# Zoran Tomljanović

Associate Professor
Department of Mathematics
Josip Juraj Strossmayer University of Osijek
Trg Ljudevita Gaja 6
Osijek, HR-31000, Croatia¸
 phone: +385-31-224-847 fax: +385-31-224-801 email: ztomljan @ mathos.hr office: E (ground floor), near student office/pored referade

## Research Interests

Numerical linear algebra
Damping optimization in mechanical systems
Control Theory

Matrix Equations

## Degrees

• PhD in Mathematics, Department of Mathematics, University of Zagreb, May 2011.

PhD Thesis: Optimal damping for vibrating systems using dimension reduction  (PDF)

• BSc in Mathematics, Department of Mathematics, University of Zagreb, Croatia, December 2005.
• 1997-2001 Mathematical Gymnasium at high school in Našice

## Publications

Journal Publications

1. N. Truhar, Z. Tomljanović, R. Li, Perturbation Theory for Hermitian Quadratic Eigenvalue Problem -- Damped and Simultaneously Diagonalizable Systems, Applied mathematics and computation 371 (2020)
The main contribution of this paper is a novel approach to the perturbation theory of a structured Hermitian quadratic eigenvalue problems \$(lambda^2 M + lambda D + K) x=0\$. We propose a new concept without linearization, considering two structures: general quadratic eigenvalue problems (QEP) and simultaneously diagonalizable quadratic eigenvalue problems (SDQEP). Our first two results are upper bounds for the difference \$left| | X_2^* M widetilde{;X};_1 |_F^2 - | X_2^* M {;X};_1 |_F^2 right|\$, and for \$| X_2^* M widetilde X_1 - X_2^* M X_1|_F\$, where the columns of \$X_1=[x_1, ldots, x_k]\$ and \$X_2=[x_{;k+1};, ldots, x_n]\$ are linearly independent right eigenvectors and \$M\$ is positive definite Hermitian matrix. As an application of these results we present an eigenvalue perturbation bound for SDQEP. The third result is a lower and an upper bound for \$|sin{;Theta(mathcal{;X};_1, widetilde{;mathcal{;X};};_1)}; |_F\$, where \$Theta\$ is a matrix of canonical angles between the eigensubspaces \$mathcal{;X};_1 \$ and \$widetilde{;mathcal{;X};};_1\$, \$mathcal{;X};_1 \$ is spanned by the set of linearly independent right eigenvectors of SDQEP and \$widetilde{;mathcal{;X};};_1\$ is spanned by the corresponding perturbed eigenvectors. The quality of the mentioned results have been illustrated by numerical examples.
2. C. Beattie, S. Gugercin, Z. Tomljanović, Sampling-free model reduction of systems with low-rank parameterization, Advances in Computational Mathematics 46/6 (2020), 1-34
We consider the reduction of parametric families of linear dynamical systems having an affine parameter dependence that allow for low-rank variation in the state matrix. Usual approaches for parametric model reduction typically involve exploring the parameter space to identify representative parameter values and the associated models become the principal focus of model reduction methodology. These models are then combined in various ways in order to interpolate the response. The initial exploration of the parameter space can be a forbiddingly expensive task. A different approach is proposed here that requires neither parameter sampling nor parameter space exploration. Instead, we represent the system response function as a composition of four subsystem response functions that are nonparametric with a purely parameter-dependent function. One may apply any one of a number of standard (non-parametric) model reduction strategies to reduce the subsystems independently, and then conjoin these reduced models with the underlying parameterization to obtain the overall parameterized response. Our approach has elements in common with the parameter mapping approach of Baur et al. (PAMM 14(1), 19–22 2014) but offers greater flexibility and potentially greater control over accuracy. In particular, a data-driven variation of our approach is described that exercises this flexibility through the use of limited frequency-sampling of the underlying non-parametric models. The parametric structure of our system representation allows for a priori guarantees of system stability in the resulting reduced models across the full range of parameter values. Incorporation of system theoretic error bounds allows us to determine appropriate approximation orders for the non-parametric systems sufficient to yield uniformly high accuracy across the parameter range. We illustrate our approach on a class of structural damping optimization problems and on a benchmark model of thermal conduction in a semiconductor chip. The parametric structure of our reduced system representation lends itself very well to the development of optimization strategies making use of efficient cost function surrogates. We discuss this in some detail for damping parameter and location optimization for vibrating structures.
3. Z. Tomljanović, M. Voigt, Semi-active H∞-norm damping optimization by adaptive interpolation, Numerical Linear Algebra with Applications 27/4 (2020), 1-17
In this work we consider the problem of semi-active damping optimization of mechanical systems with fixed damper positions. Our goal is to compute a damping that is locally optimal with respect to the H∞-norm of the transfer function from the exogenous inputs to the performance outputs. We make use of a new greedy method for computing the H∞-norm of a transfer function based on rational interpolation. In this paper, this approach is adapted to parameter-dependent transfer functions. The interpolation leads to parametric reduced-order models that can be optimized more efficiently. At the optimizers we then take new interpolation points to refine the reduced-order model and to obtain updated optimizers. In our numerical examples we show that this approach normally converges fast and thus can highly accelerate the optimization procedure. Another contribution of this work are heuristics for choosing initial interpolation points.
4. N. Truhar, Z. Tomljanović, M. Puvača, Approximation of damped quadratic eigenvalue problem by dimension reduction, Applied mathematics and computation 347 (2019), 40-53
This paper presents an approach to the efficient calculation of all or just one important part of the eigenvalues of the parameter dependent quadratic eigenvalue problem \$(lambda^2(mathbf{;v};) M + lambda(mathbf{;v};) D(mathbf{;v};) + K) x(mathbf{;v};) = 0\$, where \$M, K\$ are positive definite Hermitian \$ntimes n\$ matrices and \$D(mathbf{;v};)\$ is an \$ntimes n\$ Hermitian semidefinite matrix which depends on a damping parameter vector \$mathbf{;v};= begin{;bmatrix}; v_1 & ldots & v_k end{;bmatrix};in mathbb{;R};_+^k\$. With the new approach one can efficiently (and accurately enough) calculate all (or just part of the) eigenvalues even for the case when the parameters \$v_i\$, which in this paper represent damping viscosities, are of the modest magnitude. Moreover, we derive two types of approximations with corresponding error bounds. The quality of error bounds as well as the performance of the achieved eigenvalue tracking are illustrated in several numerical experiments.
5. I. Nakić, Z. Tomljanović, N. Truhar, Mixed control of vibrational systems, Journal of Applied Mathematics and Mechanics 99/9 (2019), 1-15
We consider new performance measures for vibrational systems based on the \$H_2\$ norm of linear time invariant systems. New measures will be used as an optimization criterion for the optimal damping of vibrational systems. We consider both theoretical and concrete cases in order to show how new measures stack up against the standard measures. The quality and advantages of new measures as well as the behaviour of optimal damping positions and corresponding damping viscosities are illustrated in numerical experiments.

