Seminar za optimizaciju i primjene

Friedrichs (1958) introduced the concept of symmetric positive systems (Friedrichs systems), in order to treat the differential equations that change their type (the Tricomi equation). The class of Friedrichs systems encompasses a wide variety of equations. A new interest arose in this area after the indroduction of operator-theoretic approach (abstract Friedrichs operators (2007)). While we continue to study Friedrichs systems using the theory of abstract Friedrichs operators, in parallel we also seek to develop the abstract theory. Recently, a von-Neumann type decomposition for the graph space of abstract Friedrichs operators has been proven.
In this talk, we present a connection of abstract Friedrichs operators with symmetric and skew-symmetric operators. One immediate consequence of this approach is a generalised von- Neumann type decomposition for symmetric operators. Briefly, we also present recent results on Friedrichs systems.

Prezentirati će se kvantitativni automati i fundamentalni problemi vezani uz njih, poput uspoređivanja, simulacija i bisimulacija, redukcija broja stanja i minimizacija, determinizacija i kanonizacija te rekonstrukcija.
Naglasak će biti stavljen na primjene u kontekstu strojnog učenja i formalne verifikacije hardverskih i softverskih sistema.

With the growing dimensionality of the data in many real-world applications, feature selection is becoming an increasingly important preprocessing step in multi-label classification. Finding a smaller subset of the most relevant features can significantly reduce resource consumption of model training, and in some cases, it can even result in a model with higher accuracy. Traditionally, feature selection has been done by employing some statistical measure to determine the most influential features, but in recent years, more and more metaheuristics have been proposed to tackle this problem more effectively. In this lecture, we will present a brief introduction to machine learning and data mining algorithms, with a focus on feature selection. We will present different approaches to feature selection, concentrating on metaheuristic wrapper methods. We will also present the results of our recent work on this topic, where we proposed the Basic Variable Neighborhood Search (BVNS) algorithm to search for the optimal subset of features, combined with a local search method based on mutual information. The algorithm can be considered a hybrid between the wrapper and filter methods, as it uses statistical knowledge about features to reduce the number of examined solutions during the local search. We compared our approach against Ant Colony Optimization (ACO) and Memetic Algorithm (MA), using the K-nearest neighbors classifier to evaluate solutions. The experiments conducted using three different metrics on a total of four benchmark datasets suggest that our approach outperforms ACO and MA.

In this paper 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 and 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 us to analyse 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.

Rational matrices R(x) arise in many applications such as in vibration analysis of machines, buildings and vehicles, in Control Theory and Linear Systems Theory, and in approximation of other nonlinear eigenvalue problems. The spectral data (poles, zeros, eigenvalues, eigenvectors, minimal bases, and minimal indices) of R(x) play a vital role in many applications. In order to provide a direct solution to the rational eigenvalue problems, in this talk, we propose a method called “Linearization” (specifically, Rosenbrock strong linearization) of rational matrices: by Rosenbrock strong linearization of a rational matrix R(x) we mean a matrix pencil L(x) preferably of smallest dimension that reveals the spectral data of R(x). Then, in order to validate our definition of linearization, we discuss the construction of various families of linearizations for structured and unstructured rational matrices. Finally, we establish the relation of spectral data between the rational matrices and the linearizations.

This lecture was organized within the Croatian Science Foundation project: Vibration Reduction in Mechanical Systems (IP-2019-04-6774, VIMS).

There are a variety of system-theoretic methods for model order reduction (MOR) of non-parametric non-stationary systems. For linear time-invariant (LTI) systems, these include, for example, balanced truncation and iterative rational Krylov algorithm. Some of these methods have been already extended to parametric non-stationary systems. In addition, recently an optimal MOR method has been developed for approximating parametric LTI systems where optimality is measured in terms of a least-squares measure both in the frequency and time domain. In this presentation, we discuss adapting these system-theoretic MOR methods to linear parametric stationary systems. In particular, we focus on the optimal parametric MOR methods. We show that incorporating the system-theoretic error measures into approximation of non-stationary problems yields more accurate reduced models compared to the traditional tools, such as POD. We also illustrate that the optimization algorithm can be performed purely in a data-driven manner using only the samples of the quantity of interests without access to full-order operators. This is joint work with Serkan Gugercin.

Composite materials are mixtures of two or more constituent materials. They are prevalent in nature and among engineered materials, probably because they often combine (desired) attributes of the materials that they are made of. We are interested in elastic composites made by mixing two isotropic (linearly) elastic phases.

One of the most prominent problems in structural mechanics is to describe all possible composites (known as the G-closure set) that can be achieved by mixing two materials (with or without prescribed ratio). While this "G-closure problem" still remains open, we have on our disposal various bounds on this set. Probably the most prominent representative among them are famous Hashin-Shtrinkman energy bounds, which have imposed themselves as the benchmark against which most experimental results are compared. While explicit calculation of bounds in two space dimensions has been done three decades ago, the procedure in three space dimensions requires a rather tedious and formidable calculations, and thus, up to date, it remained an open problem.

We shall present an explicit calculation of Hashin-Shtrikman energy bounds in three space dimensions and discus some further perspectives yielded by these result, such as application in compliance minimization optimal design problem, in the modelling of the damage accumulation, as well as the coherent phase transitions.

