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## Absolventen-Seminar • Numerische Mathematik

Verantwortliche Dozenten: |
Prof. Dr. Christian Mehl [1], Prof. Dr. Volker Mehrmann
[2] |
---|---|

Koordination: | Benjamin
Unger |

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 16.04. | 10:15 Uhr | MA
376 | Vorbesprechung | |

Matthew
Salewski | Solutions of turbulent flows
[abstract] | |||

Do
23.04. | 10:15 Uhr | MA 376 | Andreas Steinbrecher
[3] | QUALIDAES - A Software Package for the
Numerical Integration of Quasi-Linear DAEs
[abstract] |

Peter Kunkel
[4] | Optimal control for DAEs, formal adjoints
and formal optimality conditions [abstract] | |||

Mi 29.04. | 9:15 Uhr | MA 376 | Matthias Voigt [5] | On
Linear-Quadratic Optimal Control and Robustness of
Differential-Algebraic Systems [abstract] |

Do 30.04. | 10:15 Uhr | MA
376 | Balmohan V. Limaye [6] | Condition Numbers of Bases [abstract] |

Philipp Schulze [7], Benjamin
Unger | A step towards the realization problem
for retarded descriptor systems [abstract] | |||

Do 07.05. | 10:15 Uhr | MA
376 | - kein Seminar - | Conference in Honor of Volker Mehrmann on the Occasion of
his 60th Birthday [8] |

Do 14.05. | 10:15 Uhr | MA 376 | - kein
Seminar - | Christi
Himmelfahrt |

Do 21.05. | 10:15 Uhr | MA 376 | Robert
Altmann | Regularization and Simulation of
Constrained PDEs [abstract] |

Mark Embree [9] | Interpolatory
Matrix Approximations [abstract] | |||

Do 28.05. | 10:15 Uhr | MA 376 | Olga
Markova | Lengths of matrix sets with
quasi-commuting elements [abstract] |

Do 04.06. | 10:15 Uhr | MA
376 | Felix Held | Optimal control of mechanical multi-body systems
[abstract] |

Volker Mehrmann
[10] | Optimal control of delay
differential-algebraic equations [abstract] | |||

Do 11.06. | 10:15 Uhr | MA
376 | Fredy Sosa [11] | On first order asymptotic expansions for
multiplicative perturbation of eigenvalues
[abstract] |

Lia
Strenge | Modeling and simulation of a droop
controlled swarm type low voltage DC microgrid in a DAE framework
[abstract] | |||

Do
18.06. | 10:15 Uhr | MA 376 | Melina Freitag
[12] | Data assimilation as a very large
inverse problem - an introduction [abstract] |

Philipp Schulze [13] | Data-Driven Model Reduction of Linear Port-Hamiltonian
Systems [abstract] | |||

Do 25.06. | 10:15 Uhr | MA 376 | Jeroen
Stolwijk | Numerical Solution and Error
Analysis for the Euler Equations in Stationary Form
[abstract] |

Jon
Paul | Analysis and solution of dynamic flash
equations [abstract] | |||

Di 30.06 | 11:15 Uhr | MA 415 | Thomas Berger
[14] | Regularization of linear descriptor
systems [abstract] |

Zoran Tomljanović [15] | Optimization of semi-active damping and external damping
in mechanical systems with external force [abstract] |
|||

Do 02.07. | 10:15 Uhr | MA
376 | Leonhard Batzke | Low-rank perturbations of structured matrix pencils
[abstract] |

Benjamin Unger
| Delay DAEs and their regularization
[abstract] | |||

Do
09.07. | 10:15 Uhr | MA 376 | Christoph
Zimmer | On Linear
Operator-differential-algebraic Equations with
Delay [abstract] |

Daniel Bankmann | OVDBDF - A
Software Package for the Numerical Integration of Differential
Algebraic Equations [abstract] | |||

Michael Götte | Backend design of a Modelica compiler [abstract]
| |||

Do
16.07. | 10:15 Uhr | MA 376 | Ute Kandler | A certain Gram-Schmidt variant - Compensated Gram-Schmidt
[abstract] |

Helia Niroomand
Rad | A DAE Modeling Approach toward the
Crosstalk Phenomenon [abstract] |