## Projects

• Vibration Reduction in Mechanical Systems -- scientific project (IP-2019-04-6774, VIMS). This project has been fully supported by Croatian Science Foundation for the period 01.01.2020.--31.12.2023. (principal investigator)

• Control of Dynamical Systems -- scientific project (IP-2016-06-2468, ConDyS). This project has been fully supported by Croatian Science Foundation for the period 01.03.2017.--28.02.2021. (investigator)

• Robustness optimization of damped mechanical systems, -- scientific project; supported by the DAAD for period 2017--2018 (principal investigator with Matthias Voigt); cooperation with TU Berlin, Germany

• Optimization of parameter dependent mechanical systems  -- scientific project (IP-2014-09-9540; OptPDMechSys). This project has been fully supported by Croatian Science Foundation for the period 01.07.2015.--30.06.2019. (investigator)

• Damping optimization in mechanical systems excited with external force -- scientific project; supported by the J. J. Strossmayer University of Osijek for period 2015 (principal investigator)

• Mixed Integer Nonlinear Programming (MINLP) for damper optimization -- scientific project; supported by the DAAD for period 2015--2016 (investigator); cooperation with Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg

• European Model Reduction Network (EU-MORNET). Funded by: COST (European Cooperation in Science and Technology) (investigator).

• Optimization of semi-active damping in vibrational systems -- scientific project; supported by the J. J. Strossmayer University of Osijek for period 2014 (principal investigator); cooperation with Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg

• Optimal Damping of Vibrating Systems -- scientific project; supported by the DAAD for period 2013--2014 (investigator)

• Passive control of mechanical models -- scientific project No.235-2352818-1042 of the Croatian Ministry of Science, Education and Sports for period 2007.-- (investigator)

• Optimization algorithms for determination of optimal damping in mechanical systems -- scientific project; supported by the Croatian Science Foundation for period 2008--2009 (principal investigator)

## Professional Activities

Professional Societiey Membership
• International Association of Applied Mathematics and Mechanics, GAMM
• GAMM Activity Group Applied and Numerical Linear Algebra, GAMM ANLA
• Croatian Mathematical Society, HMD
• Croatian Operational Research Society, CRORS
• Society for Industrial and Applied Mathematics, SIAM