This is joint work with Ivana Crnjac and Marko Vrdoljak

Two main concerns in the problem of visual place recognition are: how to represent a place and how to match it. By using graph theory, we can construct a powerful place matching algorithm. By using optimization techniques, we can adapt a place representation. Therefore, this seminar examines numerous optimization problems and how they are incorporated in the context of visual place recognition.

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Seminar se održava online u 12:00 putem Microsoft Teams aplikacije.
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Predavanje će se održati online putem aplikacije GoToMeeting:
www.gotomeeting.com/join/570689509

This lecture was organized within the Croatian Science Foundation project: Vibration Reduction in Mechanical Systems (IP-2019-04-6774, VIMS).

Napomena: umjesto u uobičajenom terminu srijedom, seminar će se održati u utorak u 11:00 u D-9.

Moderator: Z. Tomljanović

Moderator: Z. Tomljanović

Moderator: Z. Tomljanović

Moderator: Z. Tomljanović

Moderator: D. Grahovac

We consider the computation of the H-infinity-norm for transfer
functions of a general class of linear and non-linear systems. We focus
on the case where the state-space dimension is very large. We propose a
subspace projection method to obtain approximations of the transfer
function by interpolation techniques. The H-infinity-norms are computed
for the resulting reduced functions, then the subspaces are refined by
means of the optimal points on the imaginary axis where the maximum
singular value of the reduced function is attained. In this way we
obtain much better performance compared to existing methods.

Razmatra se smetnja izoliranih vlastitih vrijednosti
Klein-Gordonovog operatora te se unatoč njegovoj nehermitičnosti
dokazuju ocjene analogne onima za hermitske operatore.

Metodologija određivanja dimenzije objekta iz neprecizne slike koja će biti prezentirana temelji se na parametarskom statističkom modelu jednostavnog slučajnog uzorka iz distribucije nastale kao suma uniformne slučajne varijable i varijable slučajne greške. Slučajna greška se modelira raznim distribucijama sa simetričnom gustoćom. Osim opisa metode i statističkih svojstava procedure bit će predstavljen i R-program za procjenu nastao na temelju ovih modela.

Previously, greedy algorithms have been shown to have good approximation
and estimation properties for superpositions of a sigmoidal function or
other ridge activation functions. In each step the parameters of a new
sigmoid are fit to the residuals of the previous sigmoids. However, it
remains a challenging task to produce guarantees for the parameter search
for each sigmoid. Here we discuss developments of two algorithms for this
task. One is an implementation of adaptive annealing, in which internal
modifications of the sigmoid is made, allowing stability of the adaptive
annealing solution. The other algorithm is a convergent nonlinear
modification of the tensor methods of Anandkumar and her colleagues which
allows optimization of the inner product of the residuals with certain
sigmoids and related activation functions. This work is joint with Jason
Klusowski.

We discuss the importance of robust and accurate implementation of core numerical linear algebra procedures in computational methods for system and control theory. In particular,
we stress the importance of error and perturbation analysis that identifies relevant condition numbers and guides computation with noisy data, and careful software implementation. The themes used as case studies include rational matrix valued least squares fitting (e.g. least squares fit to frequency response measurements of an LTI system), model order reduction issues (e.g. the Discrete Empirical Interpolation Method (DEIM)), accurate computation with structured matrices such as scaled Cauchy, Vandermonde and Hankel matrices.

Na predavanju će biti predstavljena shifted Gompertzova distribucija. Iskazat će se njezina osnovna statistička svojstva, te pokazati povezanost ove distribucije s nekim modelima difuzije novog proizvoda. Posebna pažnja bit će posvećena različitim metodama za procjenu parametara, s naglaskom na metodu najmanjih kvadrata.

Na predavanju će biti prikazana dva rezultata o globalnim minimizatorima odgovarajućih funkcija cilja u svrhu optimizacije prigušenja u mehaničkim
sustavima.

Mnoge zadaće upravljanja podrazumijevaju zadržavanje određenih signala „malima“ unatoč utjecajima raznih perturbacija. Ukoliko matricu prijenosa shvatimo kao operator sa prostora inputa u prostor outputa, tada normom tog operatora možemo opisati performanse sustava za različite klase signala. Preciznije, performanse sustava mogu se predvidjeti ukoliko su H_2 ili H_infty norme tog operatora male. Pokazat će se svojstva H_2 i H_infty normi pomoću kojih se one mogu efikasno računati, te će biti prezentirane neke metode računanja poput primjerice algoritma bisekcije za računanje H_infty norme i veza računanja H_2 norme sa Ljapunovljevom jednadžbom. Na kraju će se Ilustrirati primjene normi sustava.

Najvažniji cilj u upravljanju sustava je postići određene performanse sustava na efikasan način. Performanse sustava upravljanja možemo opisati u terminima veličine određenih signala. Primjerice, performanse sustava za praćenje mogu se izmjeriti veličinom signala koji prati pogrešku. Na predavanju će se prezentirati različiti načini opisivanja veličine signala, odnosno nekoliko normi kojima možemo prikazati različite klase signala. Uvest će se Hardyjevi prostori H_2 i H_infty koji su važni za praćenje performasi sustava, ali i mjerenje kvalitete aproksimacije kod raznih aproksimacijskih metoda u teoriji sustava. Nadalje, za različite ulazne signale prikazat će se veze između ulaznih i izlaznih podataka.