Zoran Tomljanović
Associate Professor

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.
 BSc in Mathematics, Department of Mathematics, University of Zagreb, Croatia, December 2005.
 19972001 Mathematical Gymnasium at high school in Našice
Publications
 N. Jakovčević Stor, I. Slapničar, Z. Tomljanović, Fast Computation of Optimal Damping Parameters for Linear Vibrational Systems, Mathematics 10/5 (2022), 117We 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 secondorder 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 lowrank matrix that depends on the eigenfrequencies that need to be damped. The optimization process in the standard approach requires O(n3) floatingpoint operations. In our approach, we transform the linearized matrix into an eigenvalue problem of a diagonalpluslowrank matrix whose eigenvectors have a Cauchylike structure. Our algorithm is based on a new fast eigensolver for complex symmetric diagonalplusrankone matrices and fast multiplication of linked Cauchylike 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.
 N. Jakovčević Stor, T. Mitchell, Z. Tomljanović, M. Ugrica, Fast optimization of viscosities for frequencyweighted damping of secondorder systems, Journal of Applied Mathematics and Mechanics (2022), prihvaćen za objavljivanjeWe consider frequencyweighted damping optimization for vibrating systems described by a secondorder 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 diagonalplusrankone 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, D. Tolić, Z. Tomljanović, I. Palunko, Numerically Efficient H∞ Analysis of Cooperative MultiAgent Systems, Journal of The Franklin Institute 359/16 (2022), 91109128This article proposes a numerically efficient approach for computing the maximal (or minimal) impact one agent has on the cooperative system it belongs to. For example, if one is able to disturb/bolster merely one agent in order to maximally disturb/bolster the entire team, which agent to choose? We quantify the agenttosystem impact in terms of $H_{infty}$ norm whereas output synchronization is taken as the underlying cooperative control scheme. The agent dynamics are homogeneous, second order and linear whilst communication graphs are weighted and undirected. We devise simple sufficient conditions on agent dynamics, topology and output synchronization parameters rendering all agenttosystem $H_{infty}$ norms to attain their maxima in the origin (that is, when constant disturbances are applied). Essentially, we quickly identify bottlenecks and weak/strong spots in multiagent systems without resorting to intense computations, which becomes even more important as the number of agents grows. Our analyses also provide directions towards improving communication graph design and tuning/selecting cooperative control mechanisms. Lastly, numerical examples with a large number of agents and experimental verification employing offtheshelf nano quadrotors are provided.
 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.
 C. Beattie, S. Gugercin, Z. Tomljanović, Samplingfree model reduction of systems with lowrank parameterization, Advances in Computational Mathematics 46/6 (2020), 134We consider the reduction of parametric families of linear dynamical systems having an affine parameter dependence that allow for lowrank 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 parameterdependent function. One may apply any one of a number of standard (nonparametric) 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 datadriven variation of our approach is described that exercises this flexibility through the use of limited frequencysampling of the underlying nonparametric 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 nonparametric 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.
Projects

Accelerated solution of optimal damping problems,  scientific project; supported by the DAAD for period 20212022 (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 (IP2019046774, 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 (IP2016062468, 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 20172018 (principal investigator together with Matthias Voigt); cooperation with TU Berlin, Germany

Optimization of parameter dependent mechanical systems  scientific project (IP2014099540; 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 20152016 (investigator); cooperation with Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg

European Model Reduction Network (EUMORNET). Funded by: COST (European Cooperation in Science and Technology) (investigator).

Optimization of semiactive 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 20132014 (investigator)

Passive control of mechanical models  scientific project No.23523528181042 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 20082009 (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
 Coorganizer of the 11th Conference on Applied Mathematics and Scientific Computing 59 September 2020, Brijuni, Croatia: webpage

Coorganizer of the 3rd Workshop on Optimal Control of Dynamical Systems and applications, 2831 March 2022 at Department of Mathematics, J. J. Strossmayer University of Osijek: webpage

Coorganizer of the Workshop on Optimal Control of Dynamical Systems and applications, 56 November 2020 at Department of Mathematics, J. J. Strossmayer University of Osijek: webpage

Coorganizer of the 10th Conference on Applied Mathematics and Scientific Computing 1418 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

Coorganizer of International Workshop on Optimal Control of Dynamical Systems and applications, 2022 June 2018 at Department of Mathematics, J. J. Strossmayer University of Osijek, webpage

Coorganizer of Workshop on Model Reduction Methods and Optimization, 2021 September 2016, in Opatija, Croatia, webpage

Coorganizer of The third International School on Model Reduction for Dynamical Control Systems, 5  10 October 2015, in Dubrovnik, Croatia, webpage

