LANS Informal Seminar: Johann Rudi

Seminar Title: An Optimization-Based Perturbed Newton Method for Viscoplastic Fluids with von Mises Yielding
Speaker: Johann Rudi, Wilkinson Fellow, MCS/ANL

Date/Time: 2019-05-01 10:30
Location: Bldg. 240, Rm. 4301

Description:
We target highly nonlinear fluid models with von Mises yield critera that give
rise to optimization problems with Hessians exhibiting a problematic (near)
null space upon linearization with Newton’s method.

The null space is caused by a projector-type coefficient in the Hessian, which
is created by terms in the objective functional that resemble the $L^1$-norm.
This occurs, e.g., in nonlinear incompressible Stokes flow in Earth’s mantle
with plastic yielding rheology, which effectively limits stresses in the mantle
by weakening the viscosity depending on the strain rate.
Using a standard Newton linearization for such an application is known to
produce severe Newton step length reductions due to backtracking line search
and stagnating nonlinear convergence. Additionally, these effects become
increasingly prevalent as the mesh is refined.

We analyze issues with the standard Newton linearization in an abstract setting
and propose an improved linearization, which can be applied straightforwardly
to Stokes flow with yielding and other applications as total variation
regularization.
Finally, numerical experiments compare the standard and improved Newton
linearizations in practice. When we employ our improved linearization within
our inexact Newton-Krylov method, a fast and highly robust nonlinear solver is
attained that exhibits mesh-independent convergence and scales to large numbers
of cores with high parallel efficiency.