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

1. Z. Tomljanović, $H_{\infty}$ Analysis of Cooperative Multi-Agent Systems by Adaptive Interpolation, Applications of Mathematics 70/3 (2025), 367-386
   Abstract

   We consider a projection-based model reduction approach to computing the maximal impact, one agent or a group of agents has on the cooperative system. As a cri- terion for measuring the agent-team impact on multi-agent systems, we use the H∞ norm, and output synchronization is taken as the underlying cooperative control scheme. We investigate a projection-based model reduction approach that allows efficient H∞ norm calculation. The convergence of this approach depends on initial interpolation points, so we present approaches to their determination. Since the analysis of multi-agent systems is important from different perspectives, several comparisons are presented in the section on numerical experiments. A graph Laplacian matrix of an inter-agent interaction graph is a foundational element in modeling and analyzing multi-agent systems. We consider various graph topology matrices, system parameters, and excitations of different agents. Different strategies for selecting initial interpolation points are also compared with baseline approaches for calculating the H∞ norm.
2. D. Palitta, Z. Tomljanović, I. Nakić, J. Saak, Efficient solution of sequences of parametrized Lyapunov equations with applications, Numerical Linear Algebra with Applications 32/1 (2025), 1-21
   Abstract

   Sequences of parametrized Lyapunov equations can be encountered in many application settings. Moreover, solutions of such equations are often intermediate steps of an overall procedure whose main goal is the computation of $trace(EX)$, where $X$ denotes the solution of a Lyapunov equation and $E$ is a given matrix. We are interested in addressing problems where the parameter dependency of the coefficient matrix is encoded as a low-rank modification to a seed, fixed matrix. We propose two novel numerical procedures that fully exploit such a common structure. The first one builds upon the Sherman-Morrison-Woodbury (SMW) formula and recycling Krylov techniques, and it is well-suited for small dimensional problems as it makes use of dense numerical linear algebra tools. The second algorithm can instead address large-scale problems by relying on state-of-the-art projection techniques based on the extended Krylov subspace. We test the new algorithms on several problems arising in the study of damped vibrational systems and the analyses of output synchronization problems for multi-agent systems. Our results show that the algorithms we propose are superior to state-of-the-art techniques as they are able to remarkably speed up the computation of accurate solutions.
3. 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 89/ (2024), 633-653
   Abstract

   The 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.
4. N. Truhar, Z. Tomljanović, M. Ugrica, M. Karow, Efficient approximation of novel residual bounds for a parameter dependent quadratic eigenvalue problem, Numerical Algebra, Control and Optimization (2024), 1-9
   Abstract

   This paper contributes to the perturbation theory for parameter dependent quadratic eigenvalue problems (PQEP). Specifically, we derive an approximate perturbation bound between individual unperturbed and perturbed eigenvectors for PQEP. We also derive a modified but practically more useful eigenvector perturbation bound using the Taylor expansion of eigenvalue functions. The quality of the obtained bounds is illustrated by numerical examples.
5. I. Nakić, M. Pilj Vidaković, Z. Tomljanović, Finite time horizon mixed control of vibrational systems, SIAM Journal on Scientific Computing 46/3 (2024), 280-305
   Abstract

   We 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.

6. Z. Tomljanović, Damping optimization of the excited mechanical system using dimension reduction, Mathematics and Computers in Simulation 207/ (2023), 24-40
   Abstract

   We 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.
7. 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), 1-21
   Abstract

   We 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.
8. N. Jakovčević Stor, I. Slapničar, Z. Tomljanović, Fast Computation of Optimal Damping Parameters for Linear Vibrational Systems, Mathematics 10/5 (2022), 1-17
   Abstract

   We 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.
9. I. Nakić, D. Tolić, Z. Tomljanović, I. Palunko, Numerically Efficient H∞ Analysis of Cooperative Multi-Agent Systems, Journal of The Franklin Institute 359/16 (2022), 9110-9128
   Abstract

   This 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 agent-to-system 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 agent-to-system $H_{infty}$ 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 multi-agent 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 off-the-shelf nano quadrotors are provided.
10. N. Truhar, Z. Tomljanović, R. Li, Perturbation Theory for Hermitian Quadratic Eigenvalue Problem -- Damped and Simultaneously Diagonalizable Systems, Applied mathematics and computation 371/ (2020)
    Abstract

