Efficient solution of symmetric eigenvalue problems from families of coupled systems

Efficient solution of the lowest eigenmodes is studied for a family of related eigenvalue problems with common $2\times 2$ block structure. It is assumed that the upper diagonal block varies between different versions while the lower diagonal block and the range of the coupling blocks remains unchanged. Such block structure naturally arises when studying the effect of a subsystem to the eigenmodes of the full system. The proposed method is based on interpolation of the resolvent function after some of its singularities have been removed by a spectral projection. Singular value decomposition can be used to further reduce the dimension of the computational problem. Error analysis of the method indicates exponential convergence with respect to the number of interpolation points. Theoretical results are illustrated by two numerical examples related to finite element discretisation of the Laplace operator

Distributed solution of Laplacian eigenvalue problems

The purpose of this article is to approximately compute the eigenvalues of the symmetric Dirichlet Laplacian within an interval $(0,\Lambda)$. A novel domain decomposition Ritz method, partition of unity condensed pole interpolation method, is proposed. This method can be used in distributed computing environments where communication is expensive, e.g., in clusters running on cloud computing services or networked workstations. The Ritz space is obtained from local subspaces consistent with a decomposition of the domain into subdomains. These local subspaces are constructed independently of each other, using data only related to the corresponding subdomain. Relative eigenvalue error is analysed. Numerical examples on a cluster of workstations validate the error analysis and the performance of the method.Comment: 28 page

A Novel Marker Based Method to Teeth Alignment in MRI

Magnetic resonance imaging (MRI) can precisely capture the anatomy of the vocal tract. However, the crowns of teeth are not visible in standard MRI scans. In this study, a marker-based teeth alignment method is presented and evaluated. Ten patients undergoing orthognathic surgery were enrolled. Supraglottal airways were imaged preoperatively using structural MRI. MRI visible markers were developed, and they were attached to maxillary teeth and corresponding locations on the dental casts. Repeated measurements of intermarker distances in MRI and in a replica model was compared using linear regression analysis. Dental cast MRI and corresponding caliper measurements did not differ significantly. In contrast, the marker locations in vivo differed somewhat from the dental cast measurements likely due to marker placement inaccuracies. The markers were clearly visible in MRI and allowed for dental models to be aligned to head and neck MRI scans