# Rückblick

- Absolventen Seminar WS 14/15 [16]
- Absolventen Seminar SS 14 [17]
- Absolventen Seminar WS 13/14 [18]
- Absolventen Seminar SS 13 [19]
- Absolventen Seminar WS 12/13 [20]
- Absolventen Seminar SS 12 [21]
- Absolventen Seminar WS 11/12 [22]

### Matthew Salewski (TU Berlin)

Donnerstag, 16. April 2015

**Solutions of turbulent
flows**

In the dynamical systems approach to hydrodynamics, turbulence is seen as the chaotic path that the state vector traces out in the space of allowed solutions to the equations of motion, e.g. Navier-Stokes equations. Rather than a random walk in this space, the approach contends that there are a set of equilibria and periodic orbits that organize the state-space trajectory via their unstable manifolds. From this follows a belief (or hope) that this set is approximately finite, with only a few significant elemental solutions needed to accurately capture the dynamics of turbulent flows. In my talk I will (re-)introduce these topics and place them in the context of turbulent rotating shear flows, where there is growing evidence of such solutions dictating the global statistics of the fluid dynamics even at the highest measured intensities for turbulent flows.

### Andreas Steinbrecher (TU Berlin)

Donnerstag, 23. April 2015

**QUALIDAES - A Software
Package for the Numerical Integration of Quasi-Linear DAEs
**

The behavior of dynamical systems often are modeled with
differential-algebraic equations (DAEs) in quasi-linear form

(1) E(x,t)x'=f(x,t)

as model equations
with the vector of unknowns x and the time t. Unfortunately, in
general the direct numerical integration of DAEs is not feasible due
to so-called hidden constraints

(2) 0=h(x,t).

They are contained in the DAE but not explicitly stated
as equations. The occurrence of hidden constraints leads to
difficulties like instabilities or order reduction in the numerical
integration. Hence, before a robust numerical integration is possible
it is necessary to regularize or remodel the model equations.

In this talk we will present the software package QUALIDAES for
the numerical integration of quasi-linear DAEs of the form (1) which
covers the model equations of many dynamical processes. A key point
for the robust and efficient numerical integration in QUALIDAES is the
precise consideration of the hidden constraints (2). Therefore, the
approach implemented in QUALIDAES is based on the interaction of a
regularization of the DAE (1) with an efficient numerical treatment of
the regularization in form of an overdetermined fomulation.

### Peter Kunkel (Universität Leipzig)

Donnerstag, 23. Oktober 2014

**Optimal control for DAEs, formal adjoints
and formal optimality conditions**

Deriving necessary conditions for the solution of linear-quadratic optimal control problems for differential-algebraic equations (DAEs) with arbitrary index, one should replace the given DAE by an associated index-reduced DAE. Possessing the same (smooth) solutions as the original DAE, the reduced DAE allows for a suitable solution operator for the application of abstract results from optimization theory. The resulting necessary conditions are then in terms of the reduced DAE. It was, however, observed that formally replacing the reduced DAE by the orginal DAE may still make sense to some extent. It shall therefore be discussed how this formal approach is related to the necessary conditions and to what extent this relation can be utilized.

### Matthias Voigt (TU Berlin)

Mittwoch, 29. April 2015

**On Linear-Quadratic Optimal Control and
Robustness of Differential-Algebraic Systems **

In this talk, I will practise my PhD defense presentation. The main focus will be on linear-quadratic optimal control and its relation to descriptor Kalman-Yakubovich-Popov (KYP) inequalities, Lur'e equations and even matrix pencils. Further results on robustness questions will be touched.

### Balmohan V. Limaye (Indian Institute of Technology Bombay)

Donnerstag, 30. April 2015

**Condition Numbers of Bases**

Let X=[x_1,...,x_m] be an ordered basis of a finite dimensional
subspace of L^p (1<= p <=infty). The condition number kappa_p(X)
of X is known to measure the sensitivity of computing the coefficients
in the expression of a function in L^p as a unique linear combination
of the basis elements.

We consider an ordered basis X of
an arbitrary finite dimensional normed space M, and introduce a
number cond(X) which controls `near linear dependence' of the basis
elements as well as overflow and underflow during computations
involving the basis. We also describe optimal scaling strategies for
cond(X).