    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.
11. 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
    Abstract

    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.
12. Z. Tomljanović, M. Voigt, Semi-active H∞-norm damping optimization by adaptive interpolation, Numerical Linear Algebra with Applications 27/4 (2020), 1-17
    Abstract

    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.
13. N. Truhar, Z. Tomljanović, M. Puvača, Approximation of damped quadratic eigenvalue problem by dimension reduction, Applied mathematics and computation 347/ (2019), 40-53
    Abstract

    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.
14. I. Nakić, Z. Tomljanović, N. Truhar, Mixed control of vibrational systems, Journal of Applied Mathematics and Mechanics 99/9 (2019), 1-15
    Abstract

    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.
15. Y. Kanno, M. Puvača, Z. Tomljanović, N. Truhar, Optimization Of Damping Positions In A Mechanical System, Rad HAZU, Matematičke znanosti. 23/ (2019), 141-157
    Abstract

    This paper deals with damping optimization of the mechanical system based on the minimization of the so-called "average displacement amplitude". Further, we propose three different approaches to solving this minimization problems, and present their performance on two examples.
16. Z. Tomljanović, C. Beattie, S. Gugercin, Damping optimization of parameter dependent mechanical systems by rational interpolation, Advances in Computational Mathematics 44/6 (2018), 1797-1820
    Abstract

    We consider an optimization problem related to semi-active damping of vibrating systems. The main problem is to determine the best damping matrix able to minimize influence of the input on the output of the system. We use a minimization criteria based on the $mathcal{H}_2$ system norm. The objective function is non-convex and the associated optimization problem typically requires a large number of objective function evaluations. We propose an optimization approach that calculates `interpolatory' reduced order models, allowing for significant acceleration of the optimization process. In our approach, we use parametric model reduction (PMOR) based on the Iterative Rational Krylov Algorithm, which ensures good approximations relative to the $mathcal{H}_2$ system norm, aligning well with the underlying damping design objectives. For the parameter sampling that occurs within each PMOR cycle, we consider approaches with predetermined sampling and approaches using adaptive sampling, and each of these approaches may be combined with three possible strategies for internal reduction. In order to preserve important system properties, we maintain second-order structure, which through the use of modal coordinates, allows for very efficient implementation. The methodology proposed here provides a significant acceleration of the optimization process; the gain in efficiency is illustrated in numerical experiments.
17. N. Truhar, Z. Tomljanović, M. Puvača, An Efficient Approximation For Optimal Damping In Mechanical Systems, International journal of numerical analysis and modeling 14/2 (2017), 201-217
    Abstract

    This paper is concerned with an efficient algorithm for damping optimization in mechanical systems with a prescribed structure. Our approach is based on the minimization of the total energy of the system which is equivalent to the minimization of the trace of the corresponding Lyapunov equation. Thus, the prescribed structure in our case means that a mechanical system is close to a modally damped system. Although our approach is very efficient (as expected) for mechanical systems close to modally damped system, our experiments show that for some cases when systems are not modally damped, the proposed approach provides efficient approximation of optimal damping.
18. I. Kuzmanović Ivičić, Z. Tomljanović, N. Truhar, Damping optimization over the arbitrary time of the excited mechanical system, Journal of Computational and Applied Mathematics, 304/ (2016), 120-129
    Abstract

    In this paper we consider damping optimization in mechanical system excited by an external force. We use optimization criteria based on minimizing average energy amplitude and average displacement amplitude over the arbitrary time. As the main result we derive explicit formulas for objective functions. These formulas can be implemented efficiently and accelerate optimization process significantly, which is illustrated in a numerical example.
19. P. Benner, P. Kurschner, Z. Tomljanović, N. Truhar, Semi-active damping optimization of vibrational systems using the parametric dominant pole algorithm, Journal of Applied Mathematics and Mechanics 96/5 (2016), 604-619
    Abstract

