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## Absolventen-Seminar • Numerische Mathematik

Verantwortliche Dozenten: | Prof. Dr. Christian Mehl, Prof. Dr. Volker Mehrmann |
---|---|

Koordination: | Benjamin Unger, Dr. Matthias Voigt |

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

Inhalt: | Vorträge von Diplomanden, Doktoranden, Postdocs und manchmal auch Gästen zu aktuellen Forschungsthemen |

Datum | Zeit | Raum | Vortragende(r) | Titel |
---|---|---|---|---|

Do 19.04. | 10:15 Uhr | MA 376 | Vorbesprechung | |

Do 26.04. | 10:15 Uhr | MA 376 | Yue Wu | Randomized numerical schemes for (S)ODEs/SPDEs [abstract] |

Matthias Voigt | Balanced Truncation Model Reduction for Systems with Nonzero Initial Condition [abstract] | |||

Do 03.05. | 10:15 Uhr | MA 376 | no seminar | |

Do 10.05. | 10:15 Uhr | MA 376 | no seminar | |

Do 17.05. | 10:15 Uhr | MA 376 | Benjamin Peherstorfer | Multifidelity methods and context-aware model reduction for Monte Carlo estimation and beyond [abstract] |

Murat Manguoglu | Incomplete Cholesky Factorization Based Sparse Matrix Partitioning and Its Applications [abstract] | |||

Do 24.05. | 10:15 Uhr | MA 376 | Christian Mehl | The singular generalized eigenvalue problem: perturb it! [abstract] |

Josip Tambaca | A new Naghdi type shell model [abstract] | |||

Do 31.05. | 10:15 Uhr | MA 376 | Robert Altmann | A Mathematical View on Anderson Localization [abstract] |

Christoph Zimmer | ε-Expansion of Constrained Hyperbolic PDEs [abstract] | |||

Do 07.06. | 10:15 Uhr | MA 376 | Jeroen Stolwijk | Sensitivity Analysis for the discretised Euler Equations; A Case Study on a Y-Shaped Gas Network [abstract] |

Ines Ahrens | A generalization of the Sigma method for DAEs with delay [abstract] | |||

Do 14.06. | 10:15 Uhr | MA 376 | Daniel Bankmann | On sensitivities of strangeness-free DAEs [abstract] |

Riccardo Morandin | Runge-Kutta methods for port-Hamiltonian differential-algebraic equations [abstract] | |||

Do 21.06. | 10:15 Uhr | MA 376 | Jesse Scherwitz | On a Projection Based Index Reduction Method for Differential-Algebraic Equations in Electrical Power Systems [abstract] |

Volker Mehrmann | Control and model reduction for flow problems [abstract] | |||

Do 28.06. | 10:15 Uhr | MA 376 | Heinrich Ellmann | Solution of discrete time optimal control problems using the palindromic Laub trick [abstract] |

Julian Kern | Polar Decompositions with Commuting Factors in Indefinite Inner Product Spaces: Another Approach to Uniqueness [abstract] | |||

Do 05.07. | 10:15 Uhr | MA 376 | Felix Black | A comparison of shifted proper orthogonal decomposition and symmetry reduction [abstract] |

Pia Lutum | Numerical Computation of the Real Structured Stability Radius [abstract] | |||

Do 12.07. | 10:15 Uhr | MA 376 | Marine Froidevaux | Contour integral methods for nonlinear eigenvalue problems [abstract] |

Sofia Bikopoulou | Algorithm-Based Fault Tolerance for Iterative Methods on High Performance Computing Systems [abstract] | |||

Do 19.07. | 10:15 Uhr | MA 376 | Arbi Moses Badlyan | - canceled - |

Philipp Schulze | Structure-Preserving Model Order Reduction of the Linear Advection-Diffusion Equation [abstract] |

# Rückblick

- 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

### Yue Wu (TU Berlin)

Donnerstag, 26. April 2018

**Randomized numerical schemes for (S)ODEs/SPDEs**

A wide range of applications, for instance, in the engineering and physical sciences as well as in computational finance is still spurring the demand for the development of more efficient algorithms and their theoretical justification. In particular, the current focus lies on the approximation of ODEs/S(P)DEs which cannot be treated by standard methods found in textbook.

