Felix Lucka (CI), Image Reconstruction - A Playground for Curious Applied Mathematicians
Abstract: Mathematical image reconstruction describes the process of computing images of quantities of interest from indirect observations through algorithms derived from rigorous mathematics. As the observation process can often be modeled by partial differential equations, image reconstruction problems are a classical example of inverse problems and draw from various fields of applied mathematics, including numerical analysis, Bayesian inference, variational regularization, compressed sensing, computational optimization, and machine learning. Mathematical image reconstruction became a key technique in a vast range of scientific, clinical and industrial applications. In this talk, I want to highlight some of its current trends and challenges and illustrate them by examples from my own work on biomedical imaging applications such as X-ray computed tomography (CT), photoacoustic tomography (PAT), magnetic resonance imaging (MRI) and ultrasound computed tomography (USCT).
Ronald de Wolf (A&C), Efficient algorithms for graph sparsification
Abstract: Graphs occur everywhere in discrete mathematics, but also in practical problems in logistics, the internet, social networks, etc. Sparse graphs (i.e., ones with few edges) are easier to handle than dense graphs: they take less space to store and are often cheaper to compute on. A long line of work by Karger, Spielman, Teng, and others resulted in nearly-linear-time algorithms that can sparsify any given n-vertex graph G to another n-vertex graph H whose number of edges is only O(n), while preserving many important properties of G. This then gives nearly-linear-time algorithms for solving various cut problems in graphs, for graph partitioning, and for solving Laplacian linear systems. We will describe these developments, and end with our recent work with Simon Apers showing that *quantum* algorithms can even compute such a good graph sparsification in sublinear time.