    We consider 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. The optimization approach yields a large number of Lyapunov equations which have to be solved. In this work, we propose an optimization approach that works with reduced systems which are generated using the parametric dominant pole algorithm. This optimization process is accelerated with a modal approach while the initial parameters for the parametric dominant pole algorithm are chosen in advance using residual bounds. Our approach calculates a satisfactory approximation of the impulse response energy while providing a significant acceleration of the optimization process. Numerical results illustrate the effectiveness of the proposed algorithm.
20. N. Truhar, Z. Tomljanović, K. Veselić, Damping optimization in mechanical systems with external force, Applied mathematics and computation 250/ (2015), 270-279
    Abstract

    We consider a mechanical system excited by external force. Model of such a system is described by the system of ordinary differential equations: $M ddot x(t) + D dot x(t) + K x(t) = {hat f}(t)$, where matrices $M, K$ (mass and stiffness) are positive definite and the vector ${hat f} $ corresponds to an external force. The damping matrix D is assumed to be positive semidefinite and has a small rank. We introduce two criteria that allow damping optimization of mechanical system excited by an external force. Since in general a damping optimization is a very demanding problem, we provide a new formulas which have been used for efficient damping optimization. The efficiency of new formulas is illustrated with a numerical experiment.
21. M. Rukav, K. Stražanac, N. Šuvak, Z. Tomljanović, Markov decision processes in minimization of expected costs, Croatian Operational Research Review 5/2 (2014), 247-257
    Abstract

    Basics of Markov decision processes will be introduced in order to obtain the optimization goal function for minimizing the long-run expected cost. We focus on minimization of such cost of the farmer's policy consisting of different decisions in specic states regarding both milk quality and quantity (lactation states) produced by a dairy cow. The transition probability matrix of the Markov process, used here for modeling of transitions of a dairy cow from one state to another, will be estimated from the data simulated from the lactation model that is often used in practice. We want to choose optimal actions in the states of this Markov process regarding the farmer's costs. This problem can be solved by exhaustive enumeration of all possible cases in order to obtain the optimal policy. How- ever, this is feasible only for a small number of states. Generally, this problem can be approached in the linear programming setting which yields an efficient solution. In order to demonstrate and compare these two approaches, we present an example based on the simulated data regarding milk quality and quantity.
22. P. Benner, Z. Tomljanović, N. Truhar, Optimal Damping of Selected Eigenfrequencies Using Dimension Reduction, Numerical Linear Algebra with Applications 20/1 (2013), 1-17
    Abstract

    We consider a mathematical model of a linear vibrational system described by the second-order differential equation $M ddot{x} + D dot{x} + Kx = 0$, where $M$ and $K$ are positive definite matrices, representing mass and stiffness, respectively. The damping matrix $D$ is positive semidefinite. We are interested in finding an optimal damping matrix which will damp a certain (critical) part of the undamped eigenfrequencies. For this we use an optimization criterion based on minimization of the average total energy of the system. This is equivalent to the minimization of the trace of the solution of the corresponding Lyapunov equation $A X+ X A^T =-GG^T$, where $A$ is the matrix obtained from linearizing the second-order differential equation and $G$ depends on the critical part of the eigenfrequencies to be damped. The main result is the efficient approximation and corresponding error bound for the trace of the solution of the Lyapunov equation obtained by dimension reduction, which includes the influence of the right-hand side $G G^T$ and allows us to control the accuracy of the trace approximation. This trace approximation yields a much accelerated optimization algorithm for determining the optimal damping.
23. I. Nakić, Z. Tomljanović, N. Truhar, Optimal Direct Velocity Feedback, Applied mathematics and computation 225/ (2013), 590-600
    Abstract

    We present a novel approach to the problem of Direct Velocity Feedback (DVF) optimization of vibrational structures, which treats simultaneously small as well as large gains. For that purpose, we use two different approaches. The first one is based on the gains optimization using the Lyapunov equation. In the scope of this approach we present a new formula for the optimal gain and we present a relative error for modal approximation. In addition, we present a new formula for the solution of the corresponding Lyapunov equation for the case with multiple undamped eigenfrequencies, which is a generalization of existing formulae. The second approach studies the behavior of the eigenvalues of the corresponding quadratic eigenvalue problem. Since this approach leads to the parametric eigenvalue problem we consider small and large gains separately. For the small gains, which are connected to a modal damping approximation, we present a standard approach based on Gerschgorin discs. For the large gains we present a new approach which allows us to approximate all eigenvalues very accurately and efficiently.
24. I. Kuzmanović Ivičić, Z. Tomljanović, N. Truhar, Optimization of material with modal damping, Applied mathematics and computation 218/ (2012), 7326-7338
    Abstract