We, therefore, first developed two randomized explicit Runge–Kutta schemes for ordinary differential equations (ODEs) with time-irregular coeffcient functions. In particular, the methods are applicable to ODEs of Carathéodory type, whose coeffcient functions are only integrable with respect to the time variable but are not assumed to be continuous. An important ingredient in the analysis are corresponding error bounds for the randomized Riemann sum quadrature rule.

It is demanding to approximate numerical solutions of non-autonomous SDEs where the standard smoothness and growth requirements of standard Milstein-type methods are not fulfilled. In the case of a non-differentiable drift coefficient function f, we proposed a drift-randomized Milstein method to achieve a higher order approximation and discussed the optimality of our convergence rates.

We also pushed the idea to the numerical solution of non-autonomous semilinear stochastic evolution equations (SEEs) driven by an additive Wiener noise. Usually quite restrictive smoothness requirements are imposed in order to achieve high order of convergence rate. It turns out that the resulting method converges with a higher rate with respect to the temporal discretization parameter without requiring any differentiability of the nonlinearity. Our approach also relaxes the smoothness requirements of the coefficients with respect to the time variable considerably.

### Matthias Voigt (TU Berlin)

Donnerstag, 26. April 2018

**Balanced Truncation Model Reduction for Systems with Nonzero Initial Condition**

Balanced truncation is one of the most established methods for model reduction of linear time-invariant systems. However, this method may give reduced models that result in large errors if the initial value is not equal to zero. We propose a new way of reformulating the system by shifting the state by an L_{2}-function and extending the input vector. Than classical balanced truncation can be applied and a parameter-dependent error bound is obtained. We show how reduced-order models can be practically constructed and how to efficiently determine an optimal error bound. We conclude the talk with numerical experiments and a comparison to other approaches. This is joint work with Christian Schröder.

### Benjamin Peherstorfer (University of Wisconsin-Madison)

Donnerstag, 17. Mai 2018

**Multifidelity methods and context-aware model reduction for Monte Carlo estimation and beyond**

Outer-loop applications, such as optimization, control, uncertainty quantification, and inference, form a loop around a computational model and evaluate the model in each iteration of the loop at different inputs, parameter configurations, and coefficients. Using a high-fidelity model in each iteration of the loop guarantees high accuracies but often quickly exceeds available computational resources because evaluations of high-fidelity models typically are computationally expensive. Replacing the high-fidelity model with a low-cost, low-fidelity model can lead to significant speedups but introduces an approximation error that is often hard to quantify and control. We introduce multifidelity methods that combine, instead of replace, the high-fidelity model with low-fidelity models. The overall premise of our multifidelity methods is that low-fidelity models are leveraged for speedup while occasional recourse is made to the high-fidelity model to establish accuracy guarantees. The focus of this talk is the multifidelity Monte Carlo method that samples low- and high-fidelity models to accelerate the Monte Carlo estimation of statistics of the high-fidelity model outputs. Our analysis shows that the multifidelity Monte Carlo method is optimal in the sense that the mean-squared error of the multifidelity estimator is minimized for the available computational resources. We provide a convergence analysis, prove that adapting the low-fidelity models to the Monte Carlo sampling in a context-aware sense reduces the mean-squared error, and give an outlook to multifidelity rare event simulation. Our numerical examples demonstrate that multifidelity Monte Carlo estimation provides unbiased estimators (``accuracy guarantees'') and achieves speedups of orders of magnitude compared to crude Monte Carlo estimation that uses a single model alone.

