# RFMP

## RFMP - Regularized Functional Matching Pursuit

Today, many different systems of trial functions are available for handling geoscientific problems or - mathematically formulated - for approximating functions on the sphere or the ball based on direct or indirect measurements. Spherical harmonics are a classical system of global trial functions with well-known possibilities of a physical interpretation but also with documented drawbracks e.g. in the case of heterogeneities in the data (with respect to the data quality or the data grid). On the other hand, numerous alternatives of localized trial functions have been constructed on the sphere and the ball within the last three decades like Gaussian functions, Haar functions, Slepian functions, splines, and (multiple versions of) wavelets. They all have their intrinsic advantages and disadvantages.

The Regularized Functional Matching Pursuit (RFMP) is a novel algorithm that is able to combine different basis systems for solving an ill-posed inverse problem (e.g. in the geosciences). It is based on the Matching Pursuit (MP) by Mallat and Zhang (1993) and Vincent and Bengio (2002). The RFMP extends this algorithm in several aspects: (1) inverse problems can be handled, i.e. one can incorporate an equation which has to be solved, and the data and the solution need not be elements of the same space; (2) a regularization is included to treat ill-posed problems; (3) implementations for problems on non-Euclidean domains like a sphere or a ball have been realized.

The algorithm produces a sequence of approximations which converges to the regularized normal equation. Numerical experiments show that the RFMP produces a sparse representation of the solution, i.e. it needs essentially less trial functions for achieving the same accuracy of the result than considered alternative approaches. This sparsity is achieved because the algorithm is able to combine different kinds of trial functions and because it chooses only those trial functions (out of a large overcomplete set called a dictionary) which are the best choice to reduce the (regularized) data misfit fast.

The Regularized Orthogonal Functional Matching Pursuit (ROFMP) is an enhancement of the RFMP in the sense that it yields a representation of the solution in essentially less trial functions in comparison to the RFMP. This is achieved by a particular projection procedure which compensates for redundancies in the trial functions due to their non-orthogonality. Numerical tests for a scenario of the (severely ill-posed) downward continuation of noisy gravity data from a satellite orbit to the Earth's surface demonstrated the applicability of the ROFMP and its particular features and advantages.

References:

- D. Fischer, V. Michel:
*Sparse regularization of inverse gravimetry – case study: spatial and temporal mass variations in South America*, Inverse Problems, 28 (2012), 065012 (34pp), click here for download. - D. Fischer, V. Michel:
*Automatic best-basis selection for geophysical tomographic inverse problems*, Geophysical Journal International, 193 (2013), 1291-1299. - D. Fischer, V. Michel:
*Inverting GRACE gravity data for local climate effects*, Journal of Geodetic Science, 3 (2013), 151-162. - V. Michel:
*RFMP - An Iterative Best Basis Algorithm for Inverse Problems in the Geosciences*, in: Handbook of Geomathematics (W. Freeden, M.Z. Nashed, and T. Sonar, eds.), 2nd edition, Springer, Berlin, Heidelberg, 2015, pp. 2121-2147. - V. Michel, R. Telschow:
*The regularized orthogonal functional matching pursuit for ill-posed inverse problems*, SIAM Journal on Numerical Analysis, 54 (2016), 262-287.