    This paper considers optimal parameters for modal damping $D=Mf_1(M^{-1}K;alpha_1,dots,alpha_k)+Kf_2(K^{-1}M;alpha_1,dots,alpha_k)$ in mechanical systems described by the equation $Mddot{x}+Ddot{x}+Kx=0 $, where matrices $M$ and $K$ are mass and stiffness matrices, respectively. Different models of proportional and generalized proportional damping are considered and optimal parameters with respect to different optimization criteria related to the solution of the corresponding Lyapunov equation are given. Also, some specific example problems are compared with respect to the optimal and estimated parameters.
25. P. Benner, Z. Tomljanović, N. Truhar, Dimension reduction for damping optimization in linear vibrating system, Journal of Applied Mathematics and Mechanics 91/3 (2011), 179-191
    Abstract

    We consider a mathematical model of a linear vibrational system described by the second-order differential equation $M ddot{;x}; + D dot{;x}; + Kx = 0$, where $M$ and $K$ are positive definite matrices, called mass and stiffness, espectively. We consider the case where the damping matrix $D$ is positive semidefinite. The main problem considered in the paper is the construction of efficient algorithm for calculating an optimal damping. As optimization criterion we use the minimization of the average total energy of the system which is equivalent to the minimization of the trace of the solution of the corresponding Lyapunov equation $A X+ X A^T =-I$, where $A$ is the matrix obtained from linearizing the second-order differential equation. Finding the optimal $D$ such that the trace of $X$ is minimal is a very demanding problem, caused by the large number of trace calculations, which are required for bigger matrix dimensions. We propose a dimension reduction to accelerate the optimization process. We will present an approximation of the solution of the structured Lyapunov equation and a corresponding error bound for the approximation. Our algorithm for efficient approximation of the optimal damping is based on this approximation.
26. Z. Tomljanović, N. Truhar, K. Veselić, Optimizing a damped system - a case study, International Journal of Computer Mathematics 88/7 (2011), 1533-1545
    Abstract

    We consider a second order damped-vibrational system described by the equation $ M ddot{;x}; + C(v) dot{;x}; + K x = 0 $, where $M, C(v), K$ are real, symmetric matrices of order $n$. We assume that the undamped eigenfrequencies (eigenvalues of $(lambda^2 M + K) x = 0$) $omega_1, omega_2, ldots, omega_n , $, are multiple in the sense that $omega_1 = omega_2$, $omega_3 = omega_4$, ldots, $omega_{;n-1}; = omega_n$, or are given in close pairs $omega_1 approx omega_2$, $omega_3 approx omega_4$, ldots, $omega_{;n-1}; approx omega_n$. We present a formula which gives the solution of the corresponding phase space Lyapunov equation, which then allows us to calculate the first and second derivatives of the trace of the solution, with no extra cost. This one can serve for the efficient trace minimization.
27. N. Truhar, Z. Tomljanović, R. Li, Analysis of the solution of the Sylvester equation using low-rank ADI with exact shifts, Systems and Control Letters 59/ (2010), 248-257
    Abstract

    The solution to a general Sylvester equation $AX+XB = GF^*$ with a low rank right- hand side is analyzed quantitatively through Low-rank Alternating-Directional- Implicit method (LR-ADI) with exact shifts. New bounds and perturbation bounds on X are obtained. A distinguished feature of these bounds is that they reflect the interplay between the eigenvalue decompositions of A and B and the right-hand side factors G and F. Numerical examples suggest that because of this inclusion of details, new perturbation bounds are much sharper than the existing ones.
28. N. Truhar, Z. Tomljanović, Estimation of optimal damping for mechanical vibrating systems, International Journal of Applied Mathematics and Mechanics 5/5 (2009), 14-26
    Abstract

    This paper is concerned with the efficient algorithm for dampers' and viscosity optimization in mechanical systems. Our algorithm optimize simultaneously the dampers' positions and their viscosities. For the criterion for optimization we use minimization of the average total energy of the system which can be done by the minimization of the trace of the solution of the corresponding Lyapunov equation. Efficiency of the algorithm is obtained by new heuristics for finding the optimal dampers' positions and for the approximation of the trace of the solution of the Lyapunov equation.

