Venue
Our seminar takes place on Thursdays at 10:15 AM (GMT+2) in MA 376.
Lecturers
- Prof. Dr. Tobias Breiten (write email)
- Prof. Dr. Volker Mehrmann (write email)
- Prof. Dr. Christian Mehl (write email)
Organization
Hannes Dänschel (write email)
Talks
| Date | Speaker(s) |
|---|---|
| October 20, 2022 | Volker Mehrmann |
| November 03, 2022 | Elif Yalcin |
| November 17, 2022 | Dorothea Hinsen • Bernhard Höveler |
| December 08, 2022 | Philipp Krah |
| December 15, 2022 | Shubhaditya Burela |
| January 12, 2023 | Riccardo Morandin • Paul Schwerdtner |
| January 19, 2023 | Andrea Brugnoli |
| January 26, 2023 | Karim Cherifi • Jesper Schröder |
| February 02, 2023 | Marie-Sophie Bolz |
| February 09, 2023 | Attila Karsai |
| February 16, 2023 | Victor Ion Gosea • Alessandro Borghi |
Abstracts
October 20, 2022
Volker Mehrmann: Hypocoercivity and hypocontractivity of linear evolution equations
For the class of linear semi-dissipative Hamiltonian ordinary differential equations or differential-algebraic equations (continuous or discrete time), the concepts of stability, asymptotic stability, hypocoercivity and hypocontractivity are discussed and related to concepts from control theory.
In the finite dimensional case, on the basis of staircase forms, the short-time solution behavior is characterized and connected to the hypocoercivity index of these evolution equations.
The results are applied to the analysis of flow problems.
Joint work with Franz Achleitner and Anton Arnold.
In the finite dimensional case, on the basis of staircase forms, the short-time solution behavior is characterized and connected to the hypocoercivity index of these evolution equations.
The results are applied to the analysis of flow problems.
Joint work with Franz Achleitner and Anton Arnold.
November 03, 2022
Elif Yalcin: A numerical method with the posteriori error estimates for fractional integro-differential equations
This talk is concerned with obtaining a numerical solution of fractional integro-differential equations by constructing a matrix-collocation method involving the Hermite polynomial and matrix expansions of terms. The method is equipped with the collocation points and Caputo sense is used to describe the fractional derivatives. Having gathered all matrix compounds into a matrix equation, the fractional integro-differential equation is converted to a linear system of algebraic equations including the unknown Hermite coefficients. With the solving of the system, the desired numerical solution is immediately obtained. For sake of ensuring the efficiency and precision of the method, a test case is given and the error analysis is considered.
November 17, 2022
Dorothea Hinsen: Port-Hamiltonian Systems with Time-Delays
An appealing approach to model, simulate and control complex multiphysics systems is using port-Hamiltonian (pH) systems. These systems have several advantages for example they are passive, inherently stable, robust in numerical integration and they can be easily linked together yielding a new pH system.
In the last few decades, pH models were developed for standard systems, descriptor systems and infinite dimensional systems. However, a pH formulation for systems with time delay is not yet available.
In the standard case we have an equivalence between being a pH system, being passive or fulfilling the Kalman-Yakubovich-Popov (KYP) inequality. Moreover, we can use the KYP inequality to construct a pH system explicitly. The time-delayed system can be reinterpreted as an infinite dimensional system. For which there is also a passivity equivalent KYP inequality . In this talk, we use the infinite dimensional KYP inequality to formulate a pH system for time-delayed systems in a similar way to the standard case. Furthermore, we present conditions and properties which have to hold for these systems.
In the last few decades, pH models were developed for standard systems, descriptor systems and infinite dimensional systems. However, a pH formulation for systems with time delay is not yet available.
In the standard case we have an equivalence between being a pH system, being passive or fulfilling the Kalman-Yakubovich-Popov (KYP) inequality. Moreover, we can use the KYP inequality to construct a pH system explicitly. The time-delayed system can be reinterpreted as an infinite dimensional system. For which there is also a passivity equivalent KYP inequality . In this talk, we use the infinite dimensional KYP inequality to formulate a pH system for time-delayed systems in a similar way to the standard case. Furthermore, we present conditions and properties which have to hold for these systems.
