Inverse problems

Inverse problems are the problems of finding unknown parameters or structures by indirect measurements. A typical inverse problem is the inverse conductivity problem. Its practical setting is often of the following kind: Assume that you want to find the inner structure of your torso by doing resistivity measurements at your skin.

In mathematical terms, the inverse problems usually mean finding the unknown parameter functions of a partial differential or integral equation from the knowledge of the boundary values of the solutions. In past the research of the group has been related to the following applications:

Acoustic and electromagnetic sounding and impedance tomography. In non-destructive testing, geophysical studies and medical applications, it is often desired to find out the unknown structure of an object using measurements performed outside the object, that is, non-invasively. This can be done e.g. by sounding the object with electromagnetic fields or acoustic waves. The object under study can be e.g. the human thorax, the head of a new-born, the fuselage of an airplane or the earth crust.
X-ray tomography and radar imaging. In these problems one has to reconstruct unknown structures by knowing integrals of physical functions over rays. For instance, in X-ray tomography the intensity of X-rays that have traveled through the measured object determine the total mass on given the lines of passage. This principle is applied in medical and industrial measurement devices e.g. in CT (Computerized Tomography). In recent years the measurement technology of digital X-ray sensors has greatly improved, which has caused a revolution in the medical imaging industry. This has produced a need also in Finnish industry to develop algorithms suitable for new measurement methods. Research on this area has been done on theoretical and computational questions, as well as on the level of practical applications.
Absorbing boundary conditions. The acoustic or electromagnetic scattering problem is to determine the reflection and interaction of a field hitting a given object, for instance solving the field caused by a mobile telephone in the user's body and its neighborhood. In the numerical implementation, a difficulty is caused by the unboundedness of the domain in which the solution is computed: when the calculations are implemented, the domain must be truncated to a bounded domain. A boundary condition on the boundary of the domain needs to be sought such that the boundary does not create artificial echoes. Such boundary conditions are called absorbing boundary conditions and they are related to inverse problems through optimization methods.

In general terms, inverse problems represent a field of pure and applied mathematics using various areas of mathematics. Typical methods used in the field vary from real, complex and functional analysis, theory of partial differential equations, microlocal analysis, Riemannian geometry to stochastics. The research on inverse problems have made it possible to use these tools in many applied sciences.

Personnel

Inverse problems are studied by Matti Lassas and Seppo Järvenpää and the graduate students Kenrick Bingham, Pekka Tietäväinen (supervised by Matti Lassas), and Simopekka Vänskä (supervised by Petri Ola).

Links

Selected publications and monographs

M. Lassas: Inverse boundary spectral problem for non-selfadjoint Maxwell's equations with incomplete data. Comm. Partial Differential Equations 23 (1998), 629--648.

Y. Kurylev, M. Lassas: Gelf'and Inverse Problem for a Quadratic Operator Pensil. J. Funct. Anal. , 176(2000), 247-263.

M. Cheney, D. Isaacson and M. Lassas: Optimal acoustic measurements, SIAM J. Appl. Math. 61 (2001), no. 5, 1628--1647.

A. Katchalov, Y. Kurylev, M. Lassas: Inverse Boundary Spectral Problems , Monographs and Surveys in Pure and Applied Mathematics 123, Chapman Hall/CRC-press, 2001, xi+290 pp.

M. Lassas: Inverse boundary spectral problem for non-selfadjoint Maxwell's equations with incomplete data. Comm. Partial Differential Equations 23 (1998), 629--648.

M. Lassas, J. Liukkonen, E. Somersalo: Complex Riemannian metric and absorbing boundary condition, J. Math. Pures Appl. (9) 80 (2001), no. 7, 739--768.

M. Lassas and E. Somersalo: Analysis of the PML equations in general convex geometry. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 5, 1183--1207

M. Lassas and G. Uhlmann: Determining Riemannian manifold from boundary measurements. Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 5, 771--787

Y. Kurylev and M. Lassas: Dynamical inverse problem for a hyperbolic equation and continuation of data, to appear in Proc. Roy. Soc. Edinburgh Sect. A

M. Lassas, M. Taylor, and G. Uhlmann: The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, to appear in Comm. Anal. Geom.

Theses

S. Järvenpää: Implementation of PML absorbing boundary condition for solving Maxwell's equations with Whitney elements. PhD. Dissertation, Research Reports A35, Rolf Nevanlinna Institute, Helsinki, 2001.

S. Vänskä: Determining real analytic metric by boundary measurements. Licentiate thesis.

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