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Author:Hujanen, Jaakko
Title:Inverse Problem for Quantum Mechanical Scattering in One Dimension
Yksiulotteisen kvanttimekaanisen sironnan käänteisongelma
Publication type:Master's thesis
Publication year:2014
Pages:69      Language:   eng
Department/School:Perustieteiden korkeakoulu
Main subject:Mekaniikka   (F3007)
Supervisor:Stenberg, Rolf
Instructor:Hyvönen, Nuutti ; Ojanen, Teemu
Electronic version URL: http://urn.fi/URN:NBN:fi:aalto-201410042720
OEVS:
Electronic archive copy is available via Aalto Thesis Database.
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Location:P1 Ark Aalto  1753   | Archive
Keywords:inverse problems
quantum mechanical scattering
finite element method
Gauss-Newton method
käänteisongelmat
kvanttimekaaninen sironta
elementtimenetelmä
Gauss-Newton-menetelmä
Abstract (eng):The field of inverse problem considers mathematical methods for reconstructing unknown parameters of some model from given measurements.
Such problems arise in different branches of science, e.g, in medical imaging.
In this thesis, inverse mathematics is applied to a one-dimensional scattering problem in a quantum mechanical setting: the aim is to create an algorithm to determine the potential function of a scattering interval when transmission of electrons is observed.
Since the scattering problem includes non-linear phenomena, an iterative reconstruction scheme is employed.
The implemented numerical method combines finite element method (FEM) and the Gauss-Newton optimization algorithm in a manner which utilizes the FEM solver for numerical evaluations of the Jacobian matrix.
Furthermore, the forward problem is shown to have a unique solution to reinforce credibility of the procedure.
The resulting program optimally gives good reconstructions with simulated test data, but it is sensitive to the choice of regularization.
Abstract (fin):Käänteisongelmissa tavoitteena on päätellä fysikaalisen systeemin parametreja mittausdatasta.
Tällaisia ongelmia esiintyy tieteessä laajalti, ja sovelluskohteita on esimerkiksi lääketieteellisessä kuvantamisessa.
Tämän työn tavoitteena on luoda algoritmi kvanttimekaaniselle sirontaongelmalle, jossa tutkittavan alueen potentiaalifunktion ominaisuuksia selvitetään havaituista elektronien läpäisytodennäköisyyksistä.
Tässä tapauksessa sironnan matemaattinen malli on epälineaarinen, minkä takia joudutaan käyttämään elementtimenetelmään (FEM) perustuvaa iteratiivista numeerista inversioalgoritmia.
Ratkaisin pohjautuu Gauss-Newton-optimointimenetelmään, jossa Jacobin matriisi lasketaan numeerisesti FEM-ratkaisijalla.
Suoran sirontaongelman ratkaisun olemassaolo ja yksikäsitteisyys todistetaan algoritmin toimivuuden varmistamiseksi.
Potentiaalien rekonstruktio testidatasta onnistuu hyvin, mutta menetelmä on herkkä regularisaation valinnan suhteen.
ED:2014-10-05
INSSI record number: 49781
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