Bridging adaptive discretizations and adaptive multilevel solvers

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Advanced Numerical Methods for Scientific Computing
Ludmil Zikatanov
The Pennsylvania State University, USA
Tuesday 29th November 2022
Aula Consiglio VII piano - Dipartimento di Matematica - Politecnico di Milano
Link to recording:
The talk is on the interplay (establishing a "two way street") between adaptive discretizations for partial differential equations (PDEs) and the design of algebraic multilevel solvers. In the first part of the talk we review operator preconditioning and some key results in adaptive finite element theory and practice. We then show how proconditioning in Hilbert spaces helps to construct efficient and robust residual error estimators for discretized PDEs. In the second part we show how tools from adaptive finite element methods can be applied in more general settings to design multilevel methods on graphs without referring to geometrical structures. We further present a couple of adaptive solvers which utilize multilevel hierarchies based on a Helmholtz decomposition of the underlying graph and localizing the basis for the cycle space. Contatto:
Ludmil Zikatanov holds a Full Professor position at The Pennsylvania State University, USA. He received his Ph.D. from Sofia University in 1995. His primary field of research is Numerical Analysis and encompasses the design and analysis of discretizations of partial differential equations. He is one of the experts in the theory and practice of fast multilevel solution methods for large sparse systems of linear equations and his work in this area has considerably increased the range of applicability of many other numerical methods. His recent works include other subjects at the frontier of computational mathematics, such as spectral graph theory, data compression on graphs, and their applications in machine learning. Zikatanov’s name is associated with the “XZ (Xu-Zikatanov) identity” in the theory of product iterative methods (Journal of the American Mathematical Society, 2002) and “Urschel-Zikatanov theorem” in spectral graph partitioning (Notices of the American Mathematical Society, 2016). In 2022 Zikatanov was selected as a Fulbright U.S. Scholar for academic year 2022-2023.