Poluglatka Newtonova metoda u Banachovim prostorima

master's thesis

• Author: Marin Gunja
• Mentor: Marko Vrdoljak (mentor)
• Committee member: Marko Vrdoljak (predsjednik povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Committee member: Maja Resman (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-09-29, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ukratko, u ovome radu se bavimo Newtonovom metodom i njenim raznovrsnim poopćenjima. U prvom poglavlju proučavamo standardnu Newtonovu metodu za glatku realnu funkciju realne varijable. Motiviramo metodu, opisujemo je, iznosimo teoreme konvergencije, izvodimo ocjene greške. Ukratko opisujemo kako se situacija mijenja kada imamo višestruku nultočku. Za kraj, implementiramo ovu metodu u Pythonu i navodimo još nekoliko primjera. U drugom poglavlju proširujemo Newtonovu metodu na glatke vektorske funkcije. Opet navodimo metodu i proučavamo teoreme konvergencije, s tim da se detaljnije bavimo lokalnom konvergencijom, dok globalnu konvergenciju samo ukratko opisujemo (iskazujemo jedan važan teorem i opisujemo jednu strategiju kako postići globalnu konvergenciju). Dalje opisujemo kako Newtonovom metodom možemo riješiti problem bezuvjetne optimizacije. Za kraj, implementiramo ovu metodu u Pythonu i isprobavamo implementaciju na konkretnom primjeru. U trećem poglavlju proučavamo neglatke funkcije. Uvodimo pojmove generaliziranog B-subdiferencijala i (Clarkeovog) generaliziranog diferencijala. Pomoću generaliziranog diferencijala uvodimo Newtonovu metodu za neglatke sustave. Poslije toga uvodimo pojam poluglatkih funkcija, što nam je važan pojam pri proučavanju teorema konvergencije Newtonove metode za neglatke sustave (koju nazivamo poluglatkom Newtonovom metodom kada ju primijenjujemo na poluglatke funkcije). Poslije ukratko opisujemo kako Newtonovu metodu možemo proširiti na proizvoljne Banachove prostore. Također pokazujemo kako se Newtonove metoda može iskoristiti da efikasno numerički riješimo ODJ. Dano je nekoliko primjera u kojima rješavamo ODJ prvog i drugog reda.

Abstract (english)

In this paper, we observe Newton's method and its various generalizations. In the first chapter, we observe the standard Newton's method for smooth real-valued functions of a real variable. We motivate the method, describe it, and present the convergence theorems and error estimates. We briefly describe how the situation changes when we have multiple zeros. Finally, we implement this method in Python and give few more examples. In the second chapter, we extend Newton's method so that it can be applied to smooth vector functions. Again, we describe the method and study the convergence theorems, dealing in more detail with local convergence, while we only briefly describe global convergence (we state one important theorem and describe one strategy on how to achieve global convergence). We further describe how the Newton method can be used to solve the problem of unconstrained optimization. Finally, we implement this method in Python and test the implementation on a concrete example. In the third chapter, we study nonsmooth functions. We introduce the concepts of generalized B-subdifferential and (Clarke's) generalized differential. Using the generalized differential, we introduce Newton's method for nonsmooth systems. We then introduce the notion of semismooth functions, which is an important term for us when studying the convergence theorems of Newton's method for nonsmooth systems (which we call the semismooth Newton's method when we apply it to semismooth functions). We then briefly describe how we can extend Newton's method to arbitrary Banach spaces. We also show how Newton's method can be used to efficiently numerically solve ODEs. Several examples are given in which we solve first- and second-order ODEs.

• Keywords: Newtonova metoda; Python; problem bezuvjetne optimizacije; generalizirani B-subdiferencijal; Clarkeov generalizirani diferencijal
• Keywords (english): Newton's method; Python; eneralized B-subdifferential; (Clarke's) generalized differential; the problem of unconstrained optimization
• Language: croatian
• URN:NBN: urn:nbn:hr:217:889405
• Study programme: Title: Finance and Business 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: 2020-12-10 16:06:50