Energy based modeling, why and how?

The next level of digitization will create digital twins of
every product or process. To do this in a mathematical rigorous and
risk and error controlled way, a new modeling, simulation and optimization
paradigm is needed. While automated modularized modeling is common
in some technical domains like circuit design or multi-body dynamics, it
becomes increasingly challenging  when systems or numerical solvers from
different physical domains are coupled, due to largely different scales or
modeling accuracy, and very different software technologies.

A recent system theoretic approach to address these challenges is
the use of network and energy based modeling via constrained
port-Hamiltonian (pH) systems, where the coupling is done in a physically meaningful way via
energy variables. Furthermore, for each subsystem a whole model
hierarchy can be employed  ranging from very fine grain models to highly reduced
surrogate  models arising from model reduction or data based
modeling. The model hierarchy allows adaptivity not only in the discretization but
also in the model selection.

We will present an overview over the  hierarchical pH modeling approach
and illustrate the advantages.

Condensed forms for port-Hamiltonian descriptor systems

The geometric structure of a port-Hamiltonian determined by its interconnection structures of the system allow a wide range of applicability. Specifically, the system is open to interaction with the environment and thus is receptive to control interaction. The theory of optimal control for ordinary differential equations is well established. However, the situation becomes more complicated once we include differential-algebraic constraints. Optimality conditions for general unstructured nonlinear DAEs of arbitrary index were derived by forming the derivative arrays and determine the reduced system. Following this idea, our recent research goal is to obtain optimality conditions specifically in the case of port-Hamiltonian DAEs exploiting its special structure. The first step which we will discuss in the talk, consists of formulating a more general condensed form for port-Hamiltonian DAEs.

Model Order Reduction for Parametric High Dimensional Models in the Analysis of Financial Risk

The risk analysis of financial instruments often requires the valuation of such instruments
under a wide range of future market scenarios, demanding efficient algorithms. Thus, we estab-
lish a parametric model order reduction approach based on a variant of the proper orthogonal
decomposition. The method generates small model approximations for the high dimensional
parametric convection-diffusion-reaction partial differential equations that arise in financial
risk analysis. This approach requires solving the full model at some selected parameter val-
ues to generate a reduced basis. We propose an adaptive greedy sampling technique based
on surrogate modeling for the selection of this sample parameter set. The new technique is
analyzed, implemented, and tested on industrial data of different financial instruments under
short-rate models. The results illustrate that the reduced model approach works well and
shows potential applications in historical or Monte Carlo value at risk calculations.

Dynamics and synchronization in coupled swing equations ("Kuramoto with inertia")

The subject of my talk are coupled swing equations (coupled phase oscillators with inertia, coupled second-order phase oscillators). Such systems appear as simplified models for power grids. I will discuss synchronization and certain more complicated dynamical properties of such systems. In particular, coupled swing equations can be represented as a special class of adaptive networks. As a direct consequence of this relationship, the properties and phenomenology of adaptive networks can be transferred to the above mentioned power grid models. In this way, we show new classes of multicluster states. I will also mention other possible classes of solutions that could be relevant for power grids: splay states and solitary states.

Linear Algebra properties of pencils that are posH

PosH pencils are matrix pencils whose coefficients have positive semidefinite Hermitian parts and can be seen as a generalization of dissipative Hamiltonian pencils. They appear for example in the linearization of matrix polynomials with positive semidefinite coefficients. Unlike dissipative Hamiltonian pencils, posH pencils need not be stable, but still they have some interesting Linear Algebra properties. In the talk, we discuss those properties and also how stability of posH pencils can be detected.

Riemannian optimization on a modified tensor train manifold to solve the Lyapunov operator equation

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. While this connection doesn't appear to be that useful for numeric computation at first glance, we will be able to find an approximation to the spectral decomposition for the solution to the operator equation by a Riemannian optimization scheme over a manifold that shares many properties with the tensor train manifold. Since storage and compute requirements of tensor trains only scale linearly with the dimension this perspectively allows the treatment of a much wider class of problems than most other numerical schemes.

