (PDEs), by allowing large time steps even for very restrictive CFL conditions. On the other hand, the class of PDEs with additional underling constraints (PDAEs) includes applications such as PDEs with dynamical boundary conditions or the incompressible Navier-Stokes equations.

In this talk, we construct and analyze exponential integrators for semi-linear parabolic PDAEs. Starting with semi-linear ordinary differential equations we explain the main idea behind exponential integrators. Afterwards, we extend this idea to differential-algebraic equations and PDAEs. The resulting schemes only require the solution of linear stationary saddle point problems in each time step. Further, no linearization steps or regularizations of the transient system are needed. The talk concludes with numerical examples.

This is joint work with Robert Altmann.

### Paul Schwerdtner (TU Berlin)

Donnerstag, 12. Dezember 2019

**Robust Control for Large Sparse Systems**

We present our ongoing work on the fixed-order robust controller synthesis problem for large and sparse systems.

Fixed-order methods in controller design use gradient-based optimization to compute controllers, that minimize the H-infinity norm of the resulting closed-loop transfer function. This requires many computations of the H-infinity norm of the different closed-loop transfer functions which is computationally demanding in the large scale case.

We show, how the recently developed software linorm_subsp for the computation of the H-infinity norm can be extended when used within an optimization loop to design fixed-order controllers, efficiently.

However, linorm_subsp only converges to a local maximum a given transfer function. Hence, the H-infinity norm is not always computed correctly. Therefore, we propose to complement linorm_subsp with global certification to circumvent this problem and give insights into the implementation of a global certificate.

### Rebekka Beddig (TU Berlin)

Donnerstag, 05. Dezember 2019

**H_2 x L_inf-optimal model reduction**

In this talk, we discuss H_2 x L_inf-optimal model reduction of parametric linear time-invariant systems. The H_2 x L_inf error is defined as the maximum H_2-error in the transfer function within a feasible parameter domain. We start with the computation of the H_2 x L_inf-norm using Chebychev interpolation. The next step is to minimize the error with nonsmooth constrained optimization. For the optimization process we use a gradient with respect to the matrix elements of the reduced order model. To obtain an asymptotically stable reduced system we include a stability constraint. Numerical experiments illustrate this method.

### Volker Mehrmann (TU Berlin)

Donnerstag, 28. November 2019

**Stability analysis of dissipative Hamiltonian differential-algebraic systems**

Port-Hamiltonian differential-algebraic systems are an important class of control systems that arise in all areas of science and engineering. When the system is linearized arround a stationary solution one gets a linear port-Hamiltonian differential-algebraic system. Despite the fact that the system looks very unstructured at first sight, it has remarkable properties. Stability and passivity are automatic, Jordan structures for purely imaginary eigenvalues, eigenvalues at infnity, and even singular blocks in the Kronecker canonical form are very restricted. We will show several results and then apply them to the brake squeal problem.

### Julianne Chung (TU Berlin)

Donnerstag, 14. November 2019

**Computational Methods for Large and Dynamic Inverse Problems**

In this talk, we describe efficient methods for uncertainty quantification for large, dynamic inverse problems. The first step is to compute a MAP estimate, and for this we describe efficient, iterative, matrix-free methods based on the generalized Golub-Kahan bidiagonalization. These methods can address ill-posedness and can handle many realistic scenarios, such as in passive seismic tomography or dynamic photoacoustic tomography, where the underlying parameters of interest may change during the measurement procedure. The second step is to explore the posterior distribution via sampling. We use the generalized Golub-Kahan bidiagonalization to derive an approximation of the posterior covariance matrix for "free" and describe preconditioned Lanczos methods to efficiently generate samples from the posterior distribution.

### Matthias Chung (TU Berlin)

Donnerstag, 14. November 2019

**Sampled Limited Memory Methods for Least Squares Problems with Massive Data**

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 consider randomized row-action methods that can be used to approximate the solution. We introduce a sampled limited memory row-action 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.

### Onkar Jadhav (TU Berlin)

Donnerstag, 31. Oktober 2019

**Model order reduction for parametric high dimensional interest rate models in the analysis of financial risk**

The European Parliament has introduced regulations (No 1286/2014) on packaged retail investment and insurance products (PRIIPs). According to this regulation, PRIIP manufacturers must provide a key information document (KID) describing the risk and the possible returns of these products. The formation of a KID requires expensive valuations rising the need for efficient computations. To perform such valuations efficiently, we establish a model order reduction approach based on a proper orthogonal decomposition (POD) method. The study involves the computations of high dimensional parametric convection-diffusion reaction partial differential equations. POD requires to solve the high dimensional model at some parameter values to generate a reduced-order basis. We propose a greedy sampling technique for the selection of the sample parameter set that is analyzed, implemented, and tested on the industrial data. The results obtained for the numerical example of a floater with cap and floor under the Hull-White model indicate that the MOR approach works well for the short-rate models.

### Christian Mehl (TU Berlin)

Donnerstag, 31. Oktober 2019

**Distance problems for dissipative Hamiltonian pencils and related matrix polynomials **

We investigate the distance problems to singularity, higher index, and instability for dissipative Hamiltonian systems by developing a general framework for matrix polynomials with a special symmetry and positivity structure. As we will show, the mentioned distances can then be formulated as the distance to a common kernel of some of the coefficients of the given matrix polynomial.

### Paula Klimczok (TU Berlin)

Donnerstag, 24. Oktober 2019

**Classification of Two-Variable Linear Differential Equations with Large Delays**

In this talk we will discuss the stability of linear differential equations of the form x’(t)=Ax(t)+Bx(t−τ) with a discrete delay τ and constant A and B. For a large delay τ the eigenvalues can be approximated by two sets: the asymptotic strongly unstable spectrum and the asymptotic continuous spectrum. We will characterise these sets in the case of A, B ∈ R^{2×2} and give conditions for the stability. Further, we will take a look on the computation of the eigenvalues.