### Murat Manguoglu (TU Berlin)

Donnerstag, 17. Mai 2018

**Incomplete Cholesky Factorization Based Sparse Matrix Partitioning and Its Applications**

Given a sparse indefinite matrix matrix, we propose an incomplete-Cholesky factorization based partitioning. The proposed partitioning creates a 2x2 block structure in which the (1,1) block is symmetric and positive definite. Hence, the resulting blocked matrix can be handled more efficiently than the aggregated form. The proposed partitioning scheme has applications in solving sparse linear system of equations as well as symmetric eigenvalue problems. We show the effectiveness of the method combined with a single level algebraic order reduction technique similar to Component Mode Synthesis and Algebraic Multi-level Substructuring on the Anderson Model of Localization.

### Christian Mehl (TU Berlin)

Donnerstag, 24. Mai 2018

**The singular generalized eigenvalue problem: perturb it! **

The regular generalized eigenvalue problem is a well understood and well analyzed problem and many algorithms for its solution are available. Some applications, however, lead to the necessity of solving a singular generalized eigenvalue problem, i.e., the problem of computing the eigenvalues of the regular part of the underlying singular matrix pencil.

GUPTRI is a robust software package for computing the generalized Schur decomposition of an arbitrary pencil, including singular pencils. It uses the strategy to first extract the regular part of the pencil by a staircase algorithm. During this process rank decisions have to be made and these may become critical if the pencil under consideration has large singular blocks. In this situation, it may be advantageous to have an alternative algorithm at hand.

In this talk, we develop an algorithm for the computation of the eigenvalues of the regular part of a square singular pencil that is based on applying rank-completing perturbations. These are perturbations that have a rank equal to the size of the given pencil minus its normal rank so that they ``complete'' the rank of the singular pencil to full rank. As we will see, these perturbations generically result in a regular pencil such that the regular part of the original singular pencil remains unchanged. When then the eigenvalues of the perturbed pencil are computed, we show how knowledge of perturbation theory of singular pencils can be exploited to separate the ``true'' eigenvalues from the regular part of the original singular pencil from the ``fake'' eigenvalues that are generated from the perturbed singular part.

### Josip Tambaca (University of Zagreb, Croatia)

Donnerstag, 24. Mai 2018

**A new Naghdi type shell model**

A shell model is a two-dimensional model of three-dimensional elastic body which is thin in one direction. There are several linear shell models in the mathematical literature that are rigorously justified starting from the linearized 3d elasticity. Examples are the membrane shell models, the flexural shell model and the Koiter shell model. Their application depends on the particular geometry of the shell's middle surface and the boundary condition which allow or disallow inextensional displacements.

In this talk a Naghdi type shell model will be presented and related to the classical models. This new model is given in terms of a displacement vector and the vector of infinitesimal rotation of the cross-section of the shell, both being in H^{1}. It unites different possible behaviors of the shell, it is applicable for all geometries and all boundary conditions, no complicated differential geometry is necessary for the analysis of the model and the model is also well formulated for geometries of the middle surface of the shell with corners.

### Robert Altmann (Uni Augsburg)

Donnerstag, 31. Mai 2018

**A Mathematical View on Anderson Localization**

In this talk, we prove spectral properties of linear Schrödinger operators under oscillatory high-amplitude potentials on bounded domains. More precisely, we show (depending on the degree of disorder) the existence of spectral gaps amongst the lowermost eigenvalues and the emergence of exponentially localized states.

In order to show these theoretical results, we use numerical methods. This includes the convergence theory of iterative solvers for eigenvalue problems and their optimal local preconditioning by domain decomposition.

This is joint work with P. Henning (Stockholm) and D. Peterseim (Augsburg)

### Christoph Zimmer (TU Berlin)

Donnerstag, 31. Mai 2018

**ε-Expansion of Constrained Hyperbolic PDEs **

Nowadays, automatic modeling software like Simulink or Dymola/OpenModelica is industrial standard. These tools allow to generate realistic models by interconnecting smaller physical submodels. If the submodels are hyperbolic PDEs then the generated model is a hyperbolic PDE with constraints (PDAE), where normally the interconnection causes the constraints. Classical examples of hyperbolic PDAEs can be found in electrical circuits or gas networks.

