## Seminar in Numerical Analysis: Naomi Schneider (Universität Siegen)

**Determination of a learnt best basis for two geoscientific inverse problems**

Both the approximation of the gravitational potential via the downward continuation of satellite data and of wave velocities via the travel time tomography using earthquake data are geoscientific ill- posed inverse problems. To monitor certain aspects of the system Earth, like the mass transport or its geomagnetic field, it is, however, important to tackle these challenges.

Traditionally, an approximation of such a linear(ized) inverse problem is obtained in one, a-priori chosen basis system: either a global one, e.g. spherical harmonics or polynomials on the ball, or a local one, e.g. radial basis functions and wavelets on the sphere or finite elements on the ball.

In the Geomathematics Group Siegen, we developed methods that enable us to combine different types of trial functions for such an approximation. The idea is to make the most of the benefits of different types of available trial functions. The algorithms are called the (Learning) Inverse Problem Matching Pursuits (LIPMPs). They construct an approximation iteratively from an intentionally overcomplete set of trial functions, the dictionary, such that the Tikhonov functional is reduced. Due to the learning add-on, the dictionary can very well be infinite. Moreover, the computational costs are usually decreased.

In this talk, we give details on the LIPMPs and show some current numerical results.

For further information about the seminar, please visit this webpage.

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