A Kaczmarz approach to Galactic Archaeology

We consider the problem of reconstructing a galaxy's stellar population distribution function from spectroscopy measurements. These quantities can be connected via the single-stellar population spectrum, resulting in a very large scale integral equation with a system structure. To solve this problem, we propose a projected Nesterov-Kaczmarz reconstruction (PNKR) method, which efficiently leverages the system structure and incorporates physical prior information such as smoothness and non-negativity constraints.

High-dimensional Bayesian factor analysis with application to separating the Cosmic Microwave Background

Implementation of many statistical methods for large, multivariate data sets requires one to solve a linear system that, depending on the method, is of the dimension of the number of observations or each individual data vector.  This is often the limiting factor in scaling the method with data size and complexity. In this paper we illustrate the use of Krylov subspace methods to address this issue in a statistical solution to a source separation problem in cosmology where the data size is prohibitively large for direct solution of the required system.  Two distinct approaches are described: one that uses the method of conjugate gradients directly to the Kronecker-structured problem and another that reformulates the system as a Sylvester matrix equation.  We show that both approaches produce an accurate solution within an acceptable computation time and with practical memory requirements for the data size that is currently available.

Joint work with Kirk M. Soodhalter and Dung Pham

Efficient Iterative Methods for Nonlinear Ill-Posed Problems

Iterative Methods are a popular choice for the inversion of nonlinear inverse problems, as they are both reliable, stable and frequently also fast in convergence. Besides Landweber iteration we introduce fast alternatives like Levenberg-Marquardt and Gauss-Newton as well as accelerated versions. Nesterov iteration is a popular choice in optimisation theory, and we present an analysis of the method as well as its generalisations in a next step. After some numerical examples demonstrating the efficiency of the method we introduce the nonlinear operator that describes the pyramid sensor, which is frequently used in optical devices, and iterative reconstruction methods for its data.

Ronny Ramlau & Simon Hubmer

An augmented Krylov subspace approach for solving the atmospheric tomography problem of Extremely Large Telescopes

Atmospheric tomography, i.e. the reconstruction of the turbulence profile in the atmosphere, is a challenging task for adaptive optics (AO) systems of the next generation of Extremely Large Telescopes (ELTs). Mathematically, the reconstruction of turbulent layers in the atmosphere is ill-posed, hence, limits the achievable solution accuracy. Moreover, the reconstruction has to be performed in real-time at a few hundred to thousand Hertz frame rates. Within the community of AO the first choice solver is the so called Matrix Vector Multiplication (MVM) method, which directly applies the (regularized) generalized inverse of the system operator to the data. For large systems such as the ELT of the European Southern Observatory (ESO) the atmospheric tomography problem is considerably complex and the computational efficiency becomes an issue. In this lecture we consider an iterative approach for atmospheric tomography, called augmented Finite Element Wavelet Hybrid Algorithm (FEWHA). The algorithm uses a dual domain discretization into a wavelet and finite element domain and is based on the well-known preconditioned conjugate gradient (PCG) method. A crucial indicator for the real-time performance of augmented FEWHA is the number of PCG iterations. Applying an augmented Krylov subspace method allows to decrease the number of iterations significantly. We exploit the fact that within AO we are dealing with several right-hand sides of the atmospheric tomography problem available consecutively in every time step. These right-hand sides correspond to wavefront sensor measurements and change only slightly over time. Hence, we are able to speed up the convergence of the current time step by reusing the Krylov subspace from the previous time step. We validate the quality of our algorithm by numerical simulations and demonstrate that a parallel implementation of augmented FEWHA fulfills the real-time requirements of ESO's ELT.

Unsteady multimodal water waves: A numerical approach

The talk will present some ongoing work on numerical simulation of a fluid domain with propagating multimodal and random water waves on the surface. The numerical approach presented will deal with the nonlinear free boundary problem in an irrotational setting. The nonlinear dynamic boundary condition also known as the unsteady Bernoulli's equation was be exploited for the numerical solution.