Kytkettyjen akustisten ominaisarvotehtävien ratkaiseminen

This thesis concerns the solution of finite element discretised Laplacian eigenvalue problems of coupled systems. A relevant application is studying human speech, where vowels are classified according to the lowest resonant frequencies of the vocal tract during sustained pronunciation. When computing these frequencies, the acoustic environment has to be accounted for. In this thesis, the vocal tract is considered as an interior system that is coupled with its acoustic environment, the exterior system. The coupling occurs through a fixed interface. Models for computing the resonant frequencies are validated against data consisting of simultaneous MRI images and sound recordings. As the validation requires a large number of vocal tract geometries, an automatic extraction algorithm that generates vocal tract surface triangulations from MRI data is introduced. An instrument for performing frequency sweeps on physical models printed using these geometries was also modelled as a part of this thesis. The confined space inside the head coil of the MRI machine creates an acoustic environment where mixed modes appear. That is, standing waves that oscillate both inside the oral cavity and the head coil are formed. Hence, it is important that the acoustic environment consisting of the MRI coil is accurately modelled when validating computational models. To efficiently solve relevant resonances related to different vocal tract geometries coupled with the unchanging MRI head coil, a method for reducing the computational complexity related to the fixed exterior system is introduced. The mathematical observations related to the aforementioned method were generalised from an algebraic setting to a continuous Laplace eigenvalue problem. As a result, a theory for obtaining information on an eigenfunction in a local subdomain from localised boundary data was developed. The computational realisation of this theory is a domain decomposition type eigenvalue solver where tasks related to each subdomain are mutually independent. This method can be used to approximately solve finite element discretised eigenvalue problems where the number of degrees of freedom is prohibitively large for a single workstation to compute. Such an eigenvalue solver can be used to efficiently solve large eigenvalue problems without the need for a supercomputer. Due to the independence of computations related to subdomains, tasks can be sent over a network connection, making the method suitable for cloud computing environments.Tämä väitöskirja käsittelee elementtimenetelmällä diskretoitujen Laplace -operaattorin ominaisarvotehtävien ratkaisemista toisiinsa kytketyissä akustisissa systeemeissä. Tehtävä esiintyy erityisesti puheentutkimuksessa, jossa vokaaliäänteitä voidaan luokitella ääntöväylän alimpien resonanssitaajuuksien avulla. Näitä taajuuksia laskettaessa on huomioitava akustinen ympäristö. Tässä väitöskirjassa ääntöväylää käsitellään sisäsysteeminä, joka kytketään vakiorajapinnan yli ulkosysteemiin, eli ulkotilaan. Resonanssitaajuuksien laskentaan käytettävät laskennalliset mallit validoidaan MRI-datan keräämisen yhteydessä tallennettujen ääninäytteiden avulla. Tämä vaatii suuren määrän ääntöväylägeometrioita, joten työssä esitellään automatisoitu menetelmä pinta- ja laskentaverkkojen luomiseen MRI-datasta. Lisäksi työssä mallinnetaan laite jolla voidaan mitata 3D-tulostettujen ääntöväylägeometrioiden taajuusvasteita. MRI-laitteen pääkelan ahtaudesta johtuen äänitystilanteessa esiintyy seisovia aaltoja, jotka oskilloivat sekä pääkelan että ääntöväylän sisällä. Tästä syystä ulkotilan vaikutus täytyy mallintaa tarkasti. Jotta muuttumattomana pysyvän pääkelan vaikutus siihen kytkettyihin erilaisiin ääntöväylägeometrioihin voidaan mallintaa tehokkaasti, väitöskirjassa esitellään menetelmä laskennallisen vaativuuden vähentämiseksi tämänkaltaisissa tilanteissa. Edelläkuvatun menetelmän kehittämisessä tehdyt huomiot yleistettiin algebrallisesta asetelmasta jatkuvalle Laplace -operaattorin ominaisarvotehtävälle. Näin saatiin teoreettinen menetelmä, jolla ominaisfunktion käytös paikallisessa alialueessa saadaan ratkaistua paikallisen reunakäytöksen avulla. Menetelmän laskennallinen toteutus on aluehajotukseen perustuva ominaisarvotehtävien ratkaisumenetelmä, jossa alialueisiin liittyvät laskentatyöt ovat toisistaan riippumattomia. Menetelmää voidaan käyttää elementtimenetelmällä diskretoitujen ominaisarvotehtävien approksimatiiviseen ratkaisemiseen silloin kun vapausasteiden määrä on liian suuri yksittäiselle tietokoneelle. Tällöin ei välttämättä tarvita suurteholaskentaa. Lisäksi laskentatehtävät voidaan lähettää verkon yli eri tietokoneille laskettavaksi, jolloin menetelmä soveltuu myös pilvilaskentaympäristöihin

Fourier-analyysia Heisenbergin ryhmällä

In this thesis we derive the Stone-von Neumann theorem which can be used to characterize the strongly continuous unitary representations of the Heisenberg group. We then study the group Fourier transform on the Heisenberg group. By defining the group transform on the Schwartz space of the Heisenberg group we derive an inversion theorem as well as an equivalent of Plancherel's theorem. We then extend these results to the L2-space. In addition, the Wigner transform and Wigner distribution are studied. Aside from proving some important properties of these transforms, special attention is paid to the connection between the Wigner distribution and the windowed Fourier transform.Tässä työssä johdetaan Heisenbergin ryhmän unitaariset vahvasti jatkuvat esitykset karakterisoiva Stone-von Neumannin lause, sekä tutkitaan esitysten kautta määritellyn Heisenbergin ryhmän Fourier-muunnoksen ominaisuuksia. Ryhmän Fourier-muunnos määritellään Schwartzin testifunktioille, ja sille saadaan käänteismuunnos sekä Plancherelin lausetta vastaava tulos, jonka avulla muunnos saadaan laajennettua L2-avaruudelle. Lisäksi tutkitaan Wigner-muunnosta sekä -distribuutiota, joiden tärkeimmät ominaisuudet johdetaan. Erityisesti kiinnitetään huomiota Wigner-distribuution yhteyteen ikkunoituun Fourier-muunnokseen

Efficient solution of symmetric eigenvalue problems from families of coupled systems

Efficient solution of the lowest eigenmodes is studied for a family of related eigenvalue problems with common 2 × 2 block structure. It is assumed that the upper diagonal block varies between different versions while the lower diagonal block and the range of the coupling blocks remain unchanged. Such block structure naturally arises when studying the effect of a subsystem to the eigenmodes of the full system. The proposed method is based on interpolation of the resolvent function after some of its singularities have been removed by a spectral projection. Singular value decomposition can be used to further reduce the dimension of the computational problem. Error analysis of the method indicates exponential convergence with respect to the number of interpolation points. Theoretical results are illustrated by two numerical examples related to finite element discretization of the Laplace operator.Peer reviewe