Dynamical analysis of complex systems in light of H-matrices

Predmet istraživanja ove doktorske disertacije je sistematsko povezivanje dinamičkih svojstava kompleksnih sistema (kao što su stabilnost, robusnost, reaktivnost, tranziciono ponašanje, amplifikacioni okvir) sa teorijom H-matrica, koje je motivisalo nove rezultate iz oblasti lokalizacije blok i frakcionog pseudospektra, kao i ocene norme inverzne matrice, kako u tačkastoj, tako i u blok formi, ali i primenu ovih rezultata u dinamičkoj analizi kompleksnih sistema.This thesis provides a systematic link between the dynamical properties of complex systems (such as  stability, robustness, reactivity, transient behavior, amplification envelope) and the theory of H-matrices, which has motivated new results as far as block and fractional pseudospectrum localizations are concerned, as well as some new upper bounds for the inverse of certain subclasses of H-matrices, both in the point-wise and block case, and the overall collective application of these results in the dynamical analysis of complex systems

New stability indicators for the empirical food webs

Ova doktorska disertacija uvodi nov pristup ispitivanju stabilnosti dinamičkih sistema, korišćenjem teorije pseudospektra. Na taj način se postojeći pojam stabilnosti profinjuje pojmom robusne stabilnosti, koji mnogo adekvatnije opisuje realnu ekološku stabilnost. Razvijen je nov matematički alat za izračunavanje indikatora stabilnosti, koji je zatim ilustrovan na primeru dva ekosistema tla, sa po četiri uzorka, u četiri različita stadijuma razvoja.This doctoral dissertation establishes a novel approach to the stability analysis of dynamical systems, in terms of matrix pseudospectrum. In that manner, the existing concept of stability has undergone essential refinement so as to give birth to the concept of robust stability, which has the ability to capture the ecological stability at a more adequate level. Additionally, within the framework of the dissertation, a new mathematical tool for the stability indicators computation has been developed, which has then been used to illustrate theoretical results in form of two soil ecosystems, each of them sampled four times, all of them observed in four distinct stages of evolution

On Matrix Nearness Problems: Distance to Delocalization

This paper introduces two new matrix nearness problems that are intended to generalize the distance to instability and the distance to stability. They are named the distance to delocalization and the distance to localization due to their applicability in analyzing the robustness of eigenvalues with respect to arbitrary localization sets (domains) in the complex plane. For the open left-half plane or the unit circle, the distance to the nearest unstable/stable matrix is obtained as a special case. Then, following the theoretical framework of Hermitian functions and the Lyapunov-type localization approach, we present a new Newton-type algorithm for the distance to delocalization (D2D) and study its implementations using both an explicit and an implicit computation of the desired singular values. Since our investigations are motivated by several practical applications, we will illustrate our approach on some of them. Furthermore, in the special case when the distance to delocalization becomes the distance to instability, we will validate our algorithms against the state of the art computational method

