Session: Nonsmooth PDE Constrained Optimization: Algorithms, Analysis and Applications Part 3
Chair: Harbir Antil
Cluster: PDE-constrained Optimization
Talk 1: An Adaptive Inexact Trust-Region Method for PDE-Constrained Optimization with Regularized Objectives
Speaker: Robert Baraldi
Abstract: We introduce an inexact trust-region method for efficiently solving regularized optimization problems governed by PDEs. In particular, we consider the class of problems in which the objective is the sum of a smooth, nonconvex function and nonsmooth, convex function. Such objectives are pervasive in the literature, with examples being basis pursuit, inverse problems, and topology optimization. The inclusion of nonsmooth regularizers and constraints is critical, as they often perserve physical properties or promote sparsity in the control. Enforcing these properties in an efficient manner is critical when met with computationally intense nature of solving PDEs. A common family of methods that can obtain accurate solutions with considerably smaller mesh sizes are adaptive finite element routines. They are critical in reducing error in solutions as well as mitigating numerical cost of solving the PDE. Our adaptive trust-region method solves the regularized objective while automatically refining the mesh for the PDE. Our method increases accuracy of the gradient and objective via local error estimators and our criticality measure. We present our numerical results on problems in control.
Talk 2: The SiMPL method for density-based topology optimization
Speaker: Dohyun Kim
Abstract: We introduce Sigmoidal mirror descent with a projected latent variable (SiMPL), a novel first-order optimization method for density-based topology optimization. SiMPL ensures point-wise bound preserving design updates and faster convergence than other popular first-order topology optimization methods. By leveraging the (negative) Fermi-Dirac entropy, we define a non-symmetric Bregman divergence that facilitates a simple yet effective update rule with the help of so-called latent variable. SiMPL produces a sequence of pointwise-feasible iterates even when high-order finite elements are used in the discretization. Numerical experiments demonstrates that the method outperforms other popular first-order optimization algorithms. We also present mesh- and order-independent convergence along with possible extension of this method.
Talk 3: Two-level Discretization Scheme for Total Variation in Integer Optimal Control
Speaker: Paul Manns
Abstract: We advance the discretization of the dual formulation of the total variation term with Raviart-Thomas functions which is known from literature for convex problems. Due to our integrality constraints, the previous analysis is not applicable anymore because, when considering a Γ-convergence-type argument, the recovery sequences generally need to attain non-integer, that is, infeasible, values. We overcome this problem by embedding a finer discretization of the input functions. A superlinear coupling of the mesh sizes implies an averaging on the coarser Raviart-Thomas mesh, which enables to recover the total variation of integer-valued limit functions with integer-valued, discretized input functions. In turn, we obtain a Γ-convergence-type result and convergence rates under additional regularity assumptions.
