Uploaded October 2024 | Updated September 2026, 1 week ago
Tune in to the newest episode of the SIAM Geosciences Webinar Series, focusing on adaptive regularization and linearization for nonsmooth and degenerate problems with Martin Vohralík. This series is organized by the SIAM Activity Group on Geosciences: siam.org/membership/activity-groups/detail/geosciences.
Abstract: Nonsmooth and degenerate nonlinearities are omnipresent in flow and transport in porous media. In practical simulations, they are at the origin of convergence troubles of standard iterative linearization schemes such as the Newton method. Common recipes to ensure linearization convergence include timestep cutting, damping, scheme switching to fixed-point-type or semismooth methods, or variable switching. In this talk, we describe a recipe maintaining the given iterative linearization scheme (the Newton method) with given variables at all circumstances; we instead replace the original nonsmooth and degenerate nonlinear laws by an epsilon-parametrized sequence of smooth and nondegenerate laws. For each discrete setting (given problem and numerical scheme, time step, and mesh size), we then start from a larger epsilon, apply a few Newton iterations to the arising (easy) nonlinear problem, and repeat upon decreasing the regularization parameter epsilon. The procedure is steered by an a posteriori error estimate and guarantees that the linearization estimate is below the regularization one and that the regularization estimate is below the discretization one. The overall error committed in such a numerical simulation is also certified. Numerous numerical examples for the Richards equation (a doubly nonlinear parabolic–hyperbolic and parabolic–elliptic degenerate problem), two-phase flows, and multiphase multicompositional flows (of complementarity form with phase appearance and disappearance) are presented and discussed.
#webinarseries #geophysics #seismic #environmentalscience #geosciences #dataanalysis #appliedmathematics #renewableenergy #climatechange
Tune in to the newest episode of the SIAM Geosciences Webinar Series, focusing on adaptive regularization and linearization for nonsmooth and degenerate problems with Martin Vohralík. This series is organized by the SIAM Activity Group on Geosciences: siam.org/membership/activity-groups/detail/geosciences.
Abstract: Nonsmooth and degenerate nonlinearities are omnipresent in flow and transport in porous media. In practical simulations, they are at the origin of convergence troubles of standard iterative linearization schemes such as the Newton method. Common recipes to ensure linearization convergence include timestep cutting, damping, scheme switching to fixed-point-type or semismooth methods, or variable switching. In this talk, we describe a recipe maintaining the given iterative linearization scheme (the Newton method) with given variables at all circumstances; we instead replace the original nonsmooth and degenerate nonlinear laws by an epsilon-parametrized sequence of smooth and nondegenerate laws. For each discrete setting (given problem and numerical scheme, time step, and mesh size), we then start from a larger epsilon, apply a few Newton iterations to the arising (easy) nonlinear problem, and repeat upon decreasing the regularization parameter epsilon. The procedure is steered by an a posteriori error estimate and guarantees that the linearization estimate is below the regularization one and that the regularization estimate is below the discretization one. The overall error committed in such a numerical simulation is also certified. Numerous numerical examples for the Richards equation (a doubly nonlinear parabolic–hyperbolic and parabolic–elliptic degenerate problem), two-phase flows, and multiphase multicompositional flows (of complementarity form with phase appearance and disappearance) are presented and discussed.
#webinarseries #geophysics #seismic #environmentalscience #geosciences #dataanalysis #appliedmathematics #renewableenergy #climatechange