Hybrid Projection Methods and Their Applications

IR-Tools Short Course

IR Tools is a MATLAB package of Iterative Regularization methods and test problems for linear inverse problems, which was released in Summer 2018. You can learn more about IR Tools by visiting the page: http://people.compute.dtu.dk/pcha/IRtools/. IR Tools can be downloaded from github: https://github.com/jnagy1/IRtools.

In this short course I will give a brief introduction to some of the solvers and test problems available within IR Tools and I will illustrate their usage through a set of hands-on examples and test cases.

Theoretical and practical aspects of solving an inverse problem of biomechanical parameter imaging in optical coherence elastography

Elastography is an imaging modality which can map the biomechanical properties of a given sample, and is interested in identifying the spatial distribution and values of its biomechanical parameters. In this talk, we consider the following problem from linear elastography: Given two images of a linear isotropic material sample before and after compression, quantitatively reconstruct its spatially varying Lam\'e parameters. Mathematically, this corresponds to a large-scale nonlinear inverse problem of parameter estimation type. A common approach for solving such problems is Landweber iteration, which we also adopt here. In particular, we show that the tangential cone condition is satisfied for this problem, and thus in turn convergence of Landweber iteration is guaranteed. Finally, if time allows we also touch on how to obtain physically meaningful displacement field estimates as an important prerequisite of parameter identification.

This is joint work with Lisa Krainz, Simon Hubmer, Otmar Scherzer, Wolfgang Drexler.

Computationally efficient methods for Bayesian approaches to inverse problems

In the second talk, I will discuss two methods for efficient representation of prior information in large-scale Bayesian inverse problems. The first method is applicable to parameters that can be represented using piecewise constant fields and uses multiple level sets. The second method considers Gaussian random fields with covariance operators involving non-integer powers of elliptic operators. In both these methods, I will highlight the role of Krylov subspace-based iterative methods in enabling large-scale computations.

Data driven regularization by projection

We study the solution of linear inverse problems under the premise that the forward operator is not at hand but given indirectly through some input-output training pairs. We demonstrate that regularization by projection and variational regularization can be implemented without making use of the forward operator. Convergence and stability of the regularized solutions are studied in view of a famous non-convergence statement of Seidman. We show, analytically and numerically, that regularization by projection is indeed capable of learning linear operators, such as the Radon transform.

This is joint work with A. Aspri, L. Frischauf and Yury Korolev.

Collaboration Time

Frame Decompositions and Inverse Problems

The singular-value decomposition (SVD) is an important tool for the analysis and solution of linear ill-posed problems in Hilbert spaces. However, it is often difficult to derive the SVD of a given operator explicitly, which limits its practical usefulness. An alternative in these situations are frame decompositions (FDs), which are a generalization of the SVD based on suitably connected families of functions forming frames. Similar to the SVD, these FDs encode information on the structure and ill-posedness of the problem and can be used as the basis for the design and implementation of efficient numerical solution methods. Crucially though, FDs can be derived explicitly for a wide class of operators, in particular for those satisfying a certain stability condition.

In this talk, we consider various theoretical aspects of FDs such as recipes for their construction and some properties of the reconstruction formulae induced by them. Furthermore, we present convergence and convergence rates results for continuous regularization methods based on FDs under both a-priori and a-posteriori parameter choice rules. Finally, we consider the practical utility of FDs for solving inverse problems by considering two numerical examples from computerized and atmospheric tomography.

Bayesian Inversion for Uncertainty Quantification of High Dimensional Shape-based Inverse Problems

Augmentation schemes and recycling for well- and ill-posed problems

Augmented Krylov subspace methods

In many High-Performance Computing applications, it is often of interest to accelerate an iterative Krylov subspace solver, by augmenting the Krylov subspace with some additional subspace. We discuss a new framework [Soodhalter, de Sturler, Kilmer - GAMM Mitteilungen 2020] which describes all augmented Krylov subspace methods in terms of applying a standard Krylov subspace iteration to a projected problem, and taking an additional correction to the solution from the augmentation space. We then discuss new open questions arising from this framework.

