Department of Mathematics and Systems Analysis

Research groups

Inverse problems

N. Hyvönen, A. Seppänen and S. Staboulis. Optimizing electrode positions in EIT. SIAM Journal on Applied Mathematics 2014, Vol. 74, pp. 1831-1851.

Inverse problems constitute an active and expanding research field of mathematics and its applications. Inverse problems are encountered in several areas of applied sciences such as biomedical engineering and imaging, geosciences, vulcanology, remote sensing, and non-destructive material evaluation. To put it short, a forward problem is to deduce consequences of a cause, while the corresponding inverse problem is to find the causes of a known consequence. Inverse problems are typically encountered  when one has indirect observations of the quantity of interest.

A fundamental feature of inverse problems is that they are ill-posed: Small errors in the measured data can cause arbitrarily large errors in the estimates of the parameters of interest, or can even render the problem unsolvable. It may also occur that an inverse problem does not have a unique solution, i.e., there are several different parameter values that could produce the same observed data. In consequence, to successfully tackle inverse problems, one needs to have comprehensive understanding of the uniqueness and stability of the solution as well as state-of-the-art methods for incorporating prior information into the inverse solver algorithms.

Personnel

Associate Professor Nuutti  Hyvönen
Associate Professor
Nuutti Hyvönen
M307 
Senior University Lecturer Harri  Hakula
Senior University Lecturer
Harri Hakula
M311 
Doctoral Candidate Vesa  Kaarnioja
Doctoral Candidate
Vesa Kaarnioja
M327 
Doctoral Candidate Matti  Leinonen
Doctoral Candidate
Matti Leinonen
Y249a 
Doctoral Candidate Helle  Majander
Doctoral Candidate
Helle Majander
 
Doctoral Candidate Lauri  Mustonen
Doctoral Candidate
Lauri Mustonen
Y249a 
Teaching Assistant Stratos  Staboulis
Teaching Assistant
Stratos Staboulis
Y249a 

Research topics

Diffuse tomography

The objective of diffuse tomography is to deduce information about the internal structure of a body from observations of diffuse fields at its boundary. The two main applications are

Electrical impedance tomography (EIT),
Diffuse optical tomography (DOT).

Wave field phenomena

A commonly used method for making inference of an inaccessible region is to use wave fields for sounding. The employed wave fields may be acoustic, electromagnetic or elastic. The research on wave fields is focused on

Inverse scattering problems,
Remote sensing and imaging.

Inverse source problems

The aim in an inverse source problem is to reconstruct a source based on measurements of the resulting field. Inverse source problems are encountered, e.g., in biomedical applications such as Electroencephalography/cardiography (EEG/ECG) and Magnetoencephalography/cardiography (MEG/MCG). Another area where these problems arise is seismology.

Statistical methods

A very generic and versatile method for investigating inverse problems is to recast them as Bayesian problems of statistical inference. Powerful methods for numerically solving very complicated inverse problems can be developed by using various Monte Carlo sampling algorithms. These methods are also related to dynamical problems that are traditionally referred to as filtering problems.

Research projects

The group has numerous domestic and international research contacts. Domestic collaboration is organized through the Finnish Inverse Problems Society.

Inverse problems links

Recent publications

2014 | 2013 | 2012 | 2011 | 2010 | 2009 | 2008 | 2007

2014

Harri Hakula: hp-boundary layer mesh sequences with applications to shell problems. Computers & mathematics with applications Computers & Mathematics with Applications 67(4), 2014, pp. 899–917. BibTeX

Harri Hakula, Nuutti Hyvönen, Matti Leinonen: Reconstruction algorithm based on stochastic Galerkin finite element method for electrical impedance tomography. Inverse problems 30(6), 2014. BibTeX

Nuutti Hyvönen, Aku Seppänen, Stratos Staboulis: Optimizing electrode positions in electrical impedance tomography. SIAM Journal on Applied Mathematics 74(6), 2014, pp. 1831–1851. BibTeX

Matti Leinonen, Nuutti Hyvönen, Harri Hakula: Application of stochastic Galerkin FEM to the complete electrode model of electrical impedance tomography. Journal of Computational Physics 269(1), 2014, pp. 181–200. BibTeX

Harri Hakula, Antti Rasila, Matti Vuorinen: Computation of exterior moduli of quadrilaterals. Electronic Transactions on Numerical Analysis (ETNA)(40), 2014, pp. 436–451. BibTeX

2013

Nuutti Hyvönen, Akambadath K. Nandakumaran, Hari Varma, Ram M. Vasu: Generalized eigenvalue decomposition of the field autocorrelation in correlation diffusion of photons in turbid media. Mathematical Methods in the Applied Sciences 36(11), 2013, pp. 1447–1458. BibTeX