Coorganizer of the DAAD International School on Linear Optimal Control of Dynamic Systems, 23  28 September 2013, Osijek, webpage
 Coorganizer of the Summer School on Numerical Linear Algebra for Dynamical and HighDimensional Problems, October 1015, 2011, Trogir, Croatia, webpage
Refereeing/Reviewing
Refereeing:
 Mathematical Communications
 Osječki matematički list
Workshop and Conference Talks
 Numerically Efficient AgentsToGroup H_{infty} Analysis, Mathmod2022, 2729 July 2022, Vienna, Austria
 Numerically Efficient AgentstoGroup H∞ Analysis, COST Action MatDynNet WG3+WG5 Meeting, May 1820, 2022, Namur, Belgium
 Numerically Efficient AgentstoGroup $H_{\infty}$ Analysis, 3rd Workshop on Optimal Control of Dynamical Systems and applications, 2831 March 2022 at Department of Mathematics, J. J. Strossmayer University of Osijek, Osijek, Croatia
 Optimal damping of vibrational systems using finite time energy criterion, Workshop on Control of Dynamical Systems, 1416 June 2021, Dubrovnik, Croatia
 Samplingfree model reduction of systems with lowrank parameterization, Croatian German meeting on analysis and mathematical physics, 22nd to 25th March 2021,
 Frequencyweighted damping via non smooth optimization and fast computation of QEPs, 91st GAMM Annual Meeting in Kassel, 1519 March 2021, Kassel, Germany
 Efficient approximation of novel residual bounds for parameter dependent quadratic eigenvalue problem, Tenth Conference on Applied Mathematics and Scientific Computing, 1418 September 2020, Brijuni, Croatia
 H∞ damping optimization by adaptive interpolation, The sixth Najman conference on spectral theory and differential equations, 813 September 2019, Sveti Martin na Muri, Croatia
 Samplingfree parametric model reduction of structured systems, 4th Workshop on Model Reduction of Complex Dynamical Systems, 2830 August 2019, Graz, Austria
 Perturbation Theory for Quadratic Eigenvalue Problem  Applied on Damped Mechanical Systems, The 9th International Congress on Industrial and Applied Mathematics, Valencia, 1519 July 2019, Spain
 Semiactive Hinf damping optimization by adaptive interpolation, 90th Annual Meeting GAMM 2019, Vienna, 1822 February 2019, Austria
 Samplingfree parametric model reduction of structured systems, Numerical Analysis and Scientific Computation with Applications, 26 July 2018, Kalamata, Greece