If X is an ordered basis of an inner product
space, then cond(X) can be calculated explicitly in terms of the
diagonal entries of the Gram matrix corresponding to X and the
diagonal entries of its inverse.

The number cond(X) can
be characterized in terms of the norms of the basis elements and the
norms of the elements of the ordered dual basis of M'. In fact,
cond(X) measures the sensitivity of the problem of finding the unique
ordered basis of M' which is dual to the basis X.

A
comparison of cond(X) and kappa_p(X) shows how intimately they
are related, although they measure reliability of computations
from different perspectives.

### Philipp Schulze, Benjamin Unger (TU Berlin)

Donnerstag, 30. April 2015

**A step towards the realization problem for
retarded descriptor systems **

We introduce a
method to construct low dimensional descriptor systems with retarded
argument of the form

(1a) Ex'(t) = A_1
x(t) + A_2 x(t-tau) + Bu(t),

(1b)
y(t) = Cx(t),

directly from measurements at low
computational cost. The key tool is a generalization of the Loewner
and shifted Loewner matrix to allow interpolation of the transfer
function in the delayed setting. We show that our approach extends the
well-known Loewner framework in the sense that they coincide for
vanishing delay $tau=0$. A family of reduced-order models for (1)
based on Moment Matching is introduced and we show its strong
connection to the generalized Loewner approach.

### Robert Altmann (TU Berlin)

Donnerstag, 21. Mai 2015

**Regularization and Simulation of
Constrained PDEs**

In this talk, I will practise my PhD defense presentation. The main focus will be on operator DAEs. We discuss the formulation of such systems, consistency conditions, a regularization, and the resulting advantages for the simulation of those systems.

### Mark Embree (Virginia Tech, VA)

Donnerstag, 21. Mai 2015

**Interpolatory Matrix
Approximations**

Interpolatory matrix factorizations provide alternatives to the singular value decomposition for obtaining low-rank approximations; this class includes the CUR factorization, where the C and R matrices are formed from a subset of columns and rows of the target matrix. While interpolatory approximations lack the SVD's optimality, their ingredients are easier to interpret than singular vectors: since they come directly from the matrix itself, they inherit the data's key properties (e.g., nonnegative/integer values, sparsity, etc.). We shall provide an overview of these approximate factorizations, describe how they can be analyzed using interpolatory projectors, and introduce a new method for their construction based on the Discrete Empirical Interpolation Method (DEIM). This talk describes joint work with Dan Sorensen (Rice).

### Olga Markova (Moscow State University)

Donnerstag, 28. Mai 2015

**Lengths of matrix sets
with quasi-commuting elements **

Given a finite set S of square matrices over a field,
we consider the linear span A of all products in S and define the
length of S to be the least non-negative integer k such that the
products in these generators of lengths not exceeding k span A.

In this talk we discuss the known bounds for the
lengths of matrix sets which elements pairwise quasi-commute (i.e.
commute up to a factor depending on the matrices). We also
consider the realizability problem for the lengths of such sets
consisting of two matrices.

### Volker Mehrmann (TU Berlin)

Donnerstag, 04. Juni 2015

**Optimal control of delay
differential-algebraic equations**

We discuss the solution of optimal control problems with linear delay differential-algebraic equation (DDAE) constraints. Compared to the already complicated case of standard differential-algebraic equation (DAE) constraints, several further difficulties arise.

These include the decreased regularity in the solution and the occurrence of higher derivatives of the input functions. We present the necessary optimality conditions under the assumption that the method of steps leads to a reasonable solution and discuss the algebraic properties of the optimality system.

Joint work with Peter Kunkel

### Felix Held (TU Berlin)

Donnerstag, 04. Juni 2015

**Optimal control of mechanical multi-body
systems**

In this talk, I will motivate the topic of my master thesis and outline the employed methods and goals. The mechanical model is presented and the derivation of its equations is discussed. Furthermore some remarks on their structure will be given. I also will focus on possibilities to solve these equations numerically.

