Abstract
We review two important fields of application for the method of Variable Projections (VARPRO): Seismic Prospecting and Medical Imaging. We cover the period since 2002 when a first review was published. Variable Projections is a method for the solution of nonlinear least squares problems where the model is a linear combination of nonlinear functions. Its success is based on the fact that the reduced functional, where the linear parameters are eliminated converges always faster than if the same method of solution is applied to the original problem. More importantly, many of these problems are very hard to solve in their original format and VARPRO has shown its value in many different applications.
Highlights
We consider nonlinear data fitting problems that have as their underlying model a linear combination of nonlinear functions
We review two important fields of application for the method of Variable Projections (VARPRO): Seismic Prospecting and Medical Imaging
After obtaining expressions for the identification of relaxation times associated with kinetic fluorescence decay and those associated with the dynamic evolution of fluorophores, the author of [22] suggests the use of variable projection algorithms in the evaluation of photochemical bioimaging, when the fluorophores are used as the probe molecules
Summary
We consider nonlinear data fitting problems that have as their underlying model a linear combination of nonlinear functions. In [10] the authors tackle the well-known global convergence issue associated to any full waveform inversion (FWI) approach by solving an extended-image space least-squares migration problem to remove any local minima present in the FWI objective function. They discuss the connection between the reflectivity and migration velocity inversion and show the importance of combining the two problems using one objective function. This is a non-linear optimization problem, which is solved by a fixed-point method that utilizes a variable projection method with l1 constraints on the linear parameters and bound constraints on the non-linear parameters
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