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14:00-15:00 Session 5: Inverse problems
Location: Zoom
Nonlinear Stokes Optimisation with Firedrake and ROL
PRESENTER: Angus Gibson

ABSTRACT. Mantle convection is the primary driving force for maintaining plate tectonics and shaping the Earth's surface. Recent global geodynamic models are able to incorporate many Earth-like features such as complex rheological and/or thermochemical properties. Yet, due to uncertainties in thermochemical and rheological properties of the Earth's mantle, and the fundamental lack of information of any earlier Earth-system states, the recent evolution of the mantle is poorly known. Therefore, it is important to develop methods that can reconstruct the Earth's mantle in space and time. To achieve this, we reformulate mantle flow simulations as an optimisation problem through the so-called adjoint method, where we minimise a misfit functional, representative of the difference between the present-day model state and inferences on mantle thermal structure as obtained from seismic tomography images the Earth.

We present the use of Firedrake coupled with pyadjoint to solve the forward problem, and provide an adjoint model giving gradient information for the nonlinear optimisation problem. The misfit functional and gradient are given to ROL (Rapid Optimisation Library) to perform the actual optimisation, allowing for access to a wide variety of optimisation methods. Additionally, we show the updated Python interface to ROL, allowing access to some new optimisation methods and a different way of framing optimisation problems in ROL 2.0.

PDE-based optimization of open-loop shallow geothermal systems using Firedrake and dolfin-adjoint
PRESENTER: Smajil Halilovic

ABSTRACT. Open-loop shallow geothermal systems cause thermal anomalies in the groundwater, which can reach neighboring systems and reduce their efficiency. Therefore, it is important to optimally position these systems, i.e. their extraction and injection wells, to avoid negative interactions and maximize the thermal potential of the groundwater body. Flow and heat transport in porous media are described with a system of nonlinear coupled PDEs. In addition, source terms, representing extraction and injection wells, are modeled by non-smooth Dirac delta functions. The underlying problem is a PDE-constrained optimization problem including control (spatial coordinates) and state (groundwater temperature) constraints. In this talk, we introduce an adjoint-based approach to solve this non-smooth PDE-constrained optimization problem. Dirac delta functions are approximated with smooth bump functions, which decouples source points from the mesh and enables computation of gradients. Nonlinear state constraints are incorporated using Moreau-Yosida-type regularization terms. We employ the finite-element tool Firedrake for the forward model and dolfin-adjoint for the optimization.

Solving Data-Constrained FEM Problems with Firedrake and stat-fem

ABSTRACT. The statistical finite element method is a recent theoretical development that allows for FEM models to be conditioned on sensor data. The approach casts the FEM solution as a Bayesian inference problem and allows for solving for the posterior conditioned on the observed sensor data, while rigorously taking all uncertainties in the problem into account and allowing for the possibility that the FEM model is mis-specified. The method is straightforward to parallelize and allows for efficient updating as more data observations are acquired.

This paper presents stat-fem, an implementation of the statistical FEM method that builds on Firedrake for FEM assembly and leverages PETSc to efficiently handle the required sparse matrix linear algebra manipulations underlying the posterior updating in parallel. The library is written in pure Python and requires no additional dependencies or packages beyond the standard Firedrake installation. The implementation takes advantage of ensemble parallelism to enable efficient updating with large numbers of sensor readings.

In this poster submission, I will give an overview of the statistical FEM method, highlight particular details of the implementation, and demonstrate its utility through application to a mis-specified FEM problem.

15:00-15:30Coffee Break
15:30-16:30 Session 6: New Firedrake capabilities 2
Location: Zoom
Dual Spaces in UFL and Firedrake
PRESENTER: India Marsden

ABSTRACT. The finite element method is based on finding approximate weak solutions to variational problems. These solutions live in finite element spaces. In the process of solving these problems, operators are created which map to and from the spaces dual to these finite element spaces. Assembled UFL forms are an example of such objects. Here, we present the extension of UFL and Firedrake to accommodate first class objects in the dual spaces to finite element spaces. We will show how this makes UFL a more capable language for finite element problems, and how explicit support for dual spaces facilitates further expansions into external operators and interpolation.

Updates to Interpolation in Firedrake

ABSTRACT. Interpolation is a useful and performant way of generating fields (Firedrake Functions) from UFL expressions or other Functions. Here we will discuss recent updates to the capabilities of interpolation and useful under-the-hood changes. These include new (a) parallel-compatible cross-mesh interpolation capability onto a “VertexOnlyMesh” for point evaluations, (b) functional hessian calculation in pyadjoint’s annotation of interpolation, (c) UFL language additions for expressing interpolation as dual-basis evaluation, and (d) performance enhancements from adding dual-basis evaluation capability to the FInAT element library.

With these changes new applications such as direct assimilation of point-data are now possible. A road has also been paved for much future work including (1) the extension of parallel-compatible cross-mesh interpolation, (2) optimised interpolation onto previously unsupported FInAT elements such as “EnrichedElement”, and (3) the development of UFL into a language for expressing and automating diagnostics on big field datasets produced by climate models.

External operators in Firedrake
PRESENTER: Nacime Bouziani

ABSTRACT. High level domain specific languages based on the Unified Form Language (UFL) such as Firedrake or FeniCS enable one to write down PDE-based problems in a very productive way. UFL equips Firedrake with a highly expressive interface to specify the variational forms and discrete function spaces, providing the abstractions needed for code generation. However, one of its limitations is that it does not take into account operators that are not directly expressible in the vector calculus sense. Put simply, the UFL abstraction is not rich enough to handle these operators. We refer to these operators as external operators.

This limitation is critical in many applications where PDEs are not enough to accurately describe the physical problem of interest. These applications include nonlinear implicit constitutive laws such as the Glen’s flow law for glacier flow, the use of neural networks to include features not represented in the differential equations, or closures for unresolved spatiotemporal scales. Example applications of neural networks include regularization of inverse problems such as in seismic inversion and subgrid parameterization of atmospheric or oceanographic processes like clouds or turbulence.

We present extensions to the Unified Form Language (UFL) and Firedrake that enable the inclusion of arbitrary external operators. This external operator feature composes seamlessly with the automatic differentiation capabilities of Firedrake

16:30-17:00Coffee Break
18:30-19:30After work Gather Town drinks