### Fredy Ernesto Sosa Nunez (Universidad Carlos III de Madrid)

Donnerstag, 11. Juni 2015

**On first order asymptotic expansions for
multiplicative perturbation of eigenvalues**

Let A be a complex matrix with arbitrary Jordan structure, and lambda an eigenvalue of A whose largest Jordan block has size n. Based on the use of the Newton diagram, it has been shown that for a small multiplicative perturbation hat{A}=(I+epsilon C)A(I+epsilon B) of the matrix A, the splitting of lambda under this perturbation is, generically, of order epsilon^{1/n} if lambdaneq 0. Explicit formulas for the leading coefficients are obtained, involving the perturbation matrices B and C and the eigenvectors of A. In the special case of lambda=0, similar results has been found for leading coefficients in the splitting of lambda in this case the splitting of lambda, is generically, of order epsilon^{frac{1}{n+1}}.

Joint work with Julio Moro Carreno.

Abstract as PDF [23].

### Lia Strenge (TU Berlin)

Donnerstag, 11. Juni 2015

**Modeling and simulation of a
droop controlled swarm type low voltage DC microgrid in a DAE
framework**

In this talk, I present the preliminary results of my Master's
thesis. We derive a model of a droop controlled swarm type low voltage
(LV) direct current (DC) microgrid and discuss simulation methods and
results for the derived model. This type of microgrid is a technical
solution for the bottom-up electrification of the off-grid population
in the Global South (referring to less economically developed
countries).

Mathematically, the closed-loop model is a
stiff strangeness-free differential-algebraic equation (DAE) system.
It can be of arbitrarily high (finite) dimension due to the generic
topology of the grid being modeled by an undirected graph. In
addition, the modeling allows for linear and nonlinear representation
and includes hybrid (switching) characteristics. Regarding the
simulation, the reaction of the voltages and power flows to changes in
consumption (i.e. loads) for an example topology of the grid are of
major interest to understand the behaviour of swarm type LVDC
microgrids and their control. We discuss first results on numerical
and technical level obtained with the DAE solvers DASSL (in
Dymola/Modelica) and QUALIDAES.

### Melina Freitag (University of Bath)

Freitag, 19. Juni 2015

**Data assimilation as a very large inverse
problem - an introduction**

In this talk we aim to provide a theoretical framework for data assimilation, a specific type of an inverse problem arising for example in numerical weather prediction, hydrology and geology. We consider the general mathematical theory for inverse problems and regularisation, before introducing Tikhonov regularisation as one of the most popular methods for solving inverse problems. We show that data assimilation techniques such as 3DVar and 4DVar as well as the Kalman filter and Bayes' data assimilation are, in the linear case, a form of cycled Tikhonov regularisation. We give an introduction to key data assimilation methods as currently used in practice and explain computational challenges.

### Philipp Schulze (TU Berlin)

Donnerstag, 18. Juni 2015

**Data-Driven Model Reduction of Linear
Port-Hamiltonian Systems**

The main idea of data-driven model reduction is to create a small-dimensional model from data of numerical or real-world experiments without needing a state-space representation in the beginning. However, in general the obtained realization does not exhibit any structure or related properties of the original system.

In this talk we extend the Loewner framework for data-driven model reduction to preserve the structure of linear port-Hamiltonian systems. This provides the realization with both physical meaning and mathematical properties as stability and passivity. An example illustrates how to obtain a port-Hamiltonian realization only from input-output data.

Joint work with Arjan van der Schaft

### Jeroen Stolwijk (TU Berlin)

Donnerstag, 25. Juni 2015

**Numerical Solution and Error Analysis for
the Euler Equations in Stationary Form**

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

Natural gas transportation is commonly modelled by the one-dimensional Euler equations. Although this is a widely accepted model in academia, we will see in this presentation that even the great mathematician Leonhard Euler himself encountered difficulties applying this model in a practical engineering problem.

Several simplifications of the Euler equations lead us to a system with three stationary partial differential equations together with two algebraic constraints. We will numerically solve this system using a first order forward difference scheme, Newton's method and adequate boundary and parameter values. Finally, we will perform an error analysis for the impulse equation by determining the magnitude of the discretisation, rounding and data errors as well as their amplification in the computation of the pressure.

Joint work with V. Mehrmann.

Supported by the German Research
Foundation DFG.

### Jon Paul Janet (TU Berlin)

Donnerstag, 25. Juni 2015

**Analysis and solution of
dynamic flash equations**

In this talk, I will present preliminary results from my master’s thesis as part of the COSSE program. The topic concerns the simulation and control of two-phase flash separators, which are important and ubiquitous components of many industrial processes. The topic will be motivated, and two dynamic models proposed in literature will be presented: a full model and one based on simplifying physical assumptions. In general, the system is described by a hybrid physical/empirical system of DAEs with a high degree of nonlinearity.