Comparison of Haar Wavelet Sparsity and Anisotropic Total Variation in Image Reconstruction for Cone Beam Computed Tomography

Cone beam computed tomography (CBCT) is an increasingly popular imaging modality in which X-ray data of an object is collected using a cone-shaped beam, and a reconstruction algorithm is applied to create a 3D volumetric image. CBCT has a broad range of applications, especially in medical imaging of the head and neck region and of the extremities.

Analytical reconstruction algorithms, while fast and well understood, suffer from noise and cone beam artifacts, and perform poorly with undersampled data, and there is a growing demand for new, more flexible techniques. In this work, we have developed iterative reconstruction algorithms using two different regularization techniques: the Haar wavelet transform and anisotropic total variation. The reconstruction is computed using the primal-dual fixed point (PDFP) algorithm, an effective and flexible optimization technique which allows including additional useful properties such as nonnegative X-ray attenuation. Multiple test cases for the algorithms are provided using both simulated and measured X-ray data.

Learning a microlocal prior for limited-angle tomography

Digital breast tomosynthesis is an application of limited-angle tomography, which is a highly ill-posed inverse problem. Due to the limited-angle imaging geometry, reconstructions suffer from severe stretching of features along the central direction of projections, leading to poor slice separation. We propose a method for learning a boundary estimate for features. This estimate can be presented on top of the reconstruction, indicating the true form and extent of features. Learning the boundary estimate is based on directional edge detection that is implemented using complex wavelets and morphological operations. By using deep learning, we first extract the visible part of the wavefront set and then extend it to the full domain, filling in the parts of the wavefront set that would otherwise be hidden due to the lack of measurement directions. The resulting singular support gives the boundary estimate curve.

Material-separating regularizer formulti-energy X-ray tomography

Dual-energy X-ray tomography is considered in a context where the target under imaging consists of two distinct materials. The materials are assumed to be possibly intertwined in space, but at any given location there is only one material present. Further, two X-ray energies are chosen so that there is a clear difference in the spectral dependence of the attenuation coefficients of the two materials. A novel regularizer is presented for the inverse problem of reconstructing separate tomographic images for the two materials. A combination of two things, (a) non-negativity constraint, and (b) penalty term containing the inner product between the two material images, promotes the presence of at most one material in a given pixel. A preconditioned interior point method is derived for the minimization of the regularization functional. Numerical tests with digital phantoms suggest that the new algorithm outperforms the baseline method, Joint Total Variation regularization, in terms of correctly material-characterized pixels. While the method is tested only in a two-dimensional setting with two materials and two energies, the approach readily generalizes to three dimensions and more materials. The number of materials just needs to match the number of energies used in imaging.

Using persistent homology to improve tomographic reconstructions

The limited-angle tomography has great potential in both medical and industrial imaging. However, the reconstructions have stretching artifacts, and the singularities, i.e., the boundaries of the target object, cannot be detected stably. The goal of our research is to develop a method for detecting unknown singularities from known ones using persistent homology. Thus, this study brings together inverse problems with algebraic topology.

The complex wavelets provide a method for finding stable singularities and dividing them into six subsets based on their directions. The true singularity support forms cycles, but we know only parts of the cycles when using limited angle tomography. Using prior information about directions, it is possible to estimate unknown singularities based on known singularities. Persistent homology can identify cycles when known singularities and estimated singularities occur together. This will reduce our singular support estimation and result in more accurate reconstructions.

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The entire workshop will take place in the Hamilton Building, which is located on the far east side of Trinity's main campus. Although the Hamilton Buiding has an address on Westland Row, it cannot be entered by the public from this side. Rather, one should enter campus either at the Pearse Street entrance or the Lincoln Gate entrance. From either entrance, one can enter the Hamilton Building at one of two points (here or here).

All talks will take place in the Salmon lecture theater. Salmon is on the Ground Floor roughly halfway between the two ground-floor entrances mentioned in the previous paragraph.