Committee Memberships and organization

• Upcoming: Co-organizer of the Workshop on Optimal Control of Dynamical Systems and applications, 5-6 November 2020 at Department of Mathematics, J. J. Strossmayer University of Osijek http://vims.mathos.unios.hr/home/workshop2020

• Co-organizer of the Tenth Conference on Applied Mathematics and Scientific Computing 14-18 September 2020, Brijuni, Croatia. In 2020 we have a special section on optimal control of dynamical systems and applications, coorganized with the Department of Mathematics, University of Osijek., https://applmath.math.pmf.unizg.hr/2020

• Co-organizer of International Workshop on Optimal Control of Dynamical Systems and applications, 20-22 June 2018 at Department of Mathematics, J. J. Strossmayer University of Osijek, http://workshop.mathos.unios.hr

• Co-organizer of Workshop on Model Reduction Methods and Optimization, 20-21 September 2016, in Opatija, Croatia, http://www.mathos.unios.hr/index.php/443.

• Co-organizer of The third International School on Model Reduction for Dynamical Control Systems, 5 - 10 October 2015, in Dubrovnik, Croatia http://www.mathos.unios.hr/index.php/351

• Co-organizer of the DAAD International School on Linear Optimal Control of Dynamic Systems, 23 - 28 September 2013, Osijek http://www.mathos.unios.hr/locschool2013/

• Co-organizer of the Summer School on Numerical Linear Algebra for Dynamical and High-Dimensional Problems, October 10-15, 2011, Trogir, Croatia, http://www.mpi-magdeburg.mpg.de/mpcsc/events/trogir/

Refereeing/Reviewing

Refereeing:

• Mathematical Communications
• Osječki matematički list

Workshop and Conference Talks

Seminar talks

Study Visits Abroad and Professional Improvement:

• visiting researcher at Department of Mathematics, Carlos III University of Madrid, Spain 06/09/2016 - 15/09/2016
• visiting researcher at Departments of Mathematics at Virginia Tech, USA,  21/04/2019-27/04/2019, 02/10/2017-14/10/2017, 4/11/2015-20/11/2015
• visiting researcher at TU Berlin, Germany 11/11/2018-15/11/2018, 16/07/2018-22/07/2018, 07/01/2018-14/01/2018,  21/06/2017 - 29/6/2017, 30/03/2017 - 07/04/2017, 23/02/2016 - 25/02/2016, 29/6/2015 - 9/7/2015
• visiting researcher at University in Innsbruck, Department of Mathematics 19/10/2014-22/10/2014
• visiting researcher at Max Planck Institute for Dynamics of Complex Technical Systems in Magdeburg, Germany 12/7/2018-19/7/2017, 22/11 /16 - 26/11/16, 29/06/16 - 09/07/16, 15/02/16 - 23/02/16, 04/27/15 - 05/06/15,02/04/14 - 08/4/14,   5/2/2013 - 20/2/2013, 3/7/2013 - 30/7/2013, 26/8/2012 -29/9/2012, 29/8/2010 - 19/9/2010,
• visiting researcher at TU Chemnitz, Germany 13/5/2007 - 14/6/2007, 10/4/2008 - 8/5/2008, 10/5/2009 - 10/7/2009, 24/1/2010 - 24/2/2010, 9/10/2013 - 20/10/2013

Service Activities
• Deputy Head of Department for Education and Students, since October 2017 -
• Moderator of Optimization and application seminar, since October 2017 -
• Chairman of the Mathematical Colloquium in Osijek, since January 2017 -
• Erasmus+ coordinator on Department of Mathematics, since 2013 - 2017

## Teaching

Konzultacije (Office Hours):

Termini sljedećih konzultacija:

• petak 20.11.2020. u 10 sati
• petak 27.11.2020. u 10 sati
• utorak 1.12.2020. u 10 sati

U nastavku se nalaze nazivi tema i kratki opis, a više informacija studenti mogu dobiti na konzultacijama. Mole se zaniteresirani studenti da se jave ukoliko su zainteresirani za neku od tema.