Bernhard Höveler: Spectral approximation of Lyapunov operator equations with applications in high dimensional non-linear feedback control.
Optimal feedback control for nonlinear systems is a powerful tool for many applications in engineering, physics and many other fields. The drawback of such approach is that the numerical treatment of the resulting nonlinear first order partial differential equation - the Hamiltion-Jacobi-Bellman equation (HJB) - can be difficult. One major reason for that being the high dimensionality of the state space for almost all problems of interest. In this talk it will be shown, that the HJB is linked to the operator Lyapunov equation. For this connection we'll need to define weighted Lp-spaces and take a closer look at semigroups and their properties to pave the way for a useful numerical scheme. Which will be - together with some numerical experiments - shortly introduced at the end.
December 08, 2022
Philipp Krah: Non-Linear Reduced Order Modeling for Transport Dominated Fluid Systems
In this seminar, I will practice my PhD viva and kindly ask you to give me some feedback. The talk will briefly (30min) summarize the key challenges and results of my work in the field of model order reduction for transport dominated fluid systems (TDFS). TDFS are systems for which the transported quantity changes slowly with respect to the advection speed and therefore only require a few degrees of freedom (DOF) if the system is parametrized in a reference frame that moves with the transported quantity. The talk addresses three topics:
[1] https://tubcloud.tu-berlin.de/s/zbb4tSxYedQbXrw
- adaptive multiresolution methods to reduce large-scale TDFS,
- transport compensations to enhance dimensionality reduction and
- time-parametric non-linear reduced order models for complex moving fronts.
[1] https://tubcloud.tu-berlin.de/s/zbb4tSxYedQbXrw
December 15, 2022
Shubhaditya Burela: Title to be announced ...
January 12, 2023
Riccardo Morandin: Port-Hamiltonian descriptor systems and structure-preserving time integration
Port-Hamiltonian (pH) systems arising from the modeling of complex networks often present hidden algebraic constraints, originating for example from the interconnection of different system components. This requires us to work with differential-algebraic equations (DAEs), also known as descriptor systems in the control community. While the reduction of these systems to regular ODEs is often easy to compute for specific problem instances, it is useful to have general purpose discretization schemes that can be applied directly to DAEs, preserving the algebraic constraints and the pH structure of the problems. This is especially important having in mind automatized modelling.
In this talk we present a convenient and usually sufficiently general formulation for port-Hamiltonian descriptor systems (pHDAEs). While this formulation is quite flexible and can be easily adapted to the infinite-dimensional case, in this talk we will focus primarily on finite-dimensional DAEs. Furthermore, we present several schemes for time integration that exploit the Dirac structure associated to the system to preserve some of the system properties.
The talk is based on joint work with Volker Mehrmann.
In this talk we present a convenient and usually sufficiently general formulation for port-Hamiltonian descriptor systems (pHDAEs). While this formulation is quite flexible and can be easily adapted to the infinite-dimensional case, in this talk we will focus primarily on finite-dimensional DAEs. Furthermore, we present several schemes for time integration that exploit the Dirac structure associated to the system to preserve some of the system properties.
The talk is based on joint work with Volker Mehrmann.
Paul Schwerdtner: Structured Optimization-Based Model Order Reduction, Identification, and Control
Traditional model order reduction (MOR) is used to approximate the dynamic behavior of large-scale and complex models with a low-order surrogate model. Structure-preserving MOR also ensures that this low-order surrogate model has the same physical structure as the given complex model, which is useful e.g. when system networks are considered. We present an optimization-based algorithm that performs structure-preserving MOR of large-scale models with a port-Hamiltonian structure. For this, we describe a parameterization of low-order port-Hamiltonian models and explain the optimization problem that we set up to tune the parameters in order to approximate a given model. We use an objective function that only requires evaluations of the transfer function of the given model at adaptively chosen sample points.
After describing the original method, we discuss an extension to systems with an additional parameter dependency. Finally, we explain how our method can be adapted to perform H-infinity controller synthesis for port-Hamiltonian systems.
After describing the original method, we discuss an extension to systems with an additional parameter dependency. Finally, we explain how our method can be adapted to perform H-infinity controller synthesis for port-Hamiltonian systems.
