Invited Session Fri.2.MA 415

Friday, 13:15 - 14:45 h, Room: MA 415

Cluster 19: PDE-constrained optimization & multi-level/multi-grid methods [...]

Hierarchical methods for the design of nanoporous materials


Chair: Robert Michael Lewis



Friday, 13:15 - 13:40 h, Room: MA 415, Talk 1

Paul T. Boggs
Combining multi-grid and domain decomposition as preconditioners for a class of multi-level PDE-constrained optimization problems

Coauthors: Julien Cortial, David Gay, Michael Lewis, Kevin Long, Stephen Nash


Recently we developed a multi-grid optimization (MG/Opt) strategy for
solving a class of multi-level, multi-physics problems that arise in
the design of nanoporous materials. The method works well on
moderate-sized problems, but it is clear that additional strategies
would be required for the large problems of interest. In this talk,
we discuss our use of domain decomposition (DD) to extend our MG/Opt
work to larger problems. In particular, we adopt the point of view
that DD and MG/Opt are both preconditioning strategies and we
demonstrate an effective combination of MG/Opt and DD to solve these
much larger problems. Numerical results will be presented.



Friday, 13:45 - 14:10 h, Room: MA 415, Talk 2

David M. Gay
Optimization algorithms for hierarchical problems, with application to nanoporous materials

Coauthors: Paul T. Boggs, Stewart Griffiths, Robert M. Lewis, Kevin R. Long, Stephen Nash, Robert H. Nilson


This talk concerns optimization algorithms for designing complex hierarchical systems, motivated by applications to the design of nanoporous materials. Such materials have a broad range of engineering applications, including gas storage and filtration, electrical energy storage in batteries and capacitors, and catalysis. The design of these materials involves modeling the material over many length scales, leading to a hierarchy of mathematical models. Our algorithms are also hierarchical in structure, with the goal of exploiting the model hierarchy to obtain solutions more rapidly. We discuss implementation issues and present some computational results.



Friday, 14:15 - 14:40 h, Room: MA 415, Talk 3

Robert Michael Lewis
Using inexact gradients in a multilevel optimization algorithm

Coauthor: Stephen Goptimizat Nash


Optimization algorithms typically require gradients of the objective and
constraints; however, computing accurate gradients can be computationally
expensive. We discuss the implications of using inexact gradients in the
context of the multilevel optimization algorithm MG/Opt. MG/Opt
recursively uses a hierarchy of models, of less fidelity but also less cost,
to obtain search directions for finer-level models. However,
MG/Opt requires the gradient on the fine level in order to define the
recursion. We discuss the impact of gradient errors on the multilevel
recursion in MG/Opt under various assumptions about the source of the error in
the gradients. We illustrate these impacts both analytically and numerically
for a number of model problems.


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