In this talk we will investigate hyperbolic PDAEs with a slow and a fast moving state. With the help of gas networks, we investigate how one can approximate such systems by a family of parabolic PDAEs. We prove the approximation orders of different variables under mild assumptions and show that they are optimal. Furthermore, we show how the orders improve under more regular data.

This is joint work with Robert Altmann.

### Jeroen Stolwijk (TU Berlin)

Donnerstag, 07. Juni 2018

**Sensitivity Analysis for the discretised Euler Equations; A Case Study on a Y-Shaped Gas Network**

Natural gas plays a crucial role in the energy supply of the world. After oil, it is the second most used energy supplier in Germany. The high demand for natural gas calls for an accurate, efficient and robust mathematical modelling, simulation and optimisation of the gas transport through the existing pipeline network.

The most accurate model to describe the gas flow through a pipe is given by the three-dimensional Euler equations of fluid dynamics. However, often it is not necessary to consider this highly detailed model and simplifying assumptions can be made in order to save computational cost. In this talk we analyse the three most accurate models in the resulting model hierarchy, which are discretised using the effective implicit box scheme. As a case study, we consider the gas flow through a Y-shaped network with pipes of different lengths.

After the discretisation of the models in space and time, nonlinear systems of equations are obtained for every time integration step. A boundary value perturbation analysis for these nonlinear systems is performed using componentwise relative amplification vectors and condition numbers. We conclude with a discussion and comparison of the results.

This is joint work with V. Mehrmann.

### Ines Ahrens (TU Berlin)

Donnerstag, 07. Juni 2018

**A generalization of the Sigma method for DAEs with delay**

The strangeness index for DAEs is based on the derivative array, which consists of the system itself plus its time derivatives. Index reduction is performed by selecting certain important equations from the derivative array. In a large-scale setting with high index, this might become computationally infeasible. However, if it is known a priori which equations of the original systems need to be differentiated, then the computational cost can be reduced. One way to determine these equations is by means of the Sigma method.

If the DAE features in addition a delay term then taking derivatives might not be sufficient and instead, the derivative array must additionally be shifted in time thus increasing the computational complexity even further. In this talk I will explain how one can modify the Sigma method such that it determines the equations which need to be shifted and/or differentiated.

This is joint work with Benjamin Unger.

### Daniel Bankmann (TU Berlin)

Donnerstag, 14. Juni 2018

**On sensitivities of strangeness-free DAEs**

Optimal control tasks arise in a variety of applications from, e.g., mechanical or electrical engineering, where one wants to minimize a certain cost functional with respect to some input function and the resulting state trajectory. Usually, the systems describing the dynamical behavior of these applications are governed by differential equations. Furthermore, in certain applications, e.g. humanoid locomotion, the cost functional descriptions may depend on parameters.

One can solve these problems by rewriting the optimization task into its necessary conditions, which are comprised of a boundary value problem, again parameter dependent. In this talk we first revisit basic theory for boundary value problems of ODEs. Then, we review results on adjoint sensitivity analysis of boundary value problems and analyze how these can be extended to strangeness-free DAEs.

### Riccardo Morandin (TU Berlin)

Donnerstag, 14. Juni 2018

**Runge-Kutta methods for port-Hamiltonian differential-algebraic equations**

The energy based port-Hamiltonian approach is very appealing for modeling, simulation and control of complex multiphysics systems. Furthermore, many of these systems present algebraic constraints, that should be left explicit, e.g., to control the consistency of the numerical simulations, leading to differential-algebraic equations (DAEs).

