### Zoran Tomljanović

Josip Juraj Strossmayer University of Osijek

### 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,

MSc in Mathematics, Department of Mathematics, University of Zagreb, Croatia, December 2005,

1997-2001 Mathematical Gymnasium at high school in Našice

### Publications

- K. Sabo, R. Scitovski, Š. Ungar, Z. Tomljanović, A method for searching for a globally optimal k-partition of higher-dimensional datasets, Journal of Global Optimization (2024), prihvaćen za objavljivanjeThe problem with finding a globally optimal k-partition of a set A is a very intricate optimization problem for which in general, except in the case of one-dimensional data, i.e., for data with one feature (A\subset\R), there is no method to solve. Only in the one-dimensional case there exist efficient methods that are based on the fact that the search for a globally optimal partition is equivalent to solving a global optimization problem for a symmetric Lipschitz-continuous function using the global optimization algorithm DIRECT. In the present paper, we propose a method for finding a globally optimal k-partition in the general case (A\subset \R^n, n\geq 1), generalizing an idea for solving the Lipschitz global optimization for symmetric functions. To do this, we propose a method that combines a global optimization algorithm with linear constraints and the k-means algorithm. The first of these two algorithms is used only to find a good initial approximation for the $k$-means algorithm. The method was tested on a number of artificial datasets and on several examples from the UCI Machine Learning Repository, and an application in spectral clustering for linearly non-separable datasets is also demonstrated. Our proposed method proved to be very efficient.
- Z. Tomljanović, Damping optimization of the excited mechanical system using dimension reduction, Mathematics and Computers in Simulation
**207**(2023), 24-40We consider a mechanical system excited by a periodic external force. The main problem is to determine the best damping matrix to be able to minimize the system average displacement amplitude. Damping optimization usually includes optimization of damping positions and corresponding damping viscosities. Since the objective function is non-convex, a standard optimization approach requires a large number of objective function evaluations. We first propose a dimension reduction approach that calculates approximation of the average displacement amplitude and additionally we efficiently use a low rank update structure that appears in the external damping matrix. Moreover, an error bound which allows determination of appropriate approximation orders is derived and incorporated within the optimization method. We also present a theoretical error bound that allows determination of effective damping positions. The methodology proposed here provides a significant acceleration of the optimization process. The gain in efficiency is illustrated in numerical experiments. - N. Jakovčević Stor, T. Mitchell, Z. Tomljanović, M. Ugrica, Fast optimization of viscosities for frequency-weighted damping of second-order systems, Journal of Applied Mathematics and Mechanics
**103**/5 (2023), 1-21We consider frequency-weighted damping optimization for vibrating systems described by a second-order differential equation. The goal is to determine viscosity values such that eigenvalues are kept away from certain undesirable areas on the imaginary axis. To this end, we present two complementary techniques. First, we propose new frameworks using nonsmooth constrained optimization problems, whose solutions both damp undesirable frequency bands and maintain the stability of the system. These frameworks also allow us to weight which frequency bands are the most important to damp. Second, we also propose a fast new eigensolver for the structured quadratic eigenvalue problems that appear in such vibrating systems. In order to be efficient, our new eigensolver exploits special properties of diagonal-plus-rank-one complex symmetric matrices, which we leverage by showing how each quadratic eigenvalue problem can be transformed into a short sequence of such linear eigenvalue problems. The result is an eigensolver that is substantially faster than standard techniques. By combining this new solver with our new optimization frameworks, we obtain our overall algorithm for fast computation of optimal viscosities. The efficiency and performance of our new approach are verified and illustrated on several numerical examples. - I. Nakić, M. Pilj Vidaković, Z. Tomljanović, Finite time horizon mixed control of vibrational systems, SIAM Journal on Scientific Computing (2023), prihvaćen za objavljivanjeWe consider a vibrational system control problem over a finite time horizon. The performance measure of the system is taken to be $p$-mixed $H_2$ norm which generalizes the standard $H_2$ norm. We present an algorithm for efficient calculation of this norm in the case when the system is parameter dependent and the number of inputs or outputs of the system is significantly smaller than the order of the system. Our approach is based on a novel procedure which is not based on solving Lyapunov equations and which takes into account the structure of the system. We use a characterization of the $H_2$ norm given in terms of integrals which we solve using adaptive quadrature rules. This enables us to use recycling strategies as well as parallelization. The efficiency of the new algorithm allows for an analysis of the influence of various system parameters and different finite time horizons on the value of the $p$-mixed $H_2$ norm. We illustrate our approach by numerical examples concerning an $n$-mass oscillator with one damper.
- N. Jakovčević Stor, I. Slapničar, Z. Tomljanović, Fast Computation of Optimal Damping Parameters for Linear Vibrational Systems, Mathematics
**10**/5 (2022), 1-17We propose a fast algorithm for computing optimal viscosities of dampers of a linear vibrational system. We are using a standard approach where the vibrational system is first modeled using the second-order structure. This structure yields a quadratic eigenvalue problem which is then linearized. Optimal viscosities are those for which the trace of the solution of the Lyapunov equation with the linearized matrix is minimal. Here, the free term of the Lyapunov equation is a low-rank matrix that depends on the eigenfrequencies that need to be damped. The optimization process in the standard approach requires O(n3) floating-point operations. In our approach, we transform the linearized matrix into an eigenvalue problem of a diagonal-plus-low-rank matrix whose eigenvectors have a Cauchy-like structure. Our algorithm is based on a new fast eigensolver for complex symmetric diagonal-plus-rank-one matrices and fast multiplication of linked Cauchy-like matrices, yielding computation of optimal viscosities for each choice of external dampers in O(kn2) operations, k being the number of dampers. The accuracy of our algorithm is compatible with the accuracy of the standard approach.