The index of the models will be analysed in a behaviour setting and the impact of the dependencies of the empirical terms is explored, resulting in deriving explicit conditions for both systems to be index 1. It is claimed in the literature that the simplified model has an increased d-index and this can be clearly understood from the behaviour setting.

Finally, some numerical results for a test problem are presented and analysed. Both systems and two solvers are compared.

### Thomas Berger (Universität Hamburg)

Montag, 29. Juni 2015

**Regularization of linear
descriptor systems**

For linear time-invariant
descriptor systems we consider different regularization approaches.
First, the question whether there exists a feedback which renders the
closed-loop system regular is considered. This property can be
equivalently characterized by simple algebraic and geometric
conditions in terms of the involved matrices and the augmented Wong
sequences. We also consider the slightly more general problem of
existence of a feedback such that an autonomous closed-loop system is
obtained. For systems which are not regularizable by feedback, an
additional behavioral equivalence transformation and a
reorganization of input and state variables leads to a regular system,
the index of which is at most one. This procedure is known (see
Campbell et al., Regularization of linear and nonlinear descriptor
systems, 2012) and we present a new approach which allows for a
detailed characterization of the resulting regular system. In
particular, this system is fully determined by the augmented Wong
sequences. The aforementioned result can be further improved and we
show that a feedback is actually not necessary. To this end, we
provide an algorithmic procedure for the construction of the
regularization and discuss computational aspects.

### Zoran Tomljanović (University J.J. Strossmayer in Osijek, Croatia)

Montag, 06. Juli 2015

**Optimization of
semi-active damping and external damping in mechanical system with
external force**

We present an efficient approach
for determination on an optimal semi-active damping and we also
consider damping optimization in systems excited by external
force.

First we study the problem of determining an optimal semi-active
damping of vibrating systems. For this damping optimization we use a
minimization criterion based on the impulse response energy of the
system. In this case the optimization approach yields a large number
of Lyapunov equations which have to be solved, thus we propose an
optimization approach that works with

reduced systems which
accelerate optimization process. Reduced systems are generated using
the parametric dominant pole algorithm. The optimization process is
additionally accelerated with a modal approach while the initial
parameters for the parametric dominant pole algorithm are chosen
during optimization procedure using residual bounds. Our approach
calculates a

satisfactory approximation of the impulse response
energy while providing a significant acceleration of the optimization
process.

In the second part we consider optimization of external damping in
mechanical system excited by external force. We introduce two criteria
based on the minimization of the energy functions, that allow a
damping optimization in mechanical systems with external force. This
optimization problem is a very demanding due to the numerous linear
systems that have to be solved.

For that purpose we have derived
the new formulas which allow us to calculate energy functions very
efficiently.

Numerical results illustrate the effectiveness of the
proposed approaches.

Joint work with Peter Benner, Patrick Kürschner, Krešimir Veselić and Ninoslav Truhar.

Abstract as pdf [24].

### Leonhard Batzke (TU Berlin)

Donnerstag, 02. Juli 2015

**Low-rank perturbations of
structured matrix pencils**

In this talk, I will
present several results from my thesis on generic low-rank
perturbations of structured regular matrix pencils. The focus will be
on structure-preserving rank-1 perturbations of T-alternating matrix
pencils. I will also briefly touch on other types of structured matrix
pencils and perturbations of higher rank.

### Benjamin Unger (TU Berlin)

Donnerstag, 02. Juli 2015

**Delay DAEs and their
regularization**

We recall the concept of well-posedness in terms of delay differential- algebraic equations (DDAE) and outline the need for a regularization procedure. Based on a review of the existing procedure for linear time-invariant DDAEs, introduced in [Ha, Mehrmann, 2014], we introduce a new regularization methodology that is a generalization of the strangeness index concept for DAEs [Kunkel, Mehrmann, 2006] and allows for an efficient determination of the strangeness and shift index.