Jeremi Darde, Nuutti Hyvönen, Aku Seppänen, Stratos Staboulis: {Simultaneous recovery of admittivity and body shape in electrical impedance tomography: An experimental evaluation. Inverse Problems 29(8), 2013, pp. 085004. BibTeX

Jeremi Darde, Antti Hannukainen, Nuutti Hyvönen: An Hdiv-based mixed quasi-reversibility method for solving elliptic Cauchy problems. SIAM Journal on Numerical Analysis 51(4), 2013, pp. 2123–2148. BibTeX

Roland Griesmaier, Nuutti Hyvönen, Otto Seiskari: A note on analyticity properties of far field patterns. Inverse Problems and Imaging 7(2), 2013, pp. 491–498. BibTeX

Jeremi Darde, Nuutti Hyvönen, Aku Seppänen, Stratos Staboulis: Simultaneous reconstruction of outer boundary shape and admittivity distribution in electrical impedance tomography. SIAM Journal on Imaging Sciences 6(1), 2013, pp. 176–198. BibTeX

2012

Nuutti Hyvönen, Petteri Piiroinen, Otto Seiskari: Point measurements for a Neumann-to-Dirichlet map and the Calderon problem in the plane. SIAM Journal on Mathematical Analysis 44(5), 2012, pp. 3526–3536. BibTeX

Harri Hakula, Tri Quach, Antti Rasila: Conjugate function method for numerical conformal mappings. Journal of Computational and Applied Mathematics(237), 2012, pp. 340–353. BibTeX

Jeremi Darde, Harri Hakula, Nuutti Hyvönen, Stratos Staboulis: Fine-tuning electrode information in electrical impedance tomography. Inverse Problems and Imaging 6, 2012, pp. 399–421. BibTeX

Nuutti Hyvönen, Otto Seiskari: Detection of multiple inclusions from sweep data of electrical impedance tomography. Inverse Problems 28, 2012, pp. 095014. BibTeX

Harri Hakula, Nuutti Hyvönen, Tomi Tuominen: On the hp-adaptive solution of complete electrode model forward problems of electrical impedance tomography. Journal of Computational and Applied Mathematics 235, 2012, pp. 4645–4659. BibTeX

Martin Hanke, Lauri Harhanen, Nuutti Hyvönen, Eva Schweickert: Convex source support in three dimensions. BIT Numerical Mathematics 52, 2012, pp. 45–63. BibTeX

2011

Antti Rasila, Linda Havola, Pekka Alestalo, Jarmo Malinen, Helle Majander: Matematiikan perusopetuksen kehittämistoimia ja tulosten arviointia. Tietojenkäsittelytiede(33), 2011, pp. 43–54. BibTeX

Helle Majander, Antti Rasila: Experiences of continous formative assessment in engineering mathematics. In Tutkimus suuntaamassa 2010-luvun matemaattisen aineiden opetusta, pp. 197–214. Tampereen yliopistopaino Oy- Juvenes Print, 2011. BibTeX

Linda Havola, Helle Majander, Harri Hakula, Pekka Alestalo, Antti Rasila: Aktivoiviin opetusmenetelmiin perustuvat matematiikan opetuskokeilut Aalto-yliopistossa. In Tuovi 9: Interaktiivinen tekniikka koulutuksessa 2010-konferenssin tutkijatapaamisen artikkelit, pp. 5–9. Tampereen yliopisto, 2011. BibTeX

R. Griesmaier, Nuutti Hyvönen: A regularized Newton method for locating thin tubular conductivity inhomogeneities. Inverse Problems 27(115008), 2011. BibTeX

Harri Hakula, Lauri Harhanen, Nuutti Hyvönen: Sweep data of electrical impedance tomography. Inverse Problems 27(115006), 2011. BibTeX

Hari Varma, Kuriyakkattil P. Mohanan, Nuutti Hyvönen, Akambadath K. Nandakumaran, Ram M. Vasu: Ultrasound-modulated optical tomography: recovery of amplitude of vibration in the insonified region from boundary measurement of light correlation. Journal of the Optical Society of America A 28, 2011, pp. 2322–2331. BibTeX

M. Hanke, B. Harrach, N. Hyvönen: Justification of point electrode models in electrical impedance tomography. Mathematical Models and Methods in Applied Sciences 21, 2011, pp. 1395–1413. BibTeX