1. I. Nakić, D. Tolić, I. Palunko, Z. Tomljanović, Numerically Efficient Agents-To-Group H∞ Analysis, 10th Vienna International Conference on Mathematical Modelling MATHMOD 2022, Vienna Austria, 2022, 199-204
   Abstract

   This paper proposes a numerically efficient approach for computing the maxi- mal/minimal impact a subset of agents has on the cooperative system. For instance, if one is able to disturb/bolster several agents so as to maximally disturb/bolster the entire team, which agents to choose and what kind of inputs to apply? We quantify the agents-to-team impacts in terms of H∞ norm whereas output synchronization is taken as the underlying cooperative control scheme. Sufficient conditions on agents’ parameters, synchronization gains and topology are provided such that the associated H∞ norm attains its maximum for constant agents’ disturbances. Linear second-order agent dynamics and weighted undirected topologies are considered. Our analyses also provide directions towards improving graph design and tuning/selecting cooperative control mechanisms. Lastly, numerical examples, some of which include forty thousand agents, are provided.
2. I. Kuzmanović Ivičić, Z. Tomljanović, N. Truhar, Applications of Lyapunov and T-Lyapunov equations in mechanics, Fourth Mathematical Conference of the Republic of Srpska,, Trebinje, 2014, 83-92
   Abstract

   This paper considers Lyapunov and T -Lyapunov matrix equations. Lyapunov equation is a matrix equation of the form AX + XA^T = E which plays a vital role in a number of applications, while T -Lyapunov equation is a matrix equation of the form AX +X^TA^T = E. In this paper the relation between these equations will be exploit with purpose of applying obtained results in problems regarding damping optimization in mechanical systems.
3. J. Denissen, A. Koskela, H. Mena, Z. Tomljanović, Damping optimization in vibrational systems based on amplitude, The 21th International Symposium on Mathematical Theory of Networks and Systems, Groningen, The Netherlands , 2014, 1222-1227
   Abstract

   We consider the optimal damping problem for a linear vibrational system Mx¨ + Dx˙ + Kx = 0, where M and K are positive definite matrices. For the damping optimization we use a criterion based on minimization of the integral of the solution’s amplitude over a given time interval. Finding the optimal damping D is a very demanding problem, and using this approach the computational cost comes mainly from a large number of matrix exponential computations. We propose an efficient numerical scheme to accelerate these computations. The performance of our approach is illustrated by numerical results for an n-mass oscillator.
4. P. Benner, Z. Tomljanović, N. Truhar, Damping Optimization for Linear Vibrating Systems Using Dimension Reduction, The 10th International Conference on Vibration Problems ICOVP 2011, Prag, 2011, 297-305
   Abstract

   We consider a mathematical model of a linear vibrational system described by the second-order system of differential equations $M ddot{; ; x}; ; + D dot{; ; x}; ; + Kx = 0$, where M, K and D are positive deffinite matrices, called mass, stiffness and damping, respectively. We are interested in finding an optimal damping matrix which will damp a certain part of the undamped eigenfrequencies. For this we use a minimization criterion which minimizes the average total energy of the system. This is equivalent to the minimization of the trace of the solution of a corresponding Lyapunov equation. In this paper we consider an algorithm for the efficient optimization of the damping positions based on dimension reduction techniques. Numerical results illustrate the efficiency of our approach.

