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Solving an inverse problem for an elliptic equation using a Fourier-sine series.
2019 (engelsk)Independent thesis Basic level (degree of Bachelor), 10,5 poäng / 16 hpOppgave
##### Abstract [en]

This work is about solving an inverse problem for an elliptic equation. An inverse problem is often ill-posed, which means that a small measurement error in data can yield a vigorously perturbed solution. Regularization is a way to make an ill-posed problem well-posed and thus solvable.

Two important tools to determine if a problem is well-posed or not are norms and convergence. With help from these concepts, the error of the reg- ularized function can be calculated. The error between this function and the exact function is depending on two error terms.

By solving the problem with an elliptic equation, a linear operator is eval- uated. This operator maps a given function to another function, which both can be found in the solution of the problem with an elliptic equation. This opera- tor can be seen as a mapping from the given function’s Fourier-sine coefficients onto the other function’s Fourier-sine coefficients, since these functions are com- pletely determined by their Fourier-sine series. The regularization method in this thesis, uses a chosen number of Fourier-sine coefficients of the function, and the rest are set to zero. This regularization method is first illustrated for a simpler problem with Laplace’s equation, which can be solved analytically and thereby an explicit parameter choice rule can be given.

The goal with this work is to show that the considered method is a reg- ularization of a linear operator, that is evaluated when the problem with an elliptic equation is solved. In the tests in Chapter 3 and 4, the ill-posedness of the inverse problem is illustrated and that the method does behave like a regularization is shown. Also in the tests, it can be seen how many Fourier-sine coefficients that should be considered in the regularization in different cases, to make a good approximation.

##### Abstract [sv]

Det här arbetet handlar om att lösa ett inverst problem för en elliptisk ekvation. Ett inverst problem är ofta illaställt, vilket betyder att ett litet mätfel i data kan ge en kraftigt förändrad lösning. Regularisering är ett tillvägagångssätt för att göra ett illaställt problem välställt och således lösbart.

Två viktiga verktyg för att bestämma om ett problem är välställt eller inte är normer och konvergens. Med hjälp av dessa begrepp kan felet av den regulariserade lösningen beräknas. Felet mellan den lösningen och den exakta är beroende av två feltermer.

Genom att lösa problemet med den elliptiska ekvationen, så är en linjär operator evaluerad. Denna operator avbildar en given funktion på en annan funktion, vilka båda kan hittas i lösningen till problemet med en elliptisk ekva- tion. Denna operator kan ses som en avbildning från den givna funktions Fouri- ersinuskoefficienter på den andra funktionens Fouriersinuskoefficienter, eftersom dessa funktioner är fullständigt bestämda av sina Fouriersinusserier. Regularise- ringsmetoden i denna rapport använder ett valt antal Fouriersinuskoefficienter av funktionen, och resten sätts till noll. Denna regulariseringsmetod illustreras först för ett enklare problem med Laplaces ekvation, som kan lösas analytiskt och därmed kan en explicit parametervalsregel anges.

Målet med detta arbete är att visa att denna metod är en regularisering av den linjära operator som evalueras när problemet med en elliptisk ekvation löses. I testerna i kapitel 3 och 4, illustreras illaställdheten av det inversa problemet och det visas att metoden beter sig som en regularisering. I testerna kan det också ses hur många Fouriersinuskoefficienter som borde betraktas i regulariseringen i olika fall, för att göra en bra approximation.

2019. , s. 39
##### Emneord [en]
Regularization, ill-posed problem, Fourier-sine series
##### Emneord [sv]
Regularisering, illaställt problem, Fouriersinusserie
##### Identifikatorer
ISRN: LiTH-MAT-EX–2019/04–SEOAI: oai:DiVA.org:liu-162371DiVA, id: diva2:1374134
Mathematics
##### Examiner
Tilgjengelig fra: 2019-11-29 Laget: 2019-11-29 Sist oppdatert: 2019-11-29bibliografisk kontrollert

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