Optimization and Applications seminar

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.

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

link

Seminar se održava online u 12:00 putem Microsoft Teams aplikacije.
link

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.