# Inverse Problems

2005 â€” 2007

### Overview

Inverse Problems (IP) are problems where causes for a desired or observed effect are to be determined. An important example is to determine the density distribution inside a body from measuring the attenuation of X-rays sent through this body, the problem of "X-ray tomography". The mathematical problem was studied first by Radon in 1917. Much later, pioneering work by Hounsfield and Cormack led to the first working X-ray tomography machines and later to CAT scans and was honored with the Nobel Prize for Medicine in 1979. This development revolutionized the practice of medicine. Other more recent medical imaging techniques are MRI where the effect of a strong magnetic field on the body is measured, ultrasound where sound waves are sent through the body their reflections measures and Electrical Impedance Tomography where electrical measurements are made on the boundary of the body to name just a few. Earth sciences continue to be a generator of many compelling inverse problems. All of our knowledge of the Earth's interior is indirectly derived from surface measurements, as is a great deal of what we know about the surface and the atmosphere.

Reflection seismology in oil exploration is a well-known and economically important inverse problem. Here sound waves are generated at the surface of the Earth. By looking at the reflection of these waves one would like to determine the location and character of oil deposits. From an economic perspective, seismic imaging is by far the dominant geophysical inversion technique. Seismic imaging creates images of the Earth's upper crust using seismic waves generated by artificial sources and recorded into extensive arrays of sensors (geophones or hydrophones). The technology is based on a complex, and rapidly evolving, mathematical theory that employs advanced solutions to a wave equation as tools to solve approximately the general seismic inverse problem. In the year 2000, nearly \$4-billion was spent worldwide on seismic imaging. The heterogeneity and anisotropy of the Earth's upper crust require advanced mathematics to generate wave-equation solutions suitable for seismic imaging.

Other inverse problems arise in non-destructive evaluation of materials. The structural changes due to cracks or flaws are used to identify the locations of those defects. Radar and sonar are based on inverse scattering methods. Mathematics plays a crucial role in the understanding and modeling of the inverse problem as well as in finding reconstruction algorithms. Bring the last twenty years or so there have been remarkable developments in the mathematical theory of inverse problems. These developments together with the enormous increase in computing power and new powerful numerical methods have made it possible to make significant progress on increasingly more realistic and difficult inverse problems.

Many of the physical situations indicated above are modeled by partial differential equations. The inverse problem is to determine the coefficients of the partial differential equation inside the medium from some knowledge of the solutions, usually on the boundary. Already the interaction between experts in partial differential equations and on inverse problems has produced significant advances.

### Postdoctoral Fellows

UC:

• Hugh Geiger, 2005

UW:

• Kim Knudsen, August 2004-June 2005
• Mikko Salo, February-July, 2005
• Horst Heck, August 2005-February 2006
• Xiaosheng Li (PIMS PDF), September 2005-present.

### Faculty

UBC:

U. Calgary:

U. Washington:

PIMS Distinguished Chairs and Visitors
Visitors at UBC

Visitors at UC

Visitors at UW