Matrix functions via randomized sketching

• Applicant: Dr. Marcel Schweitzer
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 538686094

The field of numerical linear algebra (NLA) is currently experiencing a shift of paradigm with the advent of fast and scalable randomized algorithms for many core linear algebra problems, the most prominent one probably being the now widely adopted randomized singular value decomposition. A specific, important tool that has emerged in recent years in the context of randomized NLA is the so-called sketch-and-solve paradigm. While initially only used—very successfully—for computations with rather crude accuracy demands, e.g., (low-rank) matrix approximation or construction of randomized preconditioners, it was rather recently discovered that it might also be applicable for high accuracy NLA computations, e.g, the solution of linear systems, eigenvalue computations and the approximation of matrix functions in explorative studies. While these show promising results, many questions regarding numerical stability, attainable accuracy and theoretical groundwork are still wide open. In this research project, we aim to make progress in answering these questions in order to obtain a reliable randomized Krylov subspace method for computing the action of large-scale matrix functions on vectors (as they appear, e.g., in the exponential time integration of differential equations). In addition to a obtaining a deeper theoretical understanding of randomized Krylov subspace methods, we want to provide a robust, scalable implementation with easy to use interfaces which practitioners can easily adopt for simulations from a wide range of application areas. While we focus specifically on matrix function computations, many developments can be expected to carry over in a similar fashion also to related problems (e.g., solution of linear systems or matrix equations).

• DFG Programme: Research Grants