arXiv subjects

David A. Ham

Publications and source records attributed to David A. Ham.

At least 19 recordsLinked to original sources

arXiv2025

An abstraction for solving multi-domain problems using finite element methods

We introduce a new abstraction for the representation and solution of multi-domain problems using finite element methods. This is an advance over previous work in that it achieves a single higher-level abstraction that represents multi-domain problems in the mixed variational problem formalism. We implemented our new abstraction in UFL and Firedrake, and validated our implementations solving a quad-triangle mixed-cell-type problem, a hex-quad mixed-cell-type problem, and a fluid-structure interaction benchmark problem.

cs.MS↗

arXiv2025

Constructing PDFs of spatially dependent fields using finite elements

A probability density function (PDF) of a spatially dependent field provides a means of calculating moments of the field or, equivalently, the proportion of a spatial domain that is mapped to a given set of values. This paper describes a finite element approach to estimating the PDF of a spatially dependent field and its numerical implementation in the Python package NumDF.

physics.flu-dyn↗

arXiv2024

Differentiable programming across the PDE and Machine Learning barrier

The combination of machine learning and physical laws has shown immense potential for solving scientific problems driven by partial differential equations (PDEs) with the promise of fast inference, zero-shot generalisation, and the ability to discover new physics. Examples include the use of fundamental physical laws as inductive bias to machine learning algorithms, also referred to as physics-driven machine learning, and the application of machine learning to represent features not represented in the differential equations such as closures for unresolved spatiotemporal scales. However, the simulation of complex physical systems by coupling advanced numerics for PDEs with state-of-the-art machine learning demands the composition of specialist PDE solving frameworks with industry-standard machine learning tools. Hand-rolling either the PDE solver or the neural net will not cut it. In this work, we introduce a generic differentiable programming abstraction that provides scientists and engineers with a highly productive way of specifying end-to-end differentiable models coupling machine learning and PDE-based components, while relying on code generation for high performance. Our interface automates the coupling of arbitrary PDE-based systems and machine learning models and unlocks new applications that could not hitherto be tackled, while only requiring trivial changes to existing code. Our framework has been adopted in the Firedrake finite-element library and supports the PyTorch and JAX ecosystems, as well as downstream libraries.

cs.LG↗

arXiv2024

Efficient N-to-M Checkpointing Algorithm for Finite Element Simulations

In this work, we introduce a new algorithm for N-to-M checkpointing in finite element simulations. This new algorithm allows efficient saving/loading of functions representing physical quantities associated with the mesh representing the physical domain. Specifically, the algorithm allows for using different numbers of parallel processes for saving and loading, allowing for restarting and post-processing on the process count appropriate to the given phase of the simulation and other conditions. For demonstration, we implemented this algorithm in PETSc, the Portable, Extensible Toolkit for Scientific Computation, and added a convenient high-level interface into Firedrake, a system for solving partial differential equations using finite element methods. We evaluated our new implementation by saving and loading data involving 8.2 billion finite element degrees of freedom using 8,192 parallel processes on ARCHER2, the UK National Supercomputing Service.

cs.DC↗

arXiv2023

Consistent Point Data Assimilation in Firedrake and Icepack

When estimating quantities and fields that are difficult to measure directly, such as the fluidity of ice, from point data sources, such as satellite altimetry, it is important to solve a numerical inverse problem that is formulated with Bayesian consistency. Otherwise, the resultant probability density function for the difficult to measure quantity or field will not be appropriately clustered around the truth. In particular, the inverse problem should be formulated by evaluating the numerical solution at the true point locations for direct comparison with the point data source. If the data are first fitted to a gridded or meshed field on the computational grid or mesh, and the inverse problem formulated by comparing the numerical solution to the fitted field, the benefits of additional point data values below the grid density will be lost. We demonstrate, with examples in the fields of groundwater hydrology and glaciology, that a consistent formulation can increase the accuracy of results and aid discourse between modellers and observationalists. To do this, we bring point data into the finite element method ecosystem as discontinuous fields on meshes of disconnected vertices. Point evaluation can then be formulated as a finite element interpolation operation (dual-evaluation). This new abstraction is well-suited to automation, including automatic differentiation. We demonstrate this through implementation in Firedrake, which generates highly optimised code for solving Partial Differential Equations (PDEs) with the finite element method. Our solution integrates with dolfin-adjoint/pyadjoint, allowing PDE-constrained optimisation problems, such as data assimilation, to be solved through forward and adjoint mode automatic differentiation.

cs.MS↗

arXiv2023

Physics-driven machine learning models coupling PyTorch and Firedrake

Partial differential equations (PDEs) are central to describing and modelling complex physical systems that arise in many disciplines across science and engineering. However, in many realistic applications PDE modelling provides an incomplete description of the physics of interest. PDE-based machine learning techniques are designed to address this limitation. In this approach, the PDE is used as an inductive bias enabling the coupled model to rely on fundamental physical laws while requiring less training data. The deployment of high-performance simulations coupling PDEs and machine learning to complex problems necessitates the composition of capabilities provided by machine learning and PDE-based frameworks. We present a simple yet effective coupling between the machine learning framework PyTorch and the PDE system Firedrake that provides researchers, engineers and domain specialists with a high productive way of specifying coupled models while only requiring trivial changes to existing code.