It is well known that geometric and symplectic integrators are especially suitable for the simulation of port-Hamiltonian systems, to preserve the qualitative behaviour of energy. For the numerical simulation of DAEs, if one wants to keep the constraints explicit, integrators for ODEs cannot usually be applied right away, and require some further development; this is e.g. the case of the Runge-Kutta methods for DAEs. Not all ODE methods can be adapted to be suitable for DAEs: additional conditions are required for the underlying system to be solvable, and to not lose order of convergence.

In this talk I will focus on the analysis of Runge-Kutta numerical integrators for a certain class of port-Hamiltonian DAEs. It is shown that, if the method is symplectic, then the discrete Hamiltonian will express the requested behaviour. Unfortunately, it is also shown that no canonical Runge-Kutta integrator can have all the good qualities. The symplecticity condition is then weakened, and a larger class of Runge-Kutta methods, that express consistent discrete Hamiltonian behaviour in the linear time-independent case, is characterized, but even in this case the requested conditions cannot be achieved at the same time. Finally, it is considered the application of partitioned Runge-Kutta methods to semi-explicit port-Hamiltonian DAEs of index 1, and it is shown that in this case some of the restrictions can be surpassed.

### Jesse Scherwitz (TU Berlin)

Donnerstag, 21. Juni 2018

**On a Projection Based Index Reduction Method for Differential-Algebraic Equations in Electrical Power Systems**

### Volker Mehrmann (TU Berlin)

Donnerstag, 21. Juni 2018

**Control and model reduction for flow problems**

We will discuss the (optimal) control of fluid flow problems, and in particular the difficulties and challenges. To address these problems, it is necessary to obtain adequate reduced order models and to use a systems based approach that studies the discretization of input-output maps.

This is a summary of current approaches and it presents results of several years of research.

### Heinrich Ellmann (TU Berlin)

Donnerstag, 28. Juni 2018

**Solution of discrete time optimal control problems using the palindromic Laub trick.**

This presentation is about the discrete time linear quadratic optimal control problem and its solution using the DARE (Discrete Time Algebraic Riccati Equation). In recent studies, the solution of the DARE was connected to the existence of deflating subspaces of a palindromic matrix pencil, using specific structure preserving properties of it. Analog to the invariant subspaces of the Hamiltonian matrix in the Laub Algorithm, deflating subspaces of the palindromic pencil can be found and used to determine a stabilizing solution of the DARE. In the Bachelor thesis, which is presented, the exactness of the computed solutions of the DARE is demonstrated for the DAREX-Benchmark examples.

### Julian Kern (TU Berlin)

Donnerstag, 28. Juni 2018

**Polar Decompositions with Commuting Factors in Indefinite Inner Product Spaces: Another Approach to Uniqueness**

In my bachelor thesis, I worked on a generalisation of polar decompositions to indefinite inner product spaces. In order to obtain a unique polar decomposition, the spectrum of the decomposition is restricted and singular matrices are often excluded from examination. I present another approach to uniqueness and illustrate some related and yet unsolved problems.

### Felix Black (TU Berlin)

Donnerstag, 05. Juli 2018

**A comparison of shifted proper orthogonal decomposition and symmetry reduction**

Transport phenomena present a challenge for common model order reduction methods such as the proper orthogonal decomposition (POD). The size of the reduced order model (ROM) obtained by POD is directly related to the decay of singular values of the snapshot matrix: A fast decay usually yields a small ROM, but if the decay is slow, the POD needs to retain many modes for an accurate ROM. Systems with advective transport such as the advection equation show the latter behaviour. In addition, the system matrix of the (spatially) semi-discretized system may be skew-symmetric and thus not asymptotically stable, which is a requirement for many MOR methods for linear systems such as balanced truncation and IRKA.

We discuss two MOR approaches, namely shifted POD and symmetry reduction. The methods are compared with respect to their theoretical framework and the steps required to obtain a ROM. The theory is illustrated with the one dimensional advection equation and supported with numerical results. We find that for the advection equation with periodic boundary conditions, shifted POD can be viewed as a discrete analogue to symmetry reduction. As a consequence of the comparison, we introduce a unifying framework for the methods.

