Model order reduction with applications

master's thesis

• Title (croatian): Metode redukcije dimenzije modela i primjene
• Author: Domagoj Bošnjak
• Mentor: Zlatko Drmač (mentor)
• Committee member: Zlatko Drmač (predsjednik povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-07-21, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Model order reduction is an integral approach to solving high-dimensional systems of ordinary and partial differential equations. As the needs for processing large amounts of data, resulting in high-order models, are only increasing, this approach will understandably stay relevant. Reduced basis methods are discussed, with the focus being on the proper orthogonal decomposition (POD). Their applications to parametric PDEs are presented as well. The encountered issue in this approach involves nonlinear systems, which are instead tackled by a combination of POD and the discrete empirical interpolation method (DEIM). Alongside them, newer versions of the DEIM algorithm are presented, namely the Q-DEIM and K-DEIM, which consist of utilizing the QR decomposition and clustering methods, respectively. Numerical experiments have shown the effectiveness of these approaches, to both synthetic and real-life problems, notably the FKPP equation, which is very important in biology. The potentially surprising results indicate that models whose dimensions are in hundreds or thousands may be accurately represented by systems of only several equations. However, the methods presented here encompass only a small part of the plethora of model order reduction methods

Abstract (croatian)

Redukcija dimenzije modela je ključan pristup rješavanju visokodimenzionalnih sustava običnih i parcijalnih diferencijalnih jednadžbi. Kako potrebe za procesiranjem velikih količina podataka koji rezultiraju u modelima visoke dimenzije rastu, ovaj pristup, razumljivo, ostaje relevantan. U ovom radu prezentirane su metode dobivanja reduciranih baza, s fokusom na POD metodu. Pokazana je i njihova primjena na parametrizirane parcijalne diferencijalne jednažbe. Problem na koji se nailazi u primjeni POD-a uključuje nelinearne parcijalne diferencijalne jednažbe, kojima se pristupa kombinacijom POD-a i diskretne empirijske interpolacijske metode(DEIM). Pored njih, dane su i novije verzije DEIM algoritma zvane Q-DEIM i K-DEIM, unutar kojih se, redom, koriste QR dekompozicija i metode klasteriranja. Numerički eksperimenti pokazali su učikovitost ovih pristupa na sintetičke i stvarne probleme, posebno na FKPP jednadžbu, vrlo važnu u biologiji. Potencijalno neočekivani rezultati ukazuju na to da se modeli čija je dimenzija u stotinama ili tisućama mogu dovoljno točno reprezentirati sistemima od svega nekoliko jednadžbi. Ipak, metode prezentirane u ovom radu obuhva´caju samo maleni dio ogromnog spektra metoda za redukciju dimenzije modela

• Keywords: reduced basis methods; proper orthogonal decomposition (POD); discrete empirical interpolation method (DEIM); FKPP equation
• Keywords (croatian): metode dobivanja reduciranih baza; POD metoda; diskretna empirijska interpolacijska metoda (DEIM); FKPP jednadžba
• Language: english
• URN:NBN: urn:nbn:hr:217:005969
• 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: 2021-09-14 12:05:04