# 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

# 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

2015 | 2014 | 2013 | 2012 | 2011 | 2010 | 2009 | 2008 | 2007

Nuutti Hyvönen, Matti Leinonen: Stochastic Galerkin finite element method with local conductivity basis for electrical impedance tomography. *SIAM/ASA Journal on Uncertainty Quantification* 3(1), 2015, pp. 998–1019. BibTeX

Lauri Harhanen, Nuutti Hyvönen, Helle Majander, Stratos Staboulis: Edge-enhancing reconstruction algorithm for three-dimensional electrical impedance tomography. *SIAM Journal on Scientific Computing* 37(1), 2015, pp. B60-B78. BibTeX

Lucas Chesnel, Nuutti Hyvönen, Stratos Staboulis: Construction of invisible conductivity perturbations for the point electrode model in electrical impedance tomography. *SIAM Journal on Applied Mathematics* 75(5), 2015, pp. 2093–2109. BibTeX

Harri Hakula, Vesa Kaarnioja, Mikael Laaksonen: Approximate methods for stochastic eigenvalue problems. *Applied Mathematics and Computation* 267, 2015, pp. 664–681. BibTeX

Harri Hakula, Mikael Laaksonen: Hybrid Stochastic Finite Element Method for Mechanical Vibration Problems. *Shock and VIbration* 2015, 2015. BibTeX

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

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

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

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

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

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

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

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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