1. M. Marković, Z. Tomljanović, Kolaborativno filtriranje, Math.e : hrvatski matematički elektronski časopis 34/1 (2018), 1-22
   Abstract

   U ovom članku opisan je jedan tip sustava za preporuke, točnije kolaborativno filtriranje bazirano na modelu orijentiranom prema proizvodu i pripadni algoritam baziran na SVD dekompoziciji matrice sustava. Ukratko je dan kratak povijesni pregled sustava za preporuke te su navedeni osnovni pojmovi i opisani su najbitniji tipovi sustava za preporuke. Zatim je iskazan centralni algoritam rada, algoritam za preporuke zasnovan na SVD dekompoziciji s redukcijom dimenzije te je obrazložena njegova korektnost. Kroz svojstva navedenog algoritma naglašene su neke korisnosti SVD dekompozicije matrice koja ima mnoge primjene u stvarnom svijetu. Nakon toga opisana su dva načina za osvježavanje sustava novim korisnicima, folding-in metoda i update metoda te su dani pripadni algoritmi. U zadnjem dijelu rada opisani su testovi provedeni na implementiranim algoritmima, algoritmu za preporuke zasnovanom na SVD dekompoziciji s redukcijom dimenzije i folding-in metodi.
2. C. Beattie, S. Gugercin, Z. Tomljanović, Sampling-free parametric model reduction of systems with structured parameter variation (2018)
3. Z. Tomljanović, N. Truhar, K. Veselić, Damping optimization in mechanical systems with external force (2015)
4. Z. Tomljanović, M. Ugrica, QR decomposition using Givens rotations and applications, Osječki matematički list 14/2 (2015), 117-141
   Abstract

   In this paper, we describe Givens rotations and their applications. We present basic properties of Givens rotation matrices and their application to calculation of QR decomposition of the given matrix which can be used for solving linear systems or the least squares problem. Givens rotations play an important role if the matrix considered has a special structure; thus, we additionally describe usage of Givens rotations for structured matrices such as tridiagonal or Hessenberg matrices. Givens rotations and their application are illustrated by examples.
5. K. Burazin, Z. Tomljanović, I. Vuksanović, Prigušivanje mehaničkih vibracija, Math.e : hrvatski matematički elektronski časopis 24/ (2014)
6. Z. Tomljanović, Hornerov algoritam i primjene, Osječki matematički list 7/ (2008), 99-106
   Abstract

   U ovom članku obrađuje se Hornerov algoritam za efikasno računanje vrijednosti polinoma u točki. Hornerov algoritam može se lako proširiti do algoritma koji daje Taylorov razvoj polinoma u okolini dane točke. Također, dani su i ilustrativni primjeri i primjene ovog algoritma.

1. R. Scitovski, N. Truhar, Z. Tomljanović, Metode optimizacije, Svučilište Josipa Jurja Strossmayera u Osijeku, Odjel za matematiku, Osijek, 2014.
   Abstract

   Namjena ovog teksta je upoznati čitatelja s glavnim metodama jednodimenzionalne i višedimenzionalne minimizacije sa i bez ograničnja. Posebna je pozornost posvećena metodama minimizacije nediferencijabilnih funkcija. Pri tome je izbjegavano dokazivanje zahtjevnih teorema, osim u slučajevima konstruktivnih dokaza koji sami po sebi upućuju na izgradnju ideja ili metoda. Navedeni optimizacijski problemi imaju veliku primjenu u raznim dijelovima života. Na primjer, često se javljaju problemi poput optimalnog oblikovanja odredenih mehaničkih sustava (oblikovanje dijelova automobilskih motora, nosivih konstrukcija u gradjevinarstvu, . . .), problem modeliranja ponašsanja tržišta, problemi iz teorije upravljanja (smirivanje sustava, optimalno upravljanje, . . . ) i mnogi drugi. Upravo činjenica da se razni problemi optimizicije pojavljuju u raznim dijelovima ljudske djelatnosti osigurava ovom tekstu široku primjenu.

1. N. Truhar, Z. Tomljanović, M. Puvača, An efficient approximation for the optimal damping in mechanical systems (2016)
   Abstract

   This paper is concerned with the efficient algorithm for damping optimization in mechanical systems with prescribed structure. Our approach is based on the minimization of the total energy of the system which is equivalent with the minimization of the trace of the corresponding Lyapunov equation. Thus, the prescribed structure in our case means that a mechanical system is close to the modally damped system. Even though our approach is very efficient (as it was expected) for the mechanical systems close to modally damped system, our experiments show that for some cases when systems are not modally damped the proposed approach provides efficient approximation of the optimal damping.
2. N. Truhar, Z. Tomljanović, Dimension reduction approach for the parameter dependent quadratic eigenvalue problem (2016)
   Abstract

