Adaptive Approximations for PDE-Constrained Parabolic Control Problems

Euler Institute

Date: 13 February 2024 / 11:00 - 12:00

USI Campus Est, room D1.14, sector D // Online on Microsoft Teams

Angela Kunoth, University of Cologne, Germany

Numerical solvers for PDEs have matured over the past decades in efficiency, largely due to the development of sophisticated algorithms based on closely intertwining theory with numerical analysis.
Consequently, systems of PDEs as they arise from optimization problems with PDE constraints also have become more and more tractable.
Optimization problems constrained by a parabolic evolution PDE are challenging from a computational point of view: They require to solve a system of PDEs coupled globally in time and space.  For their solution, time-stepping methods quickly reach their limitations due to the enormous demand for storage.  An alternative approach is a full space-time weak formulation of the parabolic PDE which allows one to treat the constraining PDE as an operator equation without distinction of the time and space variables. An optimization problem constrained by a parabolic PDE in full space-time weak form leads to a coupled system of corresponding operator equations which is, of course, still coupled globally in space and time.
For the numerical solution of such coupled PDE systems, adaptive methods appear to be most promising, as they aim at distributing the available degrees of freedom in an a-posteriori-fashion to capture singularities in the data or domain. Employing wavelet schemes, we can prove convergence and optimal complexity.
The theoretical basis for proving convergence and optimality of wavelet-based algorithms for such type of coupled PDEs is nonlinear approximation theory and the characterization of solutions of PDEs in Besov spaces.
Wavelet schemes are, however, more involved when it comes to implementations compared to finite element approximations. I will finally address corresponding ideas and results.

Angela Kunoth studied mathematics at Bielefeld University beginning in 1982, and earned a diploma there in 1990. After visiting the University of South Carolina as a Fulbright Scholar, she completed a doctorate (Dr. rer. nat.) at the Free University of Berlin in 1994. Her dissertation, Multilevel Preconditioning, was supervised by Wolfgang Dahmen. After research positions at SINTEF in Norway, at the Weierstrass Institute in Berlin, at Texas A&M University, and at RWTH Aachen University, she became an associate professor at the University of Bonn in 1999, and earned a habilitation through RWTH Aachen in 2000 with the habilitation thesis Wavelet Methods for Minimization Problems Involving Elliptic Partial Differential Equations. She moved to Paderborn University as a full professor and chair of complex systems in 2007, and at Paderborn served as director of the mathematical institute and vice-dean of the faculty for electrotechnics from 2010 to 2012.  She moved again to the University of Cologne as professor and chair for applied mathematics in 2013.
Angela Kunoth was elected to the 2023 Class of SIAM Fellows.

Host: Prof. Michael Multerer

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