2020-07-29 –, Green Track
In this talk, I present LinearMaps.jl, a well-established Julia package for handling linear maps whose action on vectors is given by the classic matrix-vector product or by the application of a function to a vector. I will focus on two recently added features, namely (diagonal) block concatenation and (higher-order) Kronecker products and sums of such abstract linear maps.
Linear maps (or operators) are ubiquitous in the mathematical sciences, modeling and scientific computing. In many problems, linear maps are not necessarily given by some matrix representation, but as programs that transform vectors in a linear fashion. Such linear programs can be used, for instance, in iterative linear algebra methods synonymously to usual matrices by wrapping them with the LinearMaps.jl package. With this package, more complicated linear maps can be constructed lazily from simple linear function maps or wrapped matrices via scaling, addition, multiplication, transposition and taking the adjoint. Recently, two further classes of such operator algebraic operations have been added to the aforementioned traditional set of operations in LinearMaps.jl: horizontal, vertical and diagonal (block) concatenation and Kronecker products, sums, and powers. These operations feature prominently in applications like structured optimization and image reconstruction. Their implementation in LinearMaps.jl facilitates top performance even in the classic matrix context due to their laziness and type-stable usage of specialized multiplication methods.
Daniel Karrasch graduated in 2012 with a PhD in Mathematics from TU Dresden, Germany. After a postdoctoral fellowship at ETH Zurich, Switzerland, he now works as a postdoctoral researcher in the Scientific Computing group at TU Munich, Germany. His research interests are in applied and computational mathematics, with a focus on dynamical systems theoretical approaches to the study of fluid motion. He is an enthusiastic contributor to the Julia language, mostly in the area of linear algebra.