Properties equivalent to port-Hamiltonian systems for ordinary differential equations and results for delay differential equations

The general concept of a port-Hamiltonian (pH) system is an energy-based dynamical system with different parts described through a state space, a Hamiltonian function and ports. The advantage of pH systems is that they can easily be interconnected via power conserving interconnections yielding a new pH system. Moreover, they are passive, inherently stable, and robust in numerical integration and maintain their structure after model reduction. For linear time-invariant systems there are three properties equivalent to pH systems, given by passivity, the Kalman-Yakubovich-Popov inequality and that the system is positive real.

In this talk I present in what cases which conditions are required so that the equivalences for ordinary differential equations apply. This part is based on joint work with Karim Cherifi and Hannes Gernandt. In addition, I present similar equivalences for delay differential equations.

Structure-Preserving LQG Balanced Truncation for Linear Port-Hamiltonian Descriptor Systems

Abstract: Linear quadratic Gaussian (LQG) balanced truncation is a model reduction technique for possibly unstable linear time-invariant systems which are to be controlled by an LQG controller. In contrast to the classical balanced truncation approach, the LQG balancing procedure is based on the closed-loop system behavior rather than on the transfer function of the open-loop system. Furthermore, LQG balanced truncation comes with an a priori error bound in the gap metric and yields both a reduced-order model (ROM) for the plant and a corresponding reduced-order controller.

In this talk, we consider the special case where the plant is described by a linear port-Hamiltonian descriptor system and demonstrate how LQG balanced truncation can be modified such that the ROM is also port-Hamiltonian. This modification still allows for an a priori error bound in the gap metric. Furthermore, we demonstrate a theoretical approach for exploiting the non-uniqueness of the port-Hamiltonian representation to achieve a faster decay of the error bound. This approach is based on a maximal solution of a Kalman-Yakubovich-Popov linear matrix inequality for descriptor systems.

This is joint work with Tobias Breiten.

On a new compact port-Hamiltonian Electrical Circuit Model

We present a new model for electrical circuits obtained by combining three themes: port-Hamiltonian energy-based modelling, structural analysis as used in the circuit world, and structural analysis of general differential-algebraic equations. The new compact port-Hamiltonian formulation has remarkable simplicity and symmetry and is proven to be always structurally amenable. Moreover, the compact port-Hamiltonian form gives rise to a differential-algebraic equation system that always has an index at most 1, and other good numerical properties. A prototype Matlab implementation shows the potential of the methodology for further developments.

On characterizations of port-Hamiltonian descriptor systems

My talk is a continuation of Dorothea's talk from two weeks ago. For linear time invariant control systems she studied the equivalence between passivity, positive realness, solutions to the KYP inequalities and the existence of a port-Hamiltonian formulation.
She elaborated that the oftentimes assumed minimality of the system is only needed to show that a positive real transfer function implies each of the above properties.
I will talk about possible extensions of the above equivalences to descriptor systems, where the situation is more involved:
For example, we allow a singular Q in the port-Hamiltonian formulation, there are several notions for controllability and observability and, in addition, the system trajectories will not attain all values of the underlying spaces which requires the solution of KYP inequalities  on a certain subspace.
The talk is based on an ongoing joint work with Karim Cherifi, Dorothea Hinsen and recent discussions with Volker Mehrmann and Philipp Schulze.