### Christoph Zimmer (TU Berlin)

Donnerstag, 16. Juli 2015

**On Linear
Operator-differential-algebraic Equations with Delay**

Constrained linear partial differential equations (lin. PDAEs) have an important role in modeling practical systems such as the incompressible linearized Navier-Stokes equation. On the other hand, time-delays occur naturally in closed-loop controlled dynamical systems, since measurements, signal transmissions, and calculations of the control require a certain time. The combination of lin. PDAEs and time-delays leads to a new mathematical object, which in includes a various number of challenges. In this talk, we investigate this kind of object in the abstract setting of linear operator-differential-algebraic equations. In particular we consider the unsteady Stokes equation with and without delay and show existence results.

### Daniel Bankmann (TU Berlin)

Donnerstag, 16. Juli 2015

**OVDBDF - A Software Package
for the Numerical Integration of Differential Algebraic
Equations**

Differential algebraic equations (DAEs) arise in many applications as multi-body systems or networks (e.g. electrical circuits) when modeling their dynamicalbehavior and can be obtained in particular via automatic modeling. We consider DAEs in the most general form

(1) F(t,x,x') = 0,

where t is the independent variable, x the state vector and x' its derivative and the partial derivative of F with respect to x' is possibly singular.

In this context the so-called hidden constraints of the system play an important role in terms of numerical robustness. These are constraints that are not explicitly given in the equations (1). Failing to provide these constraints explicitly might lead to numerical drifts. There exist various (numerical) techniques to determine these constraints, e.g. the procedure mentioned in [1] for quasi-linear DAEs. Thus, we can additionally impose

(2) G(t,x) = 0,

where G contains all the explicit and hidden constraints of the original system F, without changing the solution set.

In this talk we present the solver OVDBDF for the numerical integration of such an overdetermined system fulfilling (1) and (2) that is based on the DASSL code for square nonlinear DAEs using backward difference formulas (BDF). A similar approach has been taken by QUALIDAES for quasi-linear DAEs using Radau IIA methods. Besides a sophisticated stepsize and order control a key point for the robust and efficient numerical integration in OVDBDF is the precise consideration of the hidden constraints (2).

[1] Steinbrecher,A. Analysis of Quasi-Linear Differential-Algebraic Equations. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany, number 11-2006. 2006.

### Michael Götte (TU Berlin)

Donnerstag, 09. Juli 2015

**Backend design of a Modelica
compiler**

AMSUN is an interdisciplinary project combining modeling, compiling and simulation on the basis of Modelica. In my short talk I will present my work as a student assistant designing an C-interface for solvers like QUALIDAES and ovdBDF. The focus is dealing with automation. For that we are testing different ideas for index reduction based on the method of Pryce and automatic differentiation. This new approaches are exploiting the fact that the two solvers can deal with overdetermined systems.

### Ute Kandler (TU Berlin)

Donnerstag, 16. Juli 2015

**A certain Gram-Schmidt variant -
Compensated Gram-Schmidt **

In various applications
like tensor calculus or mixed precision arithmetic vector operations
like matrix-vector multiplication, summing, scaling and inner products
can not be evaluated exactly. We investigate the behavior of the QR
decomposition using different variants of the Gram-Schmidt
orthogonalization scheme. In particular, we introduce a variant,
called the Compensated Gram-Schmidt orthogonalization, that uses
a slightly different projector to alleviate the damage, the
perturbations inflict on the orthogonality of Q. We compare this new
variant with the well known classical and modified Gram Schmidt
methods.

### Helia Niroomand Rad (TU Berlin)

Donnerstag, 16. Juli 2015

**A DAE Modeling Approach
toward the Crosstalk Phenomenon**

In this talk, we mainly propose a modeling approach in a general framework in order to describe the crosstalk phenomenon in electro-magnetic systems, and in particular, within electrical circuits. Crosstalk in an electrical circuit, loosely speaking, refers to the undesirable disturbance coupling in between the electrical elements and its effect on the entire circuit.

We model the phenomenon by bilateral coupling of two sets of differential equations where the main set is the circuit equations formulated in the framework of modified nodal analysis, and the second set consists of the non-stationary Maxwell equations. By spacial discretization of the second set, e.g., the non-stationary Maxwell equations, we obtain a large set of differential-algebraic equations (DAEs) modeling the crosstalk phenomenon.

In the last part, we shortly focus on analysis of the DAE system corresponding to the non-stationary Maxwell equations.

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