Note: Different Date and Location
Abstract: Typically in topology optimization problems, but also in other shape optimization problems, one is interested in finding functions of bounded variation with binary values, say 0 and 1. The fact that the set of function values is discrete usually complicates the numerical treatment significantly (non-convexity, unfavorable local minima,...).
In this talk, a new relaxation concept for the above mentioned problem class is introduced and analysed. A solution of the relaxed problem attains function values in [0,1]. Based on the main analytical tool, the co-area formula, it will be shown that the relaxation is exact in the sense that a solution of the relaxed problem is also a solution of the original problem. Further, stability of the exactness property under additive perturbations in the objective is discussed, and convergence properties for a Tikhonov-type regularization (as the regularization parameter tends to zero) are addressed.
One of the main benefits of our relaxation approach is its ability of a user-friendly implementation. Besides projected gradient flows in a level set context, we also highlight semi-smooth Newton methods for a fast numerical solution of the relaxed problems. Numerical tests will be discussed.
This seminar is easily accessible to persons with disabilities. For more information or for assistance, please contact the Mathematics Department at 743-3500.