cs.LG↗

arXiv2021

Escaping the abstraction: a foreign function interface for the Unified Form Language [UFL]

High level domain specific languages for the finite element method underpin high productivity programming environments for simulations based on partial differential equations (PDE) while employing automatic code generation to achieve high performance. However, a limitation of this approach is that it does not support operators that are not directly expressible in the vector calculus. This is critical in applications where PDEs are not enough to accurately describe the physical problem of interest. The use of deep learning techniques have become increasingly popular in filling this knowledge gap, for example to include features not represented in the differential equations, or closures for unresolved spatiotemporal scales. We introduce an interface within the Firedrake finite element system that enables a seamless interface with deep learning models. This new feature composes with the automatic differentiation capabilities of Firedrake, enabling the automated solution of inverse problems. Our implementation interfaces with PyTorch and can be extended to other machine learning libraries. The resulting framework supports complex models coupling PDEs and deep learning whilst maintaining separation of concerns between application scientists and software experts.

cs.MS↗

arXiv2021

Bringing Trimmed Serendipity Methods to Computational Practice in Firedrake

We present an implementation of the trimmed serendipity finite element family, using the open source finite element package Firedrake. The new elements can be used seamlessly within the software suite for problems requiring $H^1$, \hcurl, or \hdiv-conforming elements on meshes of squares or cubes. To test how well trimmed serendipity elements perform in comparison to traditional tensor product elements, we perform a sequence of numerical experiments including the primal Poisson, mixed Poisson, and Maxwell cavity eigenvalue problems. Overall, we find that the trimmed serendipity elements converge, as expected, at the same rate as the respective tensor product elements while being able to offer significant savings in the time or memory required to solve certain problems.

math.NA↗

arXiv2021

Code generation for productive portable scalable finite element simulation in Firedrake

Creating scalable, high performance PDE-based simulations requires a suitable combination of discretizations, differential operators, preconditioners and solvers. The required combination changes with the application and with the available hardware, yet software development time is a severely limited resource for most scientists and engineers. Here we demonstrate that generating simulation code from a high-level Python interface provides an effective mechanism for creating high performance simulations from very few lines of user code. We demonstrate that moving from one supercomputer to another can require significant algorithmic changes to achieve scalable performance, but that the code generation approach enables these algorithmic changes to be achieved with minimal development effort.

cs.MS↗

arXiv2021

UFL Dual Spaces, a proposal

This white paper highlights current limitations in the algebraic closure Unified Form Language (UFL). UFL currently represents forms over finite element spaces, however finite element problems naturally result in objects in the dual to a finite element space, and operators mapping between primal and dual finite element spaces. This document sketches the relevant mathematical areas and proposes changes to the UFL language to support dual spaces as first class types in UFL.

cs.MS↗

arXiv2020

A study of vectorization for matrix-free finite element methods

Vectorization is increasingly important to achieve high performance on modern hardware with SIMD instructions. Assembly of matrices and vectors in the finite element method, which is characterized by iterating a local assembly kernel over unstructured meshes, poses difficulties to effective vectorization. Maintaining a user-friendly high-level interface with a suitable degree of abstraction while generating efficient, vectorized code for the finite element method is a challenge for numerical software systems and libraries. In this work, we study cross-element vectorization in the finite element framework Firedrake via code transformation and demonstrate the efficacy of such an approach by evaluating a wide range of matrix-free operators spanning different polynomial degrees and discretizations on two recent CPUs using three mainstream compilers. Our experiments show that our approaches for cross-element vectorization achieve 30\% of theoretical peak performance for many examples of practical significance, and exceed 50\% for cases with high arithmetic intensities, with consistent speed-up over (intra-element) vectorization restricted to the local assembly kernels.

cs.MS↗

arXiv2019

Slate: extending Firedrake's domain-specific abstraction to hybridized solvers for geoscience and beyond

Within the finite element community, discontinuous Galerkin (DG) and mixed finite element methods have become increasingly popular in simulating geophysical flows. However, robust and efficient solvers for the resulting saddle-point and elliptic systems arising from these discretizations continue to be an on-going challenge. One possible approach for addressing this issue is to employ a method known as hybridization, where the discrete equations are transformed such that classic static condensation and local post-processing methods can be employed. However, it is challenging to implement hybridization as performant parallel code within complex models, whilst maintaining separation of concerns between applications scientists and software experts. In this paper, we introduce a domain-specific abstraction within the Firedrake finite element library that permits the rapid execution of these hybridization techniques within a code-generating framework. The resulting framework composes naturally with Firedrake's solver environment, allowing for the implementation of hybridization and static condensation as runtime-configurable preconditioners via the Python interface to PETSc, petsc4py. We provide examples derived from second order elliptic problems and geophysical fluid dynamics. In addition, we demonstrate that hybridization shows great promise for improving the performance of solvers for mixed finite element discretizations of equations related to large-scale geophysical flows.

