A framework for the automation of generalised stability theory

The traditional approach to investigating the stability of a physical system is to linearise the equations about a steady base solution, and to examine the eigenvalues of the linearised operator. Over the past several decades, it has been recognised that this approach only determines the asymptotic stability of the system, and neglects the possibility of transient perturbation growth arising due to the nonnormality of the system. This observation motivated the development of a more powerful generalised stability theory (GST), which focusses instead on the singular value decomposition of the linearised propagator of the system. While GST has had significant successes in understanding the stability of phenomena in geophysical fluid dynamics, its more widespread applicability has been hampered by the fact that computing the SVD requires both the tangent linear operator and its adjoint: deriving the tangent linear and adjoint models is usually a considerable challenge, and manually embedding them inside an eigensolver is laborious. In this paper, we present a framework for the automation of generalised stability theory, which overcomes these difficulties. Given a compact high-level symbolic representation of a finite element discretisation implemented in the FEniCS system, efficient C++ code is automatically generated to assemble the forward, tangent linear and adjoint models; these models are then used to calculate the optimally growing perturbations to the forward model, and their growth rates. By automating the stability computations, we hope to make these powerful tools a more routine part of computational analysis. The efficiency and generality of the framework is demonstrated with applications drawn from geophysical fluid dynamics, phase separation and quantum mechanics.Comment: Accepted in SIS

A framework for automated PDE-constrained optimisation

A generic framework for the solution of PDE-constrained optimisation problems based on the FEniCS system is presented. Its main features are an intuitive mathematical interface, a high degree of automation, and an efficient implementation of the generated adjoint model. The framework is based upon the extension of a domain-specific language for variational problems to cleanly express complex optimisation problems in a compact, high-level syntax. For example, optimisation problems constrained by the time-dependent Navier-Stokes equations can be written in tens of lines of code. Based on this high-level representation, the framework derives the associated adjoint equations in the same domain-specific language, and uses the FEniCS code generation technology to emit parallel optimised low-level C++ code for the solution of the forward and adjoint systems. The functional and gradient information so computed is then passed to the optimisation algorithm to update the parameter values. This approach works both for steady-state as well as transient, and for linear as well as nonlinear governing PDEs and a wide range of functionals and control parameters. We demonstrate the applicability and efficiency of this approach on classical textbook optimisation problems and advanced examples

LEoPart: a particle library for FEniCS

This paper introduces LEoPart, an add-on for the open source finite element software library FEniCS to seamlessly integrate Lagrangian particle functionality with (Eulerian) mesh-based finite element (FE) approaches. LEoPart - which is so much as to say: `Lagrangian-Eulerian on Particles' - contains tools for efficient, accurate and scalable advection of Lagrangian particles on arbitrary polyhedral meshes. In addition, LEoPart comes with several projection operators for exchanging information between the scattered particles and the mesh and \textit{vice versa}. These projection operators are based on a variational framework, which allows extension to high-order accuracy. In particular, by implementing a dedicated PDE-constrained particle-mesh projection operator, LEoPart provides all the tools for diffusion-free advection, while simultaneously achieving optimal convergence and ensuring conservation of the projected particle quantities on the underlying mesh. A range of numerical examples that are prototypical to passive and active tracer methods highlight the properties and the parallel performance of the different tools in LEoPart. Finally, future developments are identified. The source code for LEoPart is actively maintained and available under an open source license at https://bitbucket.org/jakob_maljaars/leopart.Comment: 35 pages, 13 figure

A massively parallel Eulerian-Lagrangian method for advection-dominated transport in viscous fluids

Motivated by challenges in Earth mantle convection, we present a massively parallel implementation of an Eulerian-Lagrangian method for the advection-diffusion equation in the advection-dominated regime. The advection term is treated by a particle-based, characteristics method coupled to a block-structured finite-element framework. Its numerical and computational performance is evaluated in multiple, two- and three-dimensional benchmarks, including curved geometries, discontinuous solutions, pure advection, and it is applied to a coupled non-linear system modeling buoyancy-driven convection in Stokes flow. We demonstrate the parallel performance in a strong and weak scaling experiment, with scalability to up to $147,456$ parallel processes, solving for more than $5.2 \times 10^{10}$ (52 billion) degrees of freedom per time-step.Comment: 22 page