January 19, 2023
Andrea Brugnoli: Continuous and hybrid finite element discretization of port-Hamiltonian systems
Numerical methods to discretize port-Hamiltonian systems need to retain their geometric structure in order to be used for complex multiphysical applications.
In this talk, a continuous Galerkin finite element exterior calculus formulation that is able to mimetically represent coupled conservation laws is presented. The approach relies on a dual-field representation of the physical system that eliminates the need for a discrete Hodge star operator. The power balance characterizing the Stokes-Dirac structure is retrieved at the discrete level via symplectic Runge-Kutta integrators based on Gauss-Legendre collocation points. The proposed formulation is directly amenable to hybridization. The hybrid formulation is equivalent to the continuous Galerkin formulation and can be efficiently solved using a static condensation procedure in discrete time. The size reduction achieved thanks to the hybridization is considerable as only global traces of the conforming field need to be computed. Numerical experiments validate the convergence and conservation properties of the method. Furthermore, they show the equivalence of the continuous and hybrid formulation and the computational gain achieved by the latter.
In this talk, a continuous Galerkin finite element exterior calculus formulation that is able to mimetically represent coupled conservation laws is presented. The approach relies on a dual-field representation of the physical system that eliminates the need for a discrete Hodge star operator. The power balance characterizing the Stokes-Dirac structure is retrieved at the discrete level via symplectic Runge-Kutta integrators based on Gauss-Legendre collocation points. The proposed formulation is directly amenable to hybridization. The hybrid formulation is equivalent to the continuous Galerkin formulation and can be efficiently solved using a static condensation procedure in discrete time. The size reduction achieved thanks to the hybridization is considerable as only global traces of the conforming field need to be computed. Numerical experiments validate the convergence and conservation properties of the method. Furthermore, they show the equivalence of the continuous and hybrid formulation and the computational gain achieved by the latter.
January 26, 2023
Karim Cherifi: Discrete time causal port Hamiltonian systems
Most of the literature on linear port Hamiltonian systems focuses on continuous time impedance passive systems. In this talk, the focus is on discrete time causal scattering passive systems. We derive a port Hamiltonian representation for scattering passive dynamical systems and study the relation to bounded real transfer functions. The relation to passivity characterized by the Kalman-Yakubovich-Popov (KYP) inequality is also discussed. In addition, the relation between impedance and scattering passive systems is discussed. Finally, we show how the time discretization of a continuous passive system falls into the class of discrete time systems introduced.
Jesper Schröder: An approach to non-linear observer design via optimal control theory -- the Mortensen Observer
Many real processes can be described by mathematical models. Since these representations are based on idealized assumptions and simplifications, it is nearly impossible to describe any system perfectly. Another problem arises from the fact that often the state of the actual system is not available, and instead only an output can be obtained. Examples of such an output are measurements by devices like microphones and heat sensors, which are subject to additional noise. One approach to recovering an approximation of the state of the system is to design a dynamical observer. While the Kalman filter offers an efficient solution to this problem for linear systems, the design of non-linear observers is still challenging.
This talk gives an introduction to the minimum energy estimator, also called the Mortensen observer, which was proposed among others by R. E. Mortensen in 1968. The method approaches the problem via optimal control theory. We discuss feasible assumptions to ensure the well posedness of the resulting observer equation, which is achieved by a sensitivity analysis of the corresponding value function. Here, the associated Hamilton-Jacobi-Bellman equation plays an integral part. Numerical experiments illustrate the theoretical result.
This talk gives an introduction to the minimum energy estimator, also called the Mortensen observer, which was proposed among others by R. E. Mortensen in 1968. The method approaches the problem via optimal control theory. We discuss feasible assumptions to ensure the well posedness of the resulting observer equation, which is achieved by a sensitivity analysis of the corresponding value function. Here, the associated Hamilton-Jacobi-Bellman equation plays an integral part. Numerical experiments illustrate the theoretical result.
February 02, 2023
Marie-Sophie Bolz: H-skewadjoint square roots of H-selfadjoint matrices
This talk is concerned with obtaining necessary and sufficient conditions for the existence of an H-skewadjoint square root of a given H-selfadjoint matrix. The examined case is the case where H is a (skew-)symmetric matrix and consequently the studied indefinite inner product is a (skew-)symmetric bilinearform.