• Numeričko rješavanje običnih diferencijalnih jednadžbi
- obraditi osnovne metode: Eulerova i osnovne Runge Kutta metode
- implementirati ih u Matlabu ili C-u i ilustrirati efikasnost na primjerima
• QR dekompozicija s pivotiranjem
- obraditi QR dekompoziciju i QR dekompoziciju s pivotiranjem
- implementirati ju u Matlabu
- na primjerima pokazati osnovne primjene npr. na određivanje ranga matrice
• Schurova dekompozicija i primjene
- definirati definirati Schurovu dekompoziciju
- obraditi osnovna svojstva i primjene
- implementirati i ilustrirati na primjeru
• Stabilnost sustava
-obraditi glavne teorijske rezultate s naglaskom na primjenu u robotici i mehaničkim sustavima
-implementirati metode za stabilizaciju i na primjerima napraviti usporedbu i ilustriraciju
• Metoda Gaussovih eliminacija s potpunim pivotiranjem
- obraditi metodu Gaussovih eliminacija s potpunim pivotiranjem
- u Matlabu napraviti ilustraciju metode kroz vizualizaciju koraka
- napraviti vizualizaciju rjesenja dvije jednadzbe s dvije nepoznanice
• AHP metoda za odlučivanje
- izgraditi model za rješavanje višekriterijskih problema odlučivanja;
- primjeniti AHP metodu za hijerarhijsko odlučivanje na primjerima
• Rezononcija
- obraditi pojam rezonancije kod mehaničkih sustava
- na primjerima sa dvije ili tri mase ilustrirati rezonanciju
• Iterativne metode za rješavanje linearnih sustava
- napraviti osnovni pregled iterativnih metode za sustave
- imlementirati neku od metoda te napraviti ilustraraciju na numeričkim primjerima
• NP-potpuni problem i redukcija problema
- definirati pojam NP-potpnog problema
- obraditi pojam redukcije i detaljno ilustrirati redukciju na primjeru

## Nastavne aktivnosti u zimskom semestru Akademske 2020./2021.

Građevinski fakultet Osijek

Matematika na prvoj godini Arhitekture i urbanizma

utorkom od 10 do 12 sati

## Nastavne aktivnosti u ljetnom semestru Akademske 2019./2020.

konzultativno, javiti se na email

Matematička teorija računarstva, predavanja i vježbe

ponedjeljkom od 8:30 do 12 sati

Teorijske osnove računalne znanosti, predavanja i vježbe

ponedjeljkom od 8:30 sati do 12 sati

Osnove teorije upravljanja s primjenama, predavanja i seminari

utorkom od 11 do 14 sati

### Nastavne aktivnosti u prošlosti (Past Courses)

PhD study, Parameter Dependent Nonlinear Eigenvalue Problems (with Prof. N. Truhar) (summer semester 2017/2018)

Matematička teorija računarstva (Mathematical theory of computation(exercises and lectures, winter semester  13/14, 14/15, 15/16, 16/17,17/18)

Teorija odlučivanja (Decision theory) (exercises and lectures, winter semester  13/14, 14/15, 15/16, 16/17,17/18)

Metode optimizacije (Optimization methods), (vježbe, winter semestar 08/09, 09/10, 10/11, 11/12, 12/13, 13/14, 16/17)

Numerička linearna algebra (Numerical linear algebra), (exercises, winter semester 06/07, 08/09, 09/10, 10/11, 11/12, 12/13, summer semester 07/08)

Računarski praktikum, (exercises and lectures, ljetni semestar 08/09, 09/10, 11/12, 12/13, 13/14, 14/15,15/16, 16/17, 17/18)

Metodika nastave informatike, (exercises and lectures, winter semester 11/12, 12/13, 13/14, 14/15, 15/16)

Elementarna geometrija (Elementary geometry), (exercises, ljetni semestar 05/06, 06/07)

Obične diferencijalne jednadžbe (Ordinary differential equations), (exercises, zimski semestar 06/07, 07/08)

Numerička matematika (Numerical mathematics), (exercises, zimski semestar 07/08, ljetni semestar 08/09)

Integralni račun (Integral calculus), (vježbe, ljetni semestar 09/10, 10/11)

Odjel za fiziku (Department of Physics)

Diferencijalne jednadžbe (Differential equations), vježbe, ljetni semestar 06/07, 10/11, 11/12

Geometrija ravnine i prostora - uvod u algebru (Geometry of plain and space - introduction to algebra), (vježbe, zimski semestar 06/07, 07/08, 08/09, 09/10, 10/11)

Ekonomski fakultet (Faculty of Economycs)

Matematika (Mathematics), (vježbe, zimski semestar 07/09, 08/09, 09/10, 10/11, 11/12, predavanja, zimski semestar 12/13)

Prehrambeno tehnološki fakultet

Inženjerska matematika , (vježbe 08/09, 10/11, 11/12, 12/13) Primijenjena matematika (vježbe 08/09, 09/10,11/12, 12/13, 13/14)

Građevinski fakultet Osijek

Matematika (na razlikovnoj godini), exercises and lectures 12/13, 13/14,16/17

Matematika (na Arhitekturi i urbanizmu) (winter semester 16/17, 17/18)

VSPU

Poslovna matematika, predavanja i vježbe 12/13, 13/14