   This paper presents the novel approach in efficient calculation of the 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 $ntimes n$ Hermitian semidefinite matrix which depends on a parameter $mathbf{v}= begin{bmatrix} v_1 & ldots & v_k end{bmatrix}in mathbb{R}_+^k$. With the new approach one can efficiently (and accurate enough) calculate the all (or just part of the) eigenvalues even for the case when $v_i$ are of the modest magnitude. Moreover, for the both cases of approximations we have derived the corresponding upper bounds. The quality of the error bounds as well as the performance of the achieved eigenvalue tracking was illustrated in several numerical experiments.
3. N. Truhar, Z. Tomljanović, K. Veselić, Damping optimization in mechanical systems with external force (2014)
   Abstract

   We consider a mechanical system excited by external force. Model of such a system is described by the system of ordinary differential equations: $M ddot x(t) + D dot x(t) + K x(t) = {hat f}(t)$, where matrices $M, K$ (mass and stiffness) are positive definite and the vector ${hat f} $ corresponds to an external force. The damping matrix D is assumed to be positive semidefinite and has a small rank. We introduce two criteria that allow damping optimization of mechanical system excited by an external force. Since in general a damping optimization is a very demanding problem, we provide a new formulas which have been used for efficient damping optimization. The efficiency of new formulas is illustrated with a numerical experiment.

Projects

• Optimization of parameter-dependent systems with applications — scientific project (IP-2025-02-4862, OPT-PEDESTAL). This project has been fully supported by Croatian Science Foundation for the period 01.12.2025. – 30.11.2028. (principal investigator)
• 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

• 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
• Croatian association for applied and industrial mathematics CRO-MATH-IN

Committee Memberships and Organization

• UPCOMING: Co-organizer of the 5th Workshop on Optimal Control of Dynamical Systems and Applications, 10-13 May 2027 in Mon Perin resort in Istria, Croatia: webpage
• Co-organizer of the 4th Workshop on Optimal Control of Dynamical Systems and Applications, 26-28 Feb 2025 in Villany, Hungary: webpage
• 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):