### Pia Lutum (TU Berlin)

Donnerstag, 05. Juli 2018

**Numerical Computation of the Real Structured Stability Radius**

The real structured stability radius is a measurement of the robustness of a stable matrix under certain perturbations. In the talk we discuss an iterative algorithm to compute the stability radius. Based on this algorithm, we further briefly discuss an iterative way to compute the real sructured stability radius for large-scale matrices by projections.

### Marine Froidevaux (TU Berlin)

Donnerstag, 12. Juli 2018

**Contour integral methods for nonlinear eigenvalue problems**

Nowadays, a vast collection of efficient numerical methods is available to solve standard eigenvalue problems (EVPs), while nonlinear EVPs remain much more complex to solve. One of the reasons is that the linearly independence of the eigenvectors, which is a fundamental property of the linear EVP, is not guaranteed in the nonlinear case. Therefore, numerical methods using deflation of eigenspaces in order to find all eigenvalues of a linear pencil, such as e.g. the Arnoldi method, cannot be used directly on nonlinear problems.

The most well-known approaches to solve nonlinear EVPs are based on Newton’s method and on Companion form linearizations. Another approach consists in integrating the resolvent over a contour enclosing the eigenvalues of interest in order to reduce the possibly big nonlinear EVP to the solution of a standard linear EVP of smaller size.

In this talk we will consider the contour integral method described in [Bey12] and show how it can be used to detect the presence of non-trivial Jordan blocks. Moreover we will investigate some numerical examples where finite precision arithmetics can cause this method to fail. Finally, we will show how to combine the contour integral method with the Loewner framework in order to improve the numerical stability of the complete algorithm.

This is joint work with Luka Grubisic (U Zagreb), Philipp Jorkowski (TU Berlin) and Kersten Schmidt (TU Darmstadt).

[Bey12] W.-J. Beyn, An integral method for solving nonlinear eigenvalue problems, Linear Algebra and its Applications, 2012.

### Sofia Bikopoulou (TU Berlin)

Donnerstag, 12. Juli 2018

**Algorithm-Based Fault Tolerance for Iterative Methods on High Performance Computing Systems**

A key feature in the development of High Performance Computing (HPC) systems is the property of fault tolerance, i.e. the capability of a system to continue operating in a proper manner regardless of a sudden failure of one or more of its processing units. The most common practice used in HPC is checkpointing and rollback-recovery. A mass of drawbacks can be detected though, if checkpointing and rollback-recovery is used as a stand-alone technique. In spite of their common use in massively parallel platform applications, checkpointing techniques suffer from severe overhead when failures become more and more regular.

In this talk, an enhanced technique for solving large sparse linear systems merging the checkpointing and rollback-recovery method with a quadruple checksum-based fault tolerance approach is presented. The failure model proposed aims in the detection and correction of soft errors that appear due to faulty numerical computations on large-scale platforms.

### Philipp Schulze (TU Berlin)

Donnerstag, 19. Juli 2018

**Structure-Preserving Model Order Reduction of the Linear Advection-Diffusion Equation**

The advection-diffusion equation describes the advective and diffusive transport of a passive scalar and is used in many application fields including environmental science and chemical engineering. In this talk, we aim for efficient computational models for the linear advection-diffusion equation. Beside efficiency, the special focus of this talk is on deriving models that reflect important physical properties as stability and passivity.

To this end, we first employ a semi-discretization in space which yields a port-Hamiltonian system of differential-algebraic equations. Based on this high-dimensional system, we derive stable and passive reduced-order models via structure-preserving model order reduction (MOR). We compare two different MOR approaches: the proper orthogonal decomposition (POD) which is a standard method and the shifted POD which is a recently developed method especially suited for advection-dominated systems. The numerical results reveal that the shifted POD is advantageous over the classical POD, especially in the regime where the effect of diffusion is much smaller than the one of advection.