February 09, 2023
Attila Karsai: Manifold turnpikes of nonlinear port-Hamiltonian descriptor systems under minimal energy supply
Optimal control problems occur in a variety of real-world applications. In many cases, it can be observed that the optimal control makes a detour to be able to control the system with lower costs. This phenomenon is called *turnpike phenomenon*. The name is reminiscent of an observation from everyday life: when driving a long distance, it is almost always quicker to take a detour via a turnpike than to drive slowly on the country road all the time.
In the context of port-Hamiltonian systems, which stem from an energy-based modeling perspective, minimizing the supplied energy is a natural optimization objective. In this talk, turnpike phenomena of nonlinear port-Hamiltonian descriptor systems under minimal energy supply are studied. Under assumptions on the smoothness of the system nonlinearities, it is shown that the optimal control problem has a manifold dissipativity property. Using additional controllability assumptions, we observe that the optimal control problem exhibits a manifold turnpike property.
In the context of port-Hamiltonian systems, which stem from an energy-based modeling perspective, minimizing the supplied energy is a natural optimization objective. In this talk, turnpike phenomena of nonlinear port-Hamiltonian descriptor systems under minimal energy supply are studied. Under assumptions on the smoothness of the system nonlinearities, it is shown that the optimal control problem has a manifold dissipativity property. Using additional controllability assumptions, we observe that the optimal control problem exhibits a manifold turnpike property.
February 16, 2023
Victor Ion Gosea: Realization-free balanced truncation from input-output data
We propose a new, realization-free reformulation for a classical model reduction approach, i.e., for balanced truncation (BT). This is achieved by observing that BT does not make independent use of the two infinite system Gramians, which each depend on the internal states (and hence are generally inaccessible). BT rather makes use of the Gramians product instead, which preserves system invariants that do not depend on the underlying system realization. In this work, we describe how key quantities can be approximated directly from data by implicitly making use of quadrature approximations for the Gramians. Here, data are given by transfer function samples on the imaginary axis (in the frequency domain), or evaluations of the impulse response on the real axis (in the time domain). We explicitly derive reduced-order quantities using observed response data and, as a consequence, arrive at a novel, nonintrusive formulation of BT, referred to as QuadBT. We compare this to the classical (intrusive) BT approach, for various numerical test cases. If time permits, we show recent results on realization-free implementations of another classical reduction method, the singular perturbation approximation (SPA). A similar philosophy applies here as well (as for QuadBT), and hence, we derive the QuadSPA method that only requires transfer function evaluations.
Alessandro Borghi: H2 model order reduction on simply connected domains
The optimal H2 approximation has been widely used for the development of efficient model order reduction algorithms. Well known examples are IRKA and MIRIAm respectively adopted in continuous and discrete time state-space systems. These methods rely on the definition of a norm on a Hardy space from which the H2 optimal interpolation conditions follow.
One of the main assumptions for IRKA and MIRIAm is that the transfer function of the full-order model needs to be analytic on the right half complex plane and on the outside of the unit disk respectively.
However, there can be cases in which the transfer function is analytic in domains that differ from the ones above. Hence, we propose a framework to derive first-order interpolation conditions for H2 optimality in a simply connected set. The theoretical background relies on conformal maps and generalizes Hardy spaces to functions that are analytic on specific domains. The objective is to eventually develop algorithms that can be used to find a reduced order transfer function that satisfies the H2 optimality conditions in the chosen set.
One of the main assumptions for IRKA and MIRIAm is that the transfer function of the full-order model needs to be analytic on the right half complex plane and on the outside of the unit disk respectively.
However, there can be cases in which the transfer function is analytic in domains that differ from the ones above. Hence, we propose a framework to derive first-order interpolation conditions for H2 optimality in a simply connected set. The theoretical background relies on conformal maps and generalizes Hardy spaces to functions that are analytic on specific domains. The objective is to eventually develop algorithms that can be used to find a reduced order transfer function that satisfies the H2 optimality conditions in the chosen set.