• srijedom u 9:00 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
  – Većina diferencijalnih jednadžbi koje opisuju realne pojave nema rješenje u zatvorenoj formi, pa se do rješenja dolazi numerički. U ovom diplomskom radu obradit ćemo jednokoračne metode za početni problem: eksplicitnu i implicitnu Eulerovu metodu te metode Runge–Kutta, s naglaskom na klasičnu metodu četvrtog reda. Za svaku ćemo metodu izvesti lokalnu pogrešku diskretizacije, odrediti red konvergencije i ispitati područje apsolutne stabilnosti, čime se objašnjava zašto eksplicitne metode zakazuju na krutim problemima. Metode ćemo implementirati u MATLAB-u (ili C-u) i usporediti ih na primjerima različite naravi — harmonijskom oscilatoru, Van der Polovu oscilatoru i Lotka–Volterrinu modelu. Red konvergencije potvrdit ćemo logaritamskim dijagramom pogreške u ovisnosti o koraku, a efikasnost mjeriti brojem evaluacija desne strane potrebnih za zadanu točnost. Na kraju ćemo vlastite implementacije usporediti s ugrađenim rješavačima i pokazati dobitak koji donosi prilagodljiv korak.
• QR dekompozicija s pivotiranjem 
  – QR dekompozicija temelj je numerički stabilnog rješavanja problema najmanjih kvadrata, ali u slučaju matrice nepunog ranga obična dekompozicija ne otkriva koji su stupci linearno zavisni. U radu ćemo izvesti QR dekompoziciju trima pristupima — klasičnim i modificiranim Gram–Schmidtovim postupkom, Householderovim reflektorima i Givensovim rotacijama — te usporediti njihovu numeričku stabilnost i cijenu. Zatim ćemo obraditi dekompoziciju s pivotiranjem stupaca, u kojoj se stupci biraju po opadajućoj normi ostatka, pa se dobiva rastav AP = QR s dijagonalom matrice R koja ne raste. Metode ćemo implementirati u MATLAB-u i pokazati kako se iz naglog pada dijagonalnih elemenata očitava numerički rang matrice, kako se biraju najinformativniji stupci i kako se rješava problem najmanjih kvadrata nepunog ranga. Usporedbom sa singularnom dekompozicijom pokazat ćemo granice ovog pristupa, uključujući poznate primjere matrica na kojima pivotiranje ne otkriva rang.
• Računanje matrične ekponencijalne funkcije
  – Matrična eksponencijalna funkcija daje rješenje linearnog sustava diferencijalnih jednadžbi s konstantnim koeficijentima, no njezino je računanje klasičan primjer problema u kojem mnogi naizgled prirodni postupci numerički zakazuju. U radu ćemo obraditi glavne skupine metoda: sumiranje Taylorova reda, spektralnu dekompoziciju, pristup preko Schurove forme, te Padéovu aproksimaciju u kombinaciji s postupkom skaliranja i kvadriranja, koji je danas standard. Metode ćemo implementirati i usporediti na testnim matricama različite naravi — simetričnima, nenormalnima i s bliskim svojstvenim vrijednostima — mjereći točnost u odnosu na referentno rješenje i vrijeme računanja. Primjenu ćemo ilustrirati na rješavanju linearnih dinamičkih sustava.
• Metoda Gaussovih eliminacija s potpunim pivotiranjem
  – Gaussova eliminacija bez pivotiranja može se slomiti i na dobro uvjetovanim sustavima, pa je izbor stožernog elementa ključan dio metode. U radu ćemo obraditi eliminaciju s potpunim pivotiranjem, u kojoj se u svakom koraku traži element najveće apsolutne vrijednosti u cijeloj preostaloj podmatrici, što vodi na rastav PAQ = LU. Usporedit ćemo je s parcijalnim pivotiranjem po faktoru rasta i po cijeni: potpuno pivotiranje traži kvadratno mnogo usporedbi u svakom koraku, ali daje znatno bolju teorijsku ogradu na rast elemenata. Pokazat ćemo i primjere matrica na kojima parcijalno pivotiranje daje eksponencijalni rast. U MATLAB-u ćemo izraditi vizualizaciju koraka metode, u kojoj se u svakom koraku prikazuju veličine elemenata matrice i označava izabrani stožerni element, te geometrijsku vizualizaciju sustava dviju jednadžbi s dvije nepoznanice, na kojoj se vidi presjek pravaca i kako se rješenje gubi kad su pravci gotovo paralelni.
• Aproksimativna analitička rješenja harmoničkog oscilatora prigušenog kombinacijom
  linearnih i nelinearnih disipativnih sila
  – Prigušenje vibracija se najčešće opisuje silom koja je proporcionalna brzini
  (linearno prigušenje). U realnim sustavima su pored linearnih, u većoj ili manjoj mjeri,
  uvijek prisutne i nelinearne sile prigušenja, kao prigušenje proporcionalno kvadratu brzine
  (kvadratno prigušenje) i suho trenje (prigušenje neovisno o iznosu brzine, tzv.
  Coulombovo prigušenje). Već za najjednostavniji vibracijski sustav s jednim stupnjem
  slobode, tj. harmonički oscilator, nije moguće riješiti analitički jednadžbu gibanja koja
  uključuje sve tri navedene sile prigušenja. U ovom diplomskom radu ćemo, polazeći od
  veze disipacije energije i snage ukupne sile prigušenja, izvesti aproksimativna analitička
  rješenja za harmonički oscilator prigušen s tri navedene sile prigušenja. Usporedbom s
  numeričkim rješenjima, pokazat ćemo da dobivena aproksimativna analitička rješenja
  izvrsno opisuju slobodne vibracije sustava za raspon parametara (konstanti) prigušenja
  koji je relevantan za opis eksperimenata u temeljnim istraživanjima, ali i u tehnološkim
  primjenama. Demonstrirat ćemo kako se naša rješenja mogu iskoristiti da se iz
  eksperimentalnih podataka dobiju parametri svakog od tri tipa prigušenja.

Nastavne aktivnosti u zimskom semestru Akademske 2026./2027.

Linearna algebra I, predavanja

srijedom  10-12h,

Nastavne aktivnosti u ljetnom semestru Akademske 2025./2026.

Osnove teorije upravljanja s primjenama, predavanja

Utorkom 10-12 h

Redukcija modela i aproksimacijski pristupi,