cs.MS↗

arXiv2018

Thetis coastal ocean model: discontinuous Galerkin discretization for the three-dimensional hydrostatic equations

Unstructured grid ocean models are advantageous for simulating the coastal ocean and river-estuary-plume systems. However, unstructured grid models tend to be diffusive and/or computationally expensive which limits their applicability to real life problems. In this paper, we describe a novel discontinuous Galerkin (DG) finite element discretization for the hydrostatic equations. The formulation is fully conservative and second-order accurate in space and time. Monotonicity of the advection scheme is ensured by using a strong stability preserving time integration method and slope limiters. Compared to previous DG models advantages include a more accurate mode splitting method, revised viscosity formulation, and new second-order time integration scheme. We demonstrate that the model is capable of simulating baroclinic flows in the eddying regime with a suite of test cases. Numerical dissipation is well-controlled, being comparable or lower than in existing state-of-the-art structured grid models.

physics.ao-ph↗

arXiv2018

Automated shape differentiation in the Unified Form Language

We discuss automating the calculation of weak shape derivatives in the Unified Form Language (Aln{\ae}s et al., ACM Trans. Math. Softw., 2014) by introducing an appropriate additional step in the pullback from physical to reference space that computes G\^ateaux derivatives with respect to the coordinate field. We illustrate the ease of use with several examples.

math.NA↗

arXiv2017

Exposing and exploiting structure: optimal code generation for high-order finite element methods

Code generation based software platforms, such as Firedrake, have become popular tools for developing complicated finite element discretisations of partial differential equations. We extended the code generation infrastructure in Firedrake with optimisations that can exploit the structure inherent to some finite elements. This includes sum factorisation on cuboid cells for continuous, discontinuous, H(div) and H(curl) conforming elements. Our experiments confirm optimal algorithmic complexity for high-order finite element assembly. This is achieved through several novel contributions: the introduction of a more powerful interface between the form compiler and the library providing the finite elements; a more abstract, smarter library of finite elements called FInAT that explicitly communicates the structure of elements; and form compiler algorithms to automatically exploit this exposed structure.

cs.MS↗

arXiv2017

TSFC: a structure-preserving form compiler

A form compiler takes a high-level description of the weak form of partial differential equations and produces low-level code that carries out the finite element assembly. In this paper we present the Two-Stage Form Compiler (TSFC), a new form compiler with the main motivation to maintain the structure of the input expression as long as possible. This facilitates the application of optimizations at the highest possible level of abstraction. TSFC features a novel, structure-preserving method for separating the contributions of a form to the subblocks of the local tensor in discontinuous Galerkin problems. This enables us to preserve the tensor structure of expressions longer through the compilation process than other form compilers. This is also achieved in part by a two-stage approach that cleanly separates the lowering of finite element constructs to tensor algebra in the first stage, from the scheduling of those tensor operations in the second stage. TSFC also efficiently traverses complicated expressions, and experimental evaluation demonstrates good compile-time performance even for highly complex forms.

cs.MS↗

arXiv2016

A structure-exploiting numbering algorithm for finite elements on extruded meshes, and its performance evaluation in Firedrake

We present a generic algorithm for numbering and then efficiently iterating over the data values attached to an extruded mesh. An extruded mesh is formed by replicating an existing mesh, assumed to be unstructured, to form layers of prismatic cells. Applications of extruded meshes include, but are not limited to, the representation of 3D high aspect ratio domains employed by geophysical finite element simulations. These meshes are structured in the extruded direction. The algorithm presented here exploits this structure to avoid the performance penalty traditionally associated with unstructured meshes. We evaluate the implementation of this algorithm in the Firedrake finite element system on a range of low compute intensity operations which constitute worst cases for data layout performance exploration. The experiments show that having structure along the extruded direction enables the cost of the indirect data accesses to be amortized after 10-20 layers as long as the underlying mesh is well-ordered. We characterise the resulting spatial and temporal reuse in a representative set of both continuous-Galerkin and discontinuous-Galerkin discretisations. On meshes with realistic numbers of layers the performance achieved is between 70% and 90% of a theoretical hardware-specific limit.

cs.MS↗

arXiv2016

Firedrake: automating the finite element method by composing abstractions

Firedrake is a new tool for automating the numerical solution of partial differential equations. Firedrake adopts the domain-specific language for the finite element method of the FEniCS project, but with a pure Python runtime-only implementation centred on the composition of several existing and new abstractions for particular aspects of scientific computing. The result is a more complete separation of concerns which eases the incorporation of separate contributions from computer scientists, numerical analysts and application specialists. These contributions may add functionality, or improve performance. Firedrake benefits from automatically applying new optimisations. This includes factorising mixed function spaces, transforming and vectorising inner loops, and intrinsically supporting block matrix operations. Importantly, Firedrake presents a simple public API for escaping the UFL abstraction. This allows users to implement common operations that fall outside pure variational formulations, such as flux-limiters.

cs.MS↗