16:00
Multigrid Preconditioning for Space-Time Distributed Optimal Control of Parabolic Equations
Dr. Andrei Draganescu | University of Maryland Baltimore County | United States
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Dr. Andrei Draganescu | University of Maryland Baltimore County | United States
This work is concerned with designing optimal order multigrid preconditioners for space-time distributed control of parabolic equations. The focus is on the reduced problem resulted from eliminating state and adjoint variables from the KKT system. Earlier numerical experiments have shown that our ability to design optimal order preconditioners depends strongly on the discretization of the parabolic equation, with several natural discretizations leading to suboptimal preconditioners. Using a continuous-in-space-discontinuous-in-time Galerkin discretization we obtain the desired optimality.
16:20
Higher-order discretization of diffeomorphisms for PDE-constrained shape optimization
Dr. Alberto Paganini | University of Oxford | United Kingdom
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Authors:
Dr. Alberto Paganini | University of Oxford | United Kingdom
M. Sc. Florian Wechsung | Germany
Prof. Dr. Patrick E. Farrell | Germany
PDE-constrained shape optimization problems are characterized by target functionals that depend both on the shape of a domain (the control) and on the solution of a boundary value problem formulated on that domain (the state). In industrial applications, such optimization problems are formulated to improve the performance (expressed in terms of the target functional) of an initial domain.
PDE-constrained shape optimization problems can be solved numerically by meshing the initial domain and updating iteratively the coordinates of the nodes of this mesh. Steepest descent updates can be computed with approximate shape derivatives based on finite element approximations of the state.
In this work, we present a mathematical framework that generalizes standard moving mesh methods to higher-order discretizations of both computational domains and domain updates. We parametrize shapes by applying (discretized) deformation diffeomorphisms to the initial guess, and incorporate domain deformations into the PDE solver via isoparametric finite elements.
This approach allows for arbitrarily smooth representations of shapes, reduces approximation errors, and is compatible with standard finite element software.
16:40
Modified Newton methods for the fully monolithic solution of phase-field fracture propagation
Prof. Dr. Thomas Wick | École Polytechnique | France
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Prof. Dr. Thomas Wick | École Polytechnique | France
We continue our presentation from last year's OCIP conference and
provide new findings in our aim to design a fully monolithic solution
algorithm for phase-field fracture propagation. Phase-field fracture
consists of two nonlinearly-coupled partial differential equations and it is well known
that the underlying energy functional is non-convex and requires
sophisticated numerical algorithms. Most of them are based on
partitioned approaches and require many subiterations until convergence.
Moreover, the algorithms
must account for crack irreversibility, i.e., an inequality constraint in time.
In this talk, we show computational findings which show the
efficiency of an inexact augmented Lagrangian iteration using
an error-oriented Newton method as inner solver (the
method presented last year). Furthermore, we found findings
in which the error-oriented approach did not converge quite
well and we design a modified Newton method that includes
fixed-point iterations. This method seems for certain
examples more robust than the error-oriented approach.
New computational results will substantiate our algorithmic
techniques.