Algoritmi za računanje optimalnih lokalizacija Geršgorinovog tipa

There are numerous ways to localize eigenvalues. One of the best known results is that the spectrum of a given matrix ACn,n is a subset of a union of discs centered at diagonal elements whose radii equal to the sum of the absolute values of the off-diagonal elements of a corresponding row in the matrix. This result (Geršgorin's theorem, 1931) is one of the most important and elegant ways of eigenvalues localization ([63]). Among all Geršgorintype sets, the minimal Geršgorin set gives the sharpest and the most precise localization of the spectrum ([39]). In this thesis, new algorithms for computing an efficient and accurate approximation of the minimal Geršgorin set are presented.Постоје бројни начини за локализацију карактеристичних корена. Један од најчувенијих резултата је да се спектар дате матрице АCn,n налази у скупу који представља унију кругова са центрима у дијагоналним елементима матрице и полупречницима који су једнаки суми модула вандијагоналних елемената одговарајуће врсте у матрици. Овај резултат (Гершгоринова теорема, 1931.), сматра се једним од најзначајнијих и најелегантнијих начина за локализацију карактеристичних корена ([61]). Међу свим локализацијама Гершгориновог типа, минимални Гершгоринов скуп даје најпрецизнију локализацију спектра ([39]). У овој дисертацији, приказани су нови алгоритми за одређивање тачне и поуздане апроксимације минималног Гершгориновог скупа.Postoje brojni načini za lokalizaciju karakterističnih korena. Jedan od najčuvenijih rezultata je da se spektar date matrice ACn,n nalazi u skupu koji predstavlja uniju krugova sa centrima u dijagonalnim elementima matrice i poluprečnicima koji su jednaki sumi modula vandijagonalnih elemenata odgovarajuće vrste u matrici. Ovaj rezultat (Geršgorinova teorema, 1931.), smatra se jednim od najznačajnijih i najelegantnijih načina za lokalizaciju karakterističnih korena ([61]). Među svim lokalizacijama Geršgorinovog tipa, minimalni Geršgorinov skup daje najprecizniju lokalizaciju spektra ([39]). U ovoj disertaciji, prikazani su novi algoritmi za određivanje tačne i pouzdane aproksimacije minimalnog Geršgorinovog skupa

Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations

We prove asymptotic stability of shear flows close to the planar Couette flow in the 2D inviscid Euler equations on \Torus \times \Real. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L^2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically driven to small scales by a linear evolution and weakly converges as $t \rightarrow \pm\infty$. The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. This convergence was formally derived at the linear level by Kelvin in 1887 and it occurs at an algebraic rate first computed by Orr in 1907; our work appears to be the first rigorous confirmation of this behavior on the nonlinear level.Comment: 78 page

Pushing the Limits of Indoor Localization in Today’s Wi-Fi Networks

Wireless networks are ubiquitous nowadays and play an increasingly important role in our everyday lives. Many emerging applications including augmented reality, indoor navigation and human tracking, rely heavily on Wi-Fi, thus requiring an even more sophisticated network. One key component for the success of these applications is accurate localization. While we have GPS in the outdoor environment, indoor localization at a sub-meter granularity remains challenging due to a number of factors, including the presence of strong wireless multipath reflections indoors and the burden of deploying and maintaining any additional location service infrastructure. On the other hand, Wi-Fi technology has developed significantly in the last 15 years evolving from 802.11b/a/g to the latest 802.11n and 802.11ac standards. Single user multiple-input, multiple-output (SU-MIMO) technology has been adopted in 802.11n while multi-user MIMO is introduced in 802.11ac to increase throughput. In Wi-Fi’s development, one interesting trend is the increasing number of antennas attached to a single access point (AP). Another trend is the presence of frequency-agile radios and larger bandwidths in the latest 802.11n/ac standards. These opportunities can be leveraged to increase the accuracy of indoor wireless localization significantly in the two systems proposed in this thesis: ArrayTrack employs multi-antenna APs for angle-of-arrival (AoA) information to localize clients accurately indoors. It is the first indoor Wi-Fi localization system able to achieve below half meter median accuracy. Innovative multipath identification scheme is proposed to handle the challenging multipath issue in indoor environment. ArrayTrack is robust in term of signal to noise ratio, collision and device orientation. ArrayTrack does not require any offline training and the computational load is small, making it a great candidate for real-time location services. With six 8-antenna APs, ArrayTrack is able to achieve a median error of 23 cm indoors in the presence of strong multipath reflections in a typical office environment. ToneTrack is a fine-grained indoor localization system employing time difference of arrival scheme (TDoA). ToneTrack uses a novel channel combination algorithm to increase effective bandwidth without increasing the radio’s sampling rate, for higher resolution time of arrival (ToA) information. A new spectrum identification scheme is proposed to retrieve useful information from a ToA profile even when the overall profile is mostly inaccurate. The triangle inequality property is then applied to detect and discard the APs whose direct path is 100% blocked. With a combination of only three 20 MHz channels in the 2.4 GHz band, ToneTrack is able to achieve below one meter median error, outperforming the traditional super-resolution ToA schemes significantly