Promoting sparsity and localization in geophysical
inverse problems
Abstract:
Sometimes we don't have enough information, sometimes we have too much
and need to choose; sometimes our data present a highly incomplete
picture of the truth, and sometimes there is so much redundancy and
overdeterminacy that we need to cull it down somehow. In this
tutorial I will discuss several ways by which novel mathematical
tools have shed much light on problems of this nature. I will talk
about the problem of reconstructing global sea level from sparse,
uncertain, scattered indicators of local sea level (the solution
derived from Monte-Carlo techniques in a Bayesian framework). I will
introduce the design of flexible parameterizations to render inverse
problems in geodesy and geomagnetics sparse (a solution given by
linear combinations of spherical harmonics called Slepian
functions). Lastly, I will discuss the nascence of sparsity-promoting
algorithms in global seismology (under a new design that ports fast
Cartesian wavelet transforms onto the sphere and combines
least-squares data fitting with the minimization of l-1 norms). For
each of these subtopics I will briefly highlight the key mathematical
innovations and discuss the often widespread implications of the
results.
