Andreas Langer

Due to the constant improvement of hardware, the dimension of measured data generally increases. Typical examples in image processing where such large scale data appears are 4D imaging (spatial plus temporal dimensions) from functional magnetic-resonance in nuclear medical imaging, astronomical imaging and global terrestrial seismic tomography. In these applications the large amounts of data usually needs to be processed further in order to obtain the necessary or usable information. These large problems to be dealt with need to be solved in a reasonable time. This requires effective numerical methods and massively parallel solvers. One of the most successful methods to design efficient numerical solvers for large problems are subspace correction methods. Subspace correction methods are a divide-and-conquer technique originally introduced to numerically solve partial differential equations. These allow to split the computational workload and solve a sequence of smaller problems with the possibility of parallelization. We develop convergent subspace correction methods for non-smooth and non-additive optimization problems, in particular for total variation minimization, and analyze their properties and behaviour.

Related publications:

S. Hilb and A. Langer, A General Decomposition Method for a Convex Problem Related to Total Variation Minimization, submitted, 2022, 34 pp. [pdf][software]

A. Langer, Domain Decomposition for Non-smooth (in Particular TV) Minimization. In: Chen K., Schönlieb CB., Tai XC., Younces L. (eds) Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer, Cham, 2021. https://doi.org/10.1007/978-3-030-03009-4_104-1 [pdf]

A. Langer and F. Gaspoz, Overlapping domain decomposition methods for total variation denoising, SIAM Journal on Numerical Analysis, Vol. 57, No. 3, 2019, 1411-1444 [pdf] [link]

M. Hintermüller and A. Langer, Non-overlapping domain decomposition methods for dual total variation based image denoising , J. Scientific Computing, Vol 62, No. 2, 2015 pp. 456-481. [pdf]

M. Hintermüller and A. Langer, Surrogate functional based subspace correction methods for image processing, In Domain Decomposition Methods in Science and Engineering XXI, pages 829–837. Springer, 2014.[pdf]

M. Hintermüller and A. Langer, Subspace correction methods for a class of non-smooth and non-additive convex variational problems with mixed L¹/L² data-fidelity in image processing, SIAM J. Imaging Sciences, Vol 6, No 4, 2013, pp. 2134-2173 [pdf]

A. Langer, S. Osher, and C.-B. Schönlieb, Bregmanized domain decomposition for image restoration, J. Scientific Computing, Vol 54, 2013, pp. 549-576 [pdf]

M. Fornasier, Y. Kim, A. Langer, and C.-B. Schönlieb, Wavelet decomposition method for L2/TV-image deblurring, SIAM J. Imaging Sciences, Vol 5, No 3, 2012, pp. 857-885 [pdf]

M. Fornasier, A. Langer, and C.-B. Schönlieb, A convergent overlapping domain decomposition method for total variation minimization, Numer. Math., Vol 116, No 4, 2010, pp. 645-685 [pdf] [software]

M. Fornasier, A. Langer, and C.-B. Schönlieb, Domain decomposition methods for compressed sensing , Proc. Int. Conf. SampTA09, Marseilles, 2009. [pdf] or arXiv:0902.0124v1 [math.NA]

Discretization methods for non-smooth optimization problems, such as total variation (TV) minimization, are essential for accurately recovering sharp features and discontinuities in imaging and inverse problems. Due to the inherent non-differentiability of TV regularization, careful numerical treatment is required to ensure stability and convergence. In this context, we develop and study a range of discretization strategies, including finite element methods (FEM), adaptive FEM, adaptive finite difference methods (FDM), and neural network-based discretizations. These approaches aim to combine accuracy with computational efficiency across diverse applications.

In several application, for example in image processing, one solves a minimization problem of the type

where H represents a data fidelity term, which enforces the consistency between the recovered and measured data, R is an appropriate regularization term, which prevents over-fitting, and α>0 is a regularization parameter weighting the importance of the two terms. The solution of this problem clearly depends on the choice of the parameter. In particular, in image reconstruction, large α, which lead to an over-smoothed reconstruction, not only remove noise but also eliminate details in images. On the other hand, small α lead to solutions, which fit the given data properly but therefore retain noise in homogeneous regions. Hence a good reconstruction can be obtained by choosing α such that a good compromise of the aforementioned effects are made.

A scalar regularization parameter might not be the best choice for every application. For example, images usually have large homogeneous regions as well as parts with a lot of details. This motivates that α should be small in parts with small features, in order to preserve the details, and should be large in homogeneous parts to remove noise considerable. With such a choice of a spatially varying weight we expect better reconstructions than with a globally constant parameter.