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 ill-conditioned 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 ℓp-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 soft-thresholding algorithm for the case of non-separable penalty
Abstract: An explicit algorithm for the minimization of an ℓ1 penalized least squares functional
F (x)=½||Kx − y||2 + λ||Ax||1
with non-separable ℓ1 term is proposed. Each step in the iterative algorithm requires four matrix vector multiplications and a single simple projection on a convex set (or equivalently thresholding). No smoothing parameter is introduced. Convergence is proven in a finite dimensional setting for ||K||≤ √2 and ||A|| ≤ 1. A 1/N convergence rate is derived for the functional. In the special case where the matrix A in the ℓ1 term is the identity (or orthogonal), the algorithm reduces to the traditional iterative soft-thresholding algorithm. In the special case where the matrix K in the quadratic term is the identity (or orthogonal), the algorithm reduces to a gradient projection algorithm for the dual problem.

An example of such a penalty is the total variation penalty (A=grad) which promotes solutions with sparse gradients (i.e. piece-wise 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 L1-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 low-dimensional nonnegative least-squares problems. Moreover, extensive comparisons between the proposed method and the related orthogonal matching pursuit algorithm are presented.

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