European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
15:45   MS42: Large-Scale Numerical Bifurcatino Analysis
Chair: Jonas Thies
15:45
25 mins
Computing transition probabilities in stochastic ocean-climate models
Sven Baars, Daniele Castellana, Fred Wubs, Henk Dijkstra
Abstract: Transitions in an ocean-climate model may occur due to the existence of multiple steady states for the same parameter values. Because of unresolved small-scale variability, however, transitions may be observed even before a bifurcation point is reached. This unresolved variability is often represented as noise in a stochastic model. On a local scale, we can get an idea of the sensitivity to noise of a stochastic model by computing the probability density function around a steady state by means of the solution of a generalized Lyapunov equation. We can perform a continuation with these probability density functions by means of a novel solver for generalized Lyapunov equations [1]. Methods for computing actual probabilities of transitions on a global scale have so far only been used on lower dimensional problems due to the high computational cost. We present a novel projected time stepping approach, which is based on the aforementioned solution of generalized Lyapunov equations, which greatly reduces the cost of these methods for high dimensional problems. We use this approach in combination with the Trajectory-Adaptive Multilevel Sampling [3] method to compute probabilities of transitions in an idealized model of the Meridional Overturning Circulation [2]. [1] S. Baars, J. Viebahn, T. Mulder, C. Kuehn, F. Wubs, and H. Dijkstra. Continuation of Probability Density Functions Using a Generalized Lyapunov Approach. Journal of Computational Physics, 336:627–643, May 2017. doi: 10.1016/j.jcp.2017.02.021. [2] S. Baars, D. Castellana, F. Wubs, and H. A. Dijkstra. Application of Adaptive Multilevel Splitting to High-Dimensional Dynamical Systems. preprint, 2019. [3] T. Lestang, F. Ragone, C.-E. Bréhier, C. Herbert, and F. Bouchet. Computing Return Times Or Return Periods With Rare Event Algorithms. Journal of Statistical Mechanics: Theory and Experiment, 2018(4):043213, Apr 2018. doi: 10.1088/1742-5468/aab856.
16:10
25 mins
Bifurcation analysis and steady state continuations with an implicit Earth system model
Erik Mulder
Abstract: The nonlinear dynamics of the climate system are traditionally explored through explicit time integrations of the discretized equations. After a spinup phase, trajectories may end up in (statistical) steady states, periodic orbits or even chaotic motions. Often these long-term characteristics are of particular interest and especially their sensitivity to changes in environmental conditions are studied extensively. For example, estimates of the climate's equilibrium response to changes in radiative forcing are fundamental to projections of future climate change. Equilibria of large-scale dynamical systems can be computed directly through numerical steady state continuations. However, such techniques rely on fixed point iterations involving a Jacobian matrix (or at least its action). Basic differentiability requirements are often neglected in intermediate to high-resolution climate modeling approaches, which obstructs the application of continuation techniques to coupled climate problems. We therefore present a novel, fully implicit Earth system model of intermediate complexity: the I-EMIC. The I-EMIC contains a primitive equation ocean model, atmospheric heat and moisture transport, the formation of sea ice and the adjustment of albedo over snow and ice. With the I-EMIC, high-dimensional branches of bifurcation diagrams are obtained through numerical steady state continuations. Moreover, large-scale linear stability analyses are performed near major bifurcations, revealing the spatial nature of destabilizing perturbations. In addition, we will discuss a homotopy based continuation technique that allows continuations of steady states through changing topographies within the I-EMIC. Applied to problems in paleoclimatology, this approach can be used to compute deep time bifurcation diagrams and identify critical transitions due to plate tectonics. At the same time it provides an efficient alternative to obtain sequences of past equilibrium climate states by avoiding the computation of spinup trajectories.
16:35
25 mins
Automatic exploration techniques in numerical continuation illustrated by the Ginzburg-Landau equation
Michiel Wouters, Wim Vanroose
Abstract: Multiple scientific phenomena are described by dynamical systems: partial differential equations that describe a certain state in time. These systems usually contain one or multiple physical parameters, like the temperature or initial concentrations of a chemical substance. Instead of solving the equilibrium equation of such a system for a given set of parameter values, numerical continuation studies the effect of these parameters on the equilibria states. Based on the implicit function theorem, this dependency is described by continuous solution curves, approximated by the pseudo-arclength continuation algorithm. One of the main challenges of numerical continuation is the inclusion of automatic exploration techniques: techniques that allow the construction of a complete bifurcation diagram, consisting of multiple, interconnected, solution curves. These techniques typically consist of two main steps: approximating bifurcation points and analysing tangent directions to new curves that might emerge from these bifurcations. The current talk focusses on the first step. The Newton step length adaptation algorithm is used as a base for finding bifurcation points, and we introduce an alternative version of the underlying Newton algorithm to increase its efficiency. The introduced adapted Newton method is able to solve nonlinear systems with a rank-deficient Jacobian precisely near the searched solution. Nonlinear systems with this property appear naturally in the bifurcation point problem. The algorithms discussed in the talk will be illustrated by the Ginzburg-Landau equation, which describes superconductivity for certain types of material. The dependency of the state of the superconductor with respect to the magnetic field strength will be analysed by numerical continuation. Bifurcation diagrams for this problem typically consist of a high number of different solution curves, highlighting the necessity of automatic exploration techniques.
17:00
25 mins
Towards scalable automatic exploration of bifurcation diagrams for large-scale applications
Jonas Thies, Michiel Wouters, Rebekka-Sarah Hennig, Wim Vanroose
Abstract: There are several libraries for computing branches of steady states of dynamical systems, e.g. LOCA (http://www.cs.sandia.gov/LOCA/) for large- scale problems like nonlinear PDEs. The core algorithms typically are (pseudo- )arclength continuation, Newton-Krylov methods and (sparse) eigenvalue solvers. While LOCA includes some basic techniques for computing bifurcation points and switching branches, the exploration of a complete bifurcation dia- gram still takes a lot of programming e ort and manual interference. On the other hand, recent developments in algorithms for fully automatic exploration are condensed in a Python tool called PyNCT (https://pypi.org/project/PyNCT/). The scope of this algorithmically versatile software is, however, limited to relatively small (e.g. 2D) problems because of the lack of a high-performance linear algebra implementation of the numerical core. In this talk we aim to combine the best of both worlds: a high-level implementations of algorithms in PyNCT with parallel models and linear algebra implemented in Trilinos (LOCA/Epetra). PyNCT is extended to non-symmetric systems and its complete backend is replaced by the phist library (https://bitbucket.org/essex/phist), which allows straight-forwared coupling to Epetra and PDE models implemented originally for LOCA. We apply the new code to reaction-di usion and uid dynamics models in three space dimensions to demonstrate its potential. By combining state-of-the-art automatic continuation algorithms from PyNCT with high- performance solvers and preconditioners from phist and Trilinos, we show that fully automatic bifurcation analysis on HPC systems is possible.