30 research outputs found
Estimating Nuisance Parameters in Inverse Problems
Many inverse problems include nuisance parameters which, while not of direct
interest, are required to recover primary parameters. Structure present in
these problems allows efficient optimization strategies - a well known example
is variable projection, where nonlinear least squares problems which are linear
in some parameters can be very efficiently optimized. In this paper, we extend
the idea of projecting out a subset over the variables to a broad class of
maximum likelihood (ML) and maximum a posteriori likelihood (MAP) problems with
nuisance parameters, such as variance or degrees of freedom. As a result, we
are able to incorporate nuisance parameter estimation into large-scale
constrained and unconstrained inverse problem formulations. We apply the
approach to a variety of problems, including estimation of unknown variance
parameters in the Gaussian model, degree of freedom (d.o.f.) parameter
estimation in the context of robust inverse problems, automatic calibration,
and optimal experimental design. Using numerical examples, we demonstrate
improvement in recovery of primary parameters for several large- scale inverse
problems. The proposed approach is compatible with a wide variety of algorithms
and formulations, and its implementation requires only minor modifications to
existing algorithms.Comment: 16 pages, 5 figure
Influence of a Rapid Change of Temperature on the Mechanical Response of Mammalian Myocardium
On the Use of Optimal Transport Distances for a PDE-Constrained Optimization Problem in Seismic Imaging
International audienceFull waveform inversion is a PDE-constrained nonlinear least-squares problem dedicated to the estimation of the mechanical subsurface properties with high resolution. Since its introduction in the early 80s, a limitation of this method is related to the non-convexity of the misfit function which is minimized when the method is applied to the estimation of the subsurface wave velocities. Recently, the definition of an alternative misfit function based on an optimal transport distance has been proposed to mitigate this difficulty. In this study, we review the difficulties for exploiting standard optimal transport techniques for the comparison of seismic data. The main difficulty is related to the oscillatory nature of the seismic data, which requires to extend optimal transport to the transport of signed measures. We review three standard possible extensions relying on a decomposition of the data into its positive and negative part. We show how the two first might not be adapted for full waveform inversion and focus on the third one. We present a numerical strategy based on the dual formulation of a particular optimal transport distance yielding an efficient implementation. The interest of this approach is illustrated on the 2D benchmark Marmousi model