Uporaba genetskega algoritma pri inverznem problemu izospektralnih dvodimenzionalnih membran

Jozič, Primož (author), Čopar, Simon (mentor)

V magistrski nalogi preverjamo uporabo genetskega algoritma za reševanje inverznega problema izospektralnih dvodimenzionalnih membran. Ideja izhaja iz članka z naslovom 'Can One Hear the Shape of a Drum?' avtorja Marka Kaca, kjer se postavlja vprašanje, ali je mogoče iz spektra nihanja dvodimenzionalne opne določiti njeno obliko. Kasneje se izkaže, da je odgovor na to vprašanje nikalen, avtorji članka 'One Cannot Hear the Shape of a Drum' namreč pokažejo, da obstaja več različnih open, ki imajo enak spekter. V uvodnem delu naloge predstavimo teoretično ozadje, povezano z inverznim problemom in osnovami genetskih algoritmov. Poseben poudarek je na predstavitvi teorije genetskih algoritmov, ki služi kot osnova za razumevanje njihove uporabe pri reševanju inverznih problemov. Nato sistematično preizkušamo genetske algoritme na različnih primerih, ki postajajo vedno bolj kompleksni. S tem želimo raziskati in pokazati njihov potencial in učinkovitost pri reševanju inverznih problemov. Rezultati naših simulacij kažejo, da genetski algoritmi pogosto proizvedejo dobre rešitve, pri čemer se izkaže prednost v tem, da namesto ene same rešitve dobimo celo družino rešitev, ki se lahko med seboj razlikujejo. S tem pridemo do možnosti kombinacije genetskih algoritmov s klasičnimi optimizacijskimi metodami, kar lahko privede do še boljših rezultatov v krajšem času računanja. Predvsem se osredotočimo na to, da pokažemo uporabnost genetskih algoritmih in njihov potencial. Cilj naloge ni iskanje najboljšega načina za reševanje tega specifičnega problema, niti ni cilj natančno določevati parametrov genetskega algoritma za ta problem. Eden izmed ciljev naloge je tudi, da služi kot vpogled v uporabo genetskih algoritmov in kot navdih za nadaljnje delo in izboljšanje rezultatov, dobljenih v tej nalogi. Želja je, da bralec spozna možnosti, ki jih ponujajo genetski algoritmi kot drugačen pristop k reševanju problema.

• Language: Slovenian
• Keywords: inverzni problem, genetski algoritem, stohastične metode, izospektralni problem
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2024
• PID: 20.500.12556/RUL-156251
• COBISS.SI-ID: 195502595
• Publication date in RUL: 16.05.2024

JOZIČ, Primož, 2024, Uporaba genetskega algoritma pri inverznem problemu izospektralnih dvodimenzionalnih membran [online]. Master’s thesis. [Accessed 11 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=156251

Secondary language

• Language: English
• Title: The use of genetic algorithms in the inverse problem of isospectral two-dimensional membranes

Abstract

In the master’s thesis, we explore the use of genetic algorithms to solve the inverse problem of isospectral two-dimensional membranes. The idea originates from Mark Kac’s article 'Can One Hear the Shape of a Drum?', questioning whether the shape of a two-dimensional membrane can be determined from its vibration spectrum. Later, it was shown that the answer is negative, as demonstrated by the authors of 'One Cannot Hear the Shape of a Drum', who showed that multiple membranes can share the same spectrum. The introduction of the thesis presents the theoretical background related to the inverse problem and the basics of genetic algorithms, with a particular focus on the theory of genetic algorithms as a foundation for understanding their application in solving inverse problems. We then systematically test genetic algorithms on various cases, which become increasingly complex. The aim is to explore and demonstrate their potential and effectiveness in solving inverse problems. The results of our simulations often show that genetic algorithms produce good solutions, with the advantage being that instead of a single solution, we obtain an entire family of solutions. This leads to the possibility of combining genetic algorithms with classical optimization methods, which can lead to even better results in less computational time. Our main focus is to demonstrate the usefulness of genetic algorithms and their potential. The goal of the thesis is not to find the best way to solve this specific problem, nor is it to precisely determine the parameters of the genetic algorithm for this problem. One of the objectives of the thesis is also to serve as an insight into the use of genetic algorithms and as inspiration for further work and improvement of the results obtained in this thesis. The intention is for the reader to become acquainted with the possibilities offered by genetic algorithms as a different approach to problem-solving.

• Keywords: inverse problem, genetic algorithm, stochastic methods, isospectral problem