POD i DMD dekompozicije u analizi i redukciji nelinearnih dinamičkih sustava

master's thesis

• Author: Marta Kapetanović
• Mentor: Zlatko Drmač (mentor)
• Committee member: Zlatko Drmač (predsjednik povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Committee member: Miljenko Marušić (član povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-11-29, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U radu smo proučavali dvije dekompozicije dinamičkih sustava: POD i DMD. Fokus je bio na razumijevanju teorijske osnove u pozadini algoritma te primjeni u dobivanju modela nižeg reda (u slučaju POD) te rekonstrukciji i predikciji (kod DMD metode). POD dekompozicija je metoda koja za dobivanje modela nižeg reda promatranog sustava traži projekciju sustava na ortonormiranu bazu, ali takvu da je projekcija optimalna u smislu najmanjih kvadrata. Spomenutu ortonormiranu bazu nazivamo POD baza. Opisali smo računanje POD baze u nekoliko mogućih slučajeva: konačnodimenzionalni slučaj, kada se svodi na računanje SVD-a, za težinski skalarni produkt te pomoću metode uzoraka. Aproksimacijski model nižeg reda dobili smo Galerkinovom projekcijom na prostor razapet POD bazom. Ukupnu grešku reduciranog sustava dali smo u ovisnosti o greški aproksimacije. DMD metoda bazira se na pronalasku koherentnih prostornih struktura koje čine gibanje. One su dane svojstvenim vektorima matrice operatora \(A\) koji opisuje dinamiku sustava. Operator preslikavanja \(A\) općenito je nepoznat, no poznata su dva skupa uzoraka, početni i preslikani. Iz danih skupova uzoraka u stanju smo odrediti operator \(\tilde{A}_k\) koji je linearna aproksimacija od \(A\). Pokaže se da su svojstvene vrijednosti i svojstveni vektori operatora \(\tilde{A}_k\) aproksimacije svojstvenih vrijednosti i vektora od \(A\). Opisali smo tri algoritma za računanje DMD modova koji se razlikuju u ograničenjima na skup ulaznih podataka te računanju svojstvenih vektora od \(\tilde{A}\). Vremensku evoluciju sustava određuju svojstvene vrijednosti od \(A\). Potpunu rekonstukciju sustava dobivamo iz DMD modova, pripadnih svojstvenih vrijednosti i početnih amplituda. Naposljetku smo opisane metode primijenili na dinamičkim sustavima koje opisuju Navier-Stokesove jednadžbe (za DMD), te jednadžbe konvekcije difuzije reakcije (za POD).

Abstract (english)

In this thesis we have studied two decompositions of dynamic systems: POD and DMD. The focus was on understanding the theoretical background defining the algorithm and applying it to obtain the low-order model (for POD) and reconstruction and prediction (for DMD method). The POD decomposition is a method that seeks a projection of the system to an orthonormal basis to obtain a low-order model of the observed system, but such that the projection is optimal in the least-squares sense. Above mentioned orthonormal basis is called the POD basis. We have outlined the baseline POD calculation in several possible cases: in finite-dimension, when it’s as simple as calculating an SVD of a matrix, in more complex space with weighted scalar product, and by snapshot method. The low-order model was obtained by Galerkin’s projection to the space spanned by the POD basis. The total error of the reduced system is given as a function of the approximation error. The DMD method is based on finding coherent spatial structures that define the dynamics of the system. These are given by the eigenvectors of the mapping matrix \(A\). Matrix \(A\) is generally unknown, but two sets of snapshots are known. From the given sets of snaphots we can determine the operator \(\tilde{A}\), which is a linear approximation of \(A\). It can be shown that the eigenvalues and eigenvectors of \(\tilde{A}_k\) are aproximations of eigenvalues and eigenvectors of \(A\). We have provided three algorithms for computing DMD modes that differ in the constraints on the set of input data and the computation of eigenvectors of \(\tilde{A}\). The system’s time evolution is determined by eigenvalues of \(A\). Complete reconstruction of the system is obtained from DMD modes, associated eigenvalues and initial amplitudes. Finally, we applied the DMD method on dynamic system governed by Navier-Stokes equations, and POD on system governed by equations of convection diffusion reaction.

• Keywords: POD dekompozicija; DMD dekompozicija; ortonormirana baza; Galerkinova projekcija; Navier-Stokesove jednadžbe
• Keywords (english): POD decomposition; DMD decomposition; orthonormal basis; Galerkin’s projection; Navier-Stokes equations
• Language: croatian
• URN:NBN: urn:nbn:hr:217:582163
• Study programme: Title: Applied Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2018-05-25 10:07:54