Structure preserving control of port-Hamiltonian systems

Adaptive Nonlinear Optimization of Stationary District Heating Networks Based on Model and Discretization Catalogs

We propose an adaptive optimization algorithm for operating district heating networks in a stationary regime. The behavior of hot water flow in the pipe network is modeled using the incompressible Euler equations and a suitably chosen energy equation. By applying different simplifications to these equations, we derive a catalog of models. Our algorithm is based on this catalog and adaptively controls where in the network which model is used. Moreover, the granularity of the applied discretization is controlled in a similar adaptive manner. By doing so, we are able to obtain optimal solutions at low computational costs that satisfy a prescribed tolerance w.r.t. the most accurate modeling level. To adaptively control the switching between different levels and the adaptation of the discretization grids, we derive error formulas and a posteriori error estimators. Under reasonable assumptions we prove that the adaptive algorithm terminates after finitely many iterations. Our numerical results show that the algorithm is able to produce solutions for problem instances that have not been solvable before.

Computation of Spectral Intervals of linear ODE's and DAE's

The Lyapunov-intervals of linear time-varying ODE's and DAE's give informations about the exponential growth rate of the solutions. If all Lyapunov-intervals of such equations are negativ, then the (linear) equation is exponentially stable. Nevertheless in the time-varying case good perturbations, which are sufficiently smooth and more, can destroy the stability of the Lyapunov-intervals. For this reason we consider more robust spectral intervals against these pertubations, namely the Bohl- and the Sacker-Sell spectrum. One can show that the Lyapunov-spectrum lies in the Bohl and the Bohl lies in the Sacker-Sell spectrum, for sufficiently bounded systems. 

Algebraic constraints in linear, nonlinear, and boundary control, port-Hamiltonian systems

Algebraic constraints are ubiquitous in modeling complex engineering systems. From a port-based modeling perspective they are encapsulated in the Dirac structure of the port-Hamiltonian formulation. Another way of representing algebraic constraints is by replacing the expression of energy in terms of a Hamiltonian function by a general Lagrangian subspace. I will recall how both types of algebraic constraints can be combined, and how one may convert one type of algebraic constraints into the other; basically by the introduction of Lagrange multipliers. This will be first discussed for linear finite-dimensional port-Hamiltonian systems, and subsequently extended to the nonlinear case. Finally, if time permits, I will show how the same can be done for boundary control port-Hamiltonian systems described by linear pde’s (on a 1-dimensional spatial domain).

Error balancing for eigenvalue problems arising from simulation of photonic crystals

In order to analyse the propagation of light waves in photonic crystals, one typically has to approximate the solutions of multiple large-scale eigenvalue problems (EVPs). To do so, we can typically discretize the EVPs with a finite element method, which leads to a “discretization error”. The solution of the discretized EVPs can in turn be approximated with an iterative method (as the power iteration), which leads to an “iteration error”. Further, the large-scale linear system arising in every iteration may also be solved only approximately, leading to a so-called “algebraic error”. 

Nonlinear Model Reduction for an Advection-Reaction-Diffusion Equation with a Fisher term

This talk addresses model order reduction for complex moving fronts, which are transported by advection or through a reaction-diffusion process. Such systems are especially challenging for model order reduction since the transport cannot be captured by linear reduction methods.  Topological changes, such as splitting or merging of fronts pose difficulties for many nonlinear reduction methods and the small non-vanishing support of the underlying partial differential equations dynamics makes most nonlinear hyper-reduction methods infeasible.
We propose a new decomposition together with a hyper-reduction method that addresses these shortcomings. The decomposition uses a level-set function to parameterize the transport and a nonlinear activation function that captures the structure of the front. This approach shares similarities with artificial neural networks, but furthermore provides additional insights into the system, which can be used for efficient reduced order models. Hence, we illustrate that the advection equation can be solved with the same complexity as the POD-Galerkin approach while obtaining errors of less than one percent for representative examples. Furthermore, we outline a special hyper-reduction method for more complicated advection-reaction-diffusion systems. The capability of the approach is illustrated by various numerical examples in one and two spatial dimensions.

This talk is based on a manuscript that I am about to send to a journal, which is why feedback at this stage will be especially valuable. Please send me an email if you want to read the manuscript beforehand to enable more detailed comments. Thanks!

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