Harri Hakula, Antti Rasila, Matti Vuorinen: On Moduli of Rings and Quadrilaterals: Algorithms and Experiments. SIAM Journal on Scientific Computing 33(1), 2011. BibTeX

M. Hanke, N. Hyvönen, S. Reusswig: Convex backscattering support in electric impedance tomography. Numerische Mathematik 117, 2011, pp. 373–396. BibTeX

2010

Rasila, Antti, Havola, Linda, Majander, Helle, Malinen, Jarmo: Automatic assessment in engineering mathematics: evaluation of the impact. In ReflekTori 2010, Tekniikan opetuksen symposium 9.-10.12.2010, pp. 37–45. Aalto yliopisto, Koulutuskeskus Dipoli, 2010. BibTeX

Linda Blåfield, Helle Majander, Antti Rasila, Pekka Alestalo: Verkkotehtäviin pohjautuva arviointi matematiikan opetuksessa. In Tuovi 8: Interaktiivinen tekniikka koulutuksessa 2010 -konferenssin tutkijatapaamisen artikkelit, pp. 98–103. Tampereen yliopisto, 2010. BibTeX

L. Harhanen, N. Hyvönen: Convex source support in half-plane. Inverse Problems and Imaging 4, 2010, pp. 429–448. BibTeX

N. Hyvönen, M. Kalke, M. Lassas, H. Setälä, S. Siltanen: Three-dimensional dental X-ray imaging by combination of panoramic and projection data. Inverse Problems and Imaging 4, 2010, pp. 257–271. BibTeX

Nuutti Hyvönen, Kimmo Karhunen, Aku Seppänen: Frechet derivative with respect to the shape of an internal electrode in electrical impedance tomography. SIAM Journal on Applied Mathematics 70, 2010, pp. 1878–1898. BibTeX

2009

M.Hanke, N. Hyvönen, S. Reusswig: An inverse backscatter problem for electric impedance tomography. SIAM Journal on Mathematical Analysis 41, 2009, pp. 1948–1966. BibTeX

N. Hyvönen: Approximating idealized boundary data of electric impedance tomography by electrode measurements. Mathematical Models and Methods in Applied Sciences 19, 2009, pp. 1185–1202. BibTeX

N. Hyvönen: Comparison of idealized and electrode Dirichlet-to-Neumann maps in electric impedance tomography with an application to boundary determination of conductivity. Inverse Problems 25, 2009, pp. 085008. BibTeX

H. Hakula, N. Hyvönen: On computation of test dipoles for factorization method. BIT 49, 2009, pp. 75–91. BibTeX

2008

Antti H. Niemi, Juhani Pitkäranta, Harri Hakula: Point load on a shell. In Numerical mathematics and advanced applications, pp. 819–826. Springer, 2008. BibTeX

M. Hanke, N. Hyvönen, M. Lehn, S. Reusswig: Source supports in electrostatics. BIT 48, 2008, pp. 245–264. BibTeX

B. Gebauer, N. Hyvönen: Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Probl. Imaging 2, 2008, pp. 355–372. BibTeX

A. Lechleiter, N. Hyvönen, H. Hakula: The factorization method applied to the complete electrode model of impedance tomography. SIAM J. Appl. Math. 68, 2008, pp. 1097–1121. BibTeX

M. Hanke, N. Hyvönen, S. Reusswig: Convex source support and its application to electric impedance tomography. SIAM Journal on Imaging Sciences 1, 2008, pp. 364–378. BibTeX

2007

N. Hyvönen, H. Hakula, S. Pursiainen: Numerical implementation of the factorization method within the complete electrode model of electrical impedance tomography. Inverse Probl. Imaging 1, 2007, pp. 299–317. BibTeX

N. Hyvönen: Application of the factorization method to the characterization of weak inclusions in electrical impedance tomography. Adv. in Appl. Math. 39, 2007, pp. 197–221. BibTeX

N. Hyvönen: Locating Transparent Regions in Optical Absorption and Scattering Tomography. SIAM J. Appl. Math. 67, 2007, pp. 1101–1123. BibTeX

B. Gebauer, N. Hyvönen: Factorization method and irregular inclusions in electrical impedance tomography. Inverse Problems 23, 2007, pp. 2159–2170. BibTeX

N. Hyvönen: Frechet derivative with respect to the shape of a strongly convex nonscattering region in optical tomography. Inverse Problems 23, 2007, pp. 2249–2270. BibTeX

Antti H. Niemi, Juhani Pitkäranta, Harri Hakula: Benchmark computations on point-loaded shallow shells: Fourier vs. FEM. Computer Methods in Applied Mechanics and Engineering 196, 2007, pp. 894–907. BibTeX

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