SamplingFree Parametric Model Reduction Of Structured Systems, International Workshop on Optimal Control of Dynamical Systems and applications, Department of Mathematics, J. J. Strossmayer University of Osijek, 2022 June 2018, Osijek, Croatia
 Upper and Lower Bounds for Sines of Canonical Angles, SIAM Conference on Applied Linear Algebra (SIAMALA18), 48 May 2018, Hong Kong Baptist University, Hong Kong
 Samplingfree parametric model reduction of systems with structured parameter variation (poster), Model Reduction of Parametrized Systems IV, 1013 April 2018, Nantes, France
 Interpolationbased parametric model reduction for efficient damping optimization, Workshop Numerical methods for optimal control problems: algorithms, analysis and applications, 1923 June 2017, Rome, Italy
 Dimension reduction approach to the parameter dependent quadratic eigenvalue problem, 88th GAMM Annual Meeting, 610 March 2017, Weimar, Germany
 Damping optimization of parameter dependent mechanical systems by rational interpolation, 3rd Workshop on Model Reduction of Complex Dynamical Systems, Odense, Denmark, 1113 January 2017
 H_2 and H∞ semiactive damping optimization, 6TH CROATIAN MATHEMATICAL CONGRESS, June 14  17, 2016, Zagreb, Croatia
 Semiactive damping optimization using the parametric dominant pole algorithm, MatTriad'2015, 7  11 September 2015, Coimbra, Portugal
 Damping optimization in mechanical systems with external force(poster), Numerical Algebra, Matrix Theory, DifferentialAlgebraic Equations, and Control Theory, 69 May 2015, TU Berlin, Berlin, Germany
 Optimal direct velocity feedback and semiactive damping,Chemnitz–Zagreb Workshop on Harmonic Analysis for PDEs, Applications, and related topics 1st5th July 2014, TU Chemnitz, Chemnitz, Germany
 Optimal Direct Velocity Feedback, 10th International Workshop on Accurate Solution of Eigenvalue Problems, June 2  5, 2014, Dubrovnik, Croatia
 Semiactive Damping Optimization of Vibrational Systems Using Dimension Reduction, 84th Annual Meeting of the International Association of Applied Mathematics and Mechanics , March 1822, 2013, Novi Sad, Serbia
 Optimal Parameters for Damping Optimization in Linear Vibrating Systems, 9th International Workshop on Accurate Solution of Eigenvalue Problems, June 4  7, 2012, Napa Valley, USA
 Damping Optimization in Linear Vibrating Systems Using Dimension Reduction, International Conference on Vibration Problems 2011, September 58, 2011, Prague, Czech Republic
 Optimal Damping of Partial Spectra Using Dimension Reduction, 8th International Workshop on Accurate Solution of Eigenvalue Problems, June 28  July 1, 2010, Berlin, Germany
 Dimension Reduction For Damping Optimization In Linear Vibrating Systems, Applied Linear Algebra In honor of Hans Schneider , May 2428, 2010, Novi Sad, Serbia
 Damping optimization of linear vibrating systems using dimension reduction, GAMM Workshop , Applied and Numerical Linear Algebra, September 1011, 2009, ETH Zurich, Zurich, Switzerland
Seminar talks
 Numerically Efficient $H_{infty}$ Analysis of Cooperative MultiAgent Systems, ApplMath22, 59 September 2022, Brijuni, Croatia
 Numerically Efficient AgentstoGroup H∞ Analysis, COST Action MatDynNet WG3+WG5 Meeting, May 1820, 2022, Namur, Belgium
 Fast optimization of viscosities for frequencyweighted damping of secondorder systems, Applied numerical analysis seminar,17 Sep 2021, Virginia Polytechnic Institute and State University, Blacksburg, USA
 Damping optimization in mechanical systems using parametric model reduction, Seminar za primijenjenu matematiku i teoriju upravljanja, University of Dubrovnik, 5 February 2021
 Samplingfree model reduction of systems with lowrank parameterization, Optimization and application seminar, 25. November 2020, J. J. Strossmayer University of Osijek Department of Mathematics
 Damping optimization in mechanical systems using samplingfree model reduction, MAgdeburg Lectures on Optimization and Control, September 25, 2020, Magdeburg, Germany
 Hinf damping optimization by adaptive interpolation, Applied Numerical Analysis Seminar, 26 April 2019, Departments of Mathematics at Virginia Tech, USA
 Approximation bounds for parameter dependent quadratic eigenvalue problem, Applied Numerical Analysis Seminar, 4 Oct 2017, Departments of Mathematics at Virginia Tech, USA
 Optimization of semiactive damping and viscous damping in excited mechanical systems, Matrix Computation Seminar, 10 November 2015, Departments of Mathematics at Virginia Tech, USA
 Optimization of semiactive damping and external damping in mechanical systems with external force, AbsolventenSeminar  Numerische Mathematik, 30 June 2015, Berlin, Germany
 Damping optimization in mechanical systems with external force, CSC
Seminar at MPI Magdeburg, Germany, 5 May 2015
 Optimizacija prigušenja u mehaničkim sustavima sa vanjskom silom, Department of Mathematics, University of Osijek, Optmization and applications seminar, 15 April 2015
 Optimization of semiactive damping and viscous damping in mechanical systems with external force, October 21, 2014, University in Innsbruck, Department of Mathematics, Austria
 Optimization of mechanical systems using dimension reduction, October 17, 2013, Forschungsseminar Scientific Computing, Technische Universitat Chemnitz, Germany
 Optimalno prigušenje kod vibracijskih sistema koristeći redukciju dimenzije, March 17, 2011, Seminar za numeričku matematiku i računarstvo, PMFMathematical Departments, University of Zagreb, Croatia
 Optimizacija prigušenja u vibracijskim sistemima pomoću redukcije dimenzije sistema, April 29, 2009,Optmization and applications seminar, University of Osijek, Department of Mathematics, Croatia
 An efficient algorithm for solving and optimizing some types of Lyapunov, May 15, 2007, TU Chemnitz, Research Seminar Numerics, Germany
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, 21/04/201927/04/2019, 02/10/201714/10/2017, 4/11/201520/11/2015
 visiting researcher at TU Berlin, Germany 11/11/201815/11/2018, 16/07/201822/07/2018, 07/01/201814/01/2018, 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 19/10/201422/10/2014
 visiting researcher at Max Planck Institute for Dynamics of Complex Technical Systems in Magdeburg, Germany 12/7/201819/7/2017, 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 at J. J. Strossmayer University of Osijek, Department of Mathematics, since October 2017  2021
 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 (ured 18 u prizemlju Odjela za matematiku):.
 srijedom u 10 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 Cu 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  Udaljenost do neupravljivog sustava
obraditi pojam upravljivosti i važnost udaljenosti do neupravljivog sustava
obraditi glavne teorijske rezultate s naglaskom na primjenu u robotici i mehaničkim sustavima
implementirati metodu za udaljenost do neupravljivog sustava te ju na primjerima ilustrirati  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  Iterativne metode za rješavanje linearnih sustava
 napraviti osnovni pregled iterativnih metode za sustave
 imlementirati neku od metoda te napraviti ilustraraciju na numeričkim primjerima  NPpotpuni problem i redukcija problema (zauzeto)
 definirati pojam NPpotpnog problema
 obraditi pojam redukcije i detaljno ilustrirati redukciju na primjeru
Nastavne aktivnosti u zimskom semestru Akademske 2022./2023.
Lijearna algebra I, predavanja
srijedom od 810,
Nastavne aktivnosti u ljetnom semestru Akademske 2022./2023.
Osnove teorije upravljanja s primjenama, predavanja
Teorijske osnove računalne znanosti, predavanja i vježbe
Personal