Mid-Term Report Summary 2 - INVPROB (Inverse Problems)
The project Inverse Problems (InvProb) started in 1.3.2011 and the PI of this project is prof. Lassi Päivärinta. The project operates in the borderline of pure and applied mathematics and the main theme is the mathematical framework of inverse problems.
A common goal in all inverse problems is to recover information from indirect, incomplete, or noisy observations by using mathematical models. These problems arise in practical situations such as medical imaging, exploration geophysics, and non-destructive testing of mechanical parts where measurements made in the exterior of a body are used to deduce properties of the inaccessible interior.
One of the paradigmatic inverse problems is the inverse conductivity problem posed by A.P. Calderón. Here one wishes to determine the electrical conductivity of a body by making current and voltage measurements at the boundary. This inverse problem has attracted a great deal of interest in recent years, and both its theoretical and applied aspects have been under intense study. A positive answer to Calderón's problem means that an object inside is visible from the boundary. Recently, one aspect of the inverse conductivity problem has attracted a great deal of attention in the context of invisibility and cloaking. The invisibility cloak is a structure that would not only make an object invisible but also undetectable to electromagnetic waves, thus making it cloaked. This means that invisibility cloak is a specific counterexample to the Calderón's problem, since a cloacked object cannot be detected at all from the boundary.
We will mention two highlight achievements of the project so far. The project has produced many other interesting results in addition to the following two new scientific breakthroughs.
A one of the achievements of this project, PI with collaborators K. Astala and M. Lassas have exactly analysed the regions of visibility (when the Calderón's problem has postive answer) and the region of cloacking counterexamples. An unexpected result obtained in this study also shows that there are other kinds of counterexamples to visibility than the cloacking counterexamples. These counterexamples are called electric holograms, since they create an illusion of a non-existing body.
Another recent development concerns scattering of waves. In quantum mechanics and in acoustics one interesting question is whether one can probe materials with waves producing no response at all. In such a case the scattered wave vanishes. It turns out that this requires these wavelengths to correspond to so called transmission eigenvalues. The study of transmission eigenvalues goes back to 80's and just recently in 2008 the PI together with J. Sylvester showed the existence of transmission eigenvalues in general. This in turn implied that there might also be non-scattering waves in general. However, one of the very recent achievements in the project is that PI together with a team member E. Blåsten and the collaborator J. Sylvester showed that if a penetrable scatterer has a straight angle corner, it will always scatter, even though it has infinitely many transmission eigenvalues.
A common goal in all inverse problems is to recover information from indirect, incomplete, or noisy observations by using mathematical models. These problems arise in practical situations such as medical imaging, exploration geophysics, and non-destructive testing of mechanical parts where measurements made in the exterior of a body are used to deduce properties of the inaccessible interior.
One of the paradigmatic inverse problems is the inverse conductivity problem posed by A.P. Calderón. Here one wishes to determine the electrical conductivity of a body by making current and voltage measurements at the boundary. This inverse problem has attracted a great deal of interest in recent years, and both its theoretical and applied aspects have been under intense study. A positive answer to Calderón's problem means that an object inside is visible from the boundary. Recently, one aspect of the inverse conductivity problem has attracted a great deal of attention in the context of invisibility and cloaking. The invisibility cloak is a structure that would not only make an object invisible but also undetectable to electromagnetic waves, thus making it cloaked. This means that invisibility cloak is a specific counterexample to the Calderón's problem, since a cloacked object cannot be detected at all from the boundary.
We will mention two highlight achievements of the project so far. The project has produced many other interesting results in addition to the following two new scientific breakthroughs.
A one of the achievements of this project, PI with collaborators K. Astala and M. Lassas have exactly analysed the regions of visibility (when the Calderón's problem has postive answer) and the region of cloacking counterexamples. An unexpected result obtained in this study also shows that there are other kinds of counterexamples to visibility than the cloacking counterexamples. These counterexamples are called electric holograms, since they create an illusion of a non-existing body.
Another recent development concerns scattering of waves. In quantum mechanics and in acoustics one interesting question is whether one can probe materials with waves producing no response at all. In such a case the scattered wave vanishes. It turns out that this requires these wavelengths to correspond to so called transmission eigenvalues. The study of transmission eigenvalues goes back to 80's and just recently in 2008 the PI together with J. Sylvester showed the existence of transmission eigenvalues in general. This in turn implied that there might also be non-scattering waves in general. However, one of the very recent achievements in the project is that PI together with a team member E. Blåsten and the collaborator J. Sylvester showed that if a penetrable scatterer has a straight angle corner, it will always scatter, even though it has infinitely many transmission eigenvalues.