### Projects

- Accelerated solution of optimal damping problems, — scientific project; supported by the DAAD for period 2021–2022 (principal investigator together with Jens Saak); cooperation with Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany
- 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 together 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 8th Croatian Mathematical Congress in Osijek , to be held on July 2 – 5, 2024, at the School of Applied Mathematics and Informatics in Osijek.- Co-organizer of the Winter School on Model Reduction for Optimization and Control that will be held on 19 – 23 February 2024 in Dubrovnik, Croatia: webpage
- Co-organizer of the 3rd Workshop on Optimal Control of Dynamical Systems and applications, 28-31 March 2022 at Department of Mathematics, J. J. Strossmayer University of Osijek: webpage
- 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: webpage
- Co-organizer of the 10th 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, webpage
- 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, webpage
- Co-organizer of Workshop on Model Reduction Methods and Optimization, 20-21 September 2016, in Opatija, Croatia, webpage
- Co-organizer of The third International School on Model Reduction for Dynamical Control Systems, 5 – 10 October 2015, in Dubrovnik, Croatia, webpage
- Co-organizer of the DAAD International School on Linear Optimal Control of Dynamic Systems, 23 – 28 September 2013, Osijek, webpage
- Co-organizer of the Summer School on Numerical Linear Algebra for Dynamical and High-Dimensional Problems, October 10-15, 2011, Trogir, Croatia, webpage

### Teaching

**Konzultacije (Office Hours):**

Termini sljedećih konzultacija (ured 18 u prizemlju Odjela za matematiku):

- utorkom u 12:00 sati

**Teme diplomskih i završnih radova:**

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**Interpolacijske metoda za redukciju reda modela**

– obraditi osnovne metode za radukciju i usprediti ih sa nekim direktnim pristupima

– implementirati i ilustrirati na primjerima**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**Iterativne metode za rješavanje linearnih sustava**

– napraviti osnovni pregled iterativnih metode za sustave

– imlementirati neku od metoda te napraviti ilustraraciju na numeričkim primjerima

## Nastavne aktivnosti u zimskom semestru Akademske 2023./2024.

Linearna algebra I, predavanja

srijedom 8-10h,

Redukcija modela i aproksimacijski pristupi,

utorkom 8-10h

## Nastavne aktivnosti u ljetnom semestru Akademske 2023./2024.

Osnove teorije upravljanja s primjenama, predavanja

Utorkom 10-12 h