Minisymposia Abstracts
Speaker:  Matti Lassas University of Helsinki 
Title:  Sparsity in inverse problems and the discretization of continuous inverse problems 
Abstract: 
We consider the question how inverse problems posed for continuous objects, for instance for continuous functions, can be discretized. This means the approximation of the problem by finite dimensional inverse problems. We will consider a linear inverse problem m = Au + ε. Here function m is the measurement, A is a illconditioned linear operator,
u is an unknown function, and ε is random noise. The inverse problem means the determination of u when m is given. The traditional solutions for the problem include the generalized Tikhonov regularization and the estimation of u using Bayesian methods.
To solve the problem in practice u and m are discretized, that is, approximated
by vectors in a finite dimensional vector space.
In this talk, we consider sparsity promoting regularization and Bayesian methods, such as the use of the ℓ^{p}priors, the total variation (TV) prior, and the Besov space priors. We show positive results when the finite dimensional approximation of a continuous problem can be done successfully and present some negative results when the finite dimensional problems fail to converge to the expected limit. 
Speaker:  Ignace Loris Universite Libre de Bruxelles 
Title:  On a generalization of the iterative softthresholding algorithm for the case of nonseparable penalty 
Abstract: 
An explicit algorithm for the minimization of an ℓ_{1}
penalized least squares functional
An example of such a penalty is the total variation penalty (A=grad) which promotes solutions with sparse gradients (i.e. piecewise constant solutions). Joint work with Caroline Verhoeven (ULB). 
Speaker:  Marta Betcke University College London 
Title:  Applications of Mesh Wavelets in Diffuse Optical Tomography 
Abstract: 
In diffuse optical tomography (DOT) the near infrared light is used to probe
the optical properties of the tissue such as absorption and scattering. The
value of those parameters can be related to oxygenation levels of the tissue
and hence provides a functional imaging modality with main applications being
neonatal brain imaging and breast imaging.
In DOT light scattering is assumed to be the dominant process and hence the forward problem is well described by the diffusion equation, which is commonly solved by Finite Element methods. The inverse problem amounts to inverting the sensitivity map. Since the sensitivity map itself depends on the optical coefficients, DOT is a nonlinear problem and hence it requires an iterative solution. The forward problem in DOT naturally lives on the finite element mesh. However, for the solution of the inverse problem the sensitivity map is usually mapped to a regular voxel grid, which are particularly appealing if hierarchical basis can be found, to benefit from some sort of sparsity in this basis. While simplifying the handling of the inverse problem, this step introduces additional errors due to the quite arbitrary choice of the mapping between the two grids. In this contribution we discuss a use of a wavelet basis defined directly on the finite element mesh for the solution of the inverse problem in DOT. Such basis allows for using the same sparsity techniques as available on the regular grid, without the need of remapping the solution between the mesh and the regular voxel grid. 
Speaker:  Martin Benning Westfaelische Wilhelms Universitaet Muenster 
Title:  An Adaptive Inverse Scale Space Method for Compressed Sensing 
Abstract:  In this talk a novel adaptive approach for solving L^{1}minimization problems as frequently
arising in compressed sensing is introduced, which is based on the recently introduced inverse scale space
method. The scheme allows to efficiently compute minimizers by solving a sequence of lowdimensional nonnegative
leastsquares problems. Moreover, extensive comparisons between the proposed method and the related orthogonal
matching pursuit algorithm are presented.

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