title: Computational methods for nonlocal PDE-constrained optimisation: applications in ecology, image processing, and phase separation
Creators: name: Miniguano Trujillo, Andrés
Publisher: name: The University of Edinburgh
Year: 2025
Subjects: subject: Nonlocal
subject: Optimal Control
subject: Parameter Learning
subject: Numerical Methods for PDEs
subject: Dynamical Density Functional Theory
subject: Optimisation
subject: Spectral Methods
subject: Variational Imaging
subject: SIRD model
subject: Mathematical Epidemiology
subject: PDE-Constrained Optimization
subject: Optimisation of systems of ODEs
subject: Phase Separation
subject: Cahn-Hilliard Equation
subject: Singular Convolution
subject: Preconditioning
subject: Bilevel Optimisation
subject: Differential Algebraic Solvers
subject: Optimise-then-discretise
subject: Numerical Linear Algebra
subject: ANOVA Kernel
subject: Fast Summation
subject: Non-equispaced Fast Fourier Transform
subject: Spectral Analysis
subject: Population Models
subject: Graph Laplacian

description: Complex systems of interacting quantities commonly arise when studying physical and ecological processes. The toolbox of a mathematical modeller provides many approximations to capture the intricate nature of such interactions. In particular, the field of differential equations (DEs) offers a rich set of modelling tools that, when complemented with the backbone of numerical linear algebra, can effectively capture and offer detailed insights and predictions about a system of interest. A DE system describes a number of interacting quantities related through terms involving one or several partial derivatives and possible nonlocalities. A nonlocal interaction is, in essence, an integro-differential quantity that relates a pointwise quantity with one or several clouds of points that are not necessarily close to the point of interest. As the name suggests, nonlocalities generalise local evaluations (e.g., partial derivatives or pointwise evaluations) to gain a domain-informed perspective of a function. As a result, combining local and nonlocal operators offers a powerful, general-purpose, modelling tool. Such a combination is regulated by a set of parameters that are either provided by observation or set using the knowledge of the modeller. However, measuring parameters for complex differential models is not a straightforward task. Nonlinearity, nonlocality, and (dis)continuous dependencies impede experimental model calibration, which also lacks theoretical guarantees for a given fit. Optimal control theory offers an effective framework for data-driven parameter identification problems governed by DEs. An optimal control problem aims to optimise a quantity (e.g., a model fit) defined on a suitable function space, which is governed by a set of parameters (or control variables) that determine the profile of a given response (or state variables) modelled by a differential system. By generalising concepts of classical optimisation theory to abstract function spaces, optimal control provides existence and optimality certificates that allow us to design effective numerical schemes targeted for model calibration. This work presents three parameter identification problems: a disease spread model, an image restoration problem, and a phase separation model. These problems arise in different application settings but share one common aspect: they contain a nonlocal term either in their formulation or as a by-product of the optimality conditions for the calibration process. Nonlocalities add additional complexity, as existence results are limited and problem-specific, while their discretisation often leads to dense and ill-conditioned systems. These challenges inform our approach, guiding the different analytical and numerical methods we develop for each problem. Our baseline for the parametric study of disease evolution is the classical Susceptible, Infected, Recovered, and Deceased (SIRD) model. The SIRD, a nonlinear system of ordinary differential equations, provides a comprehensive understanding of an outbreak under mild assumptions. The system is governed by three parameters: the contact rate, the recovery rate, and the mortality rate. Determining these parameters is critical to understanding the evolution of the disease and predicting scenarios that can guide disease prevention and control policies. We present an optimal control framework for parameter identification of the SIRD from observed data. We derive existence and uniqueness results alongside optimality conditions for the problem under a constant-in-time assumption for the controls. The conditions on the control yield a nonlocality in the optimality system. We present a numerical study of tailored iterative optimisation routines to obtain efficient numerical approximations. We demonstrate the effectiveness of the optimal control tools under extensive scenario tests. Moreover, we show how this approach can extend to more complex systems that allow spatial interactions. Practical image-acquisition methods introduce artefacts, such as noise, that corrupt the digital output of a ground observation. In particular, the stochastic nature of noise makes its identification and removal a particular ill-posed problem. Variational models offer a theory-rich and mesh-independent solution approach for the image denoising problem. This perspective relies on the combination of a regularising term modulated by a fidelity term to produce a new image with flattened regions that resemble the observed image. Here, the choice of a regulariser determines which information is removed. Nonlocal image denoising offers a class of regularisers by taking into account the redundancy of information in an image. This class of methods has been particularly effective at recovering images featuring textures and highly-varying regions. Bilevel optimisation and optimal control theory can determine an appropriate fidelity weight. We focus on finding optimal fidelity weights for a quadratic nonlocal regularising term. Specifically, we present the unnormalised extended Gaussian ANOVA kernel as an extension of the classical nonlocal means kernel. Naïve discretisations of the nonlocal kernel can be carried out in O(N²) operations, where N is the number of pixels in the image. Nonetheless, computing the ANOVA kernel can be done via the nonequispaced fast Fourier transform (NFFT), yielding a reduced O(N) complexity. Thus, we obtain efficient evaluations of the nonlocal kernel, which is crucial when solving the Euler-Lagrange system for the denoising problem. The discretised system is characterised by a graph Laplacian, for which we present a spectral study. We propose a family of diagonal preconditioners and dense variants to accelerate matrix-free iterative methods, that perform impressively under a suitable change of basis. Extensive numerical tests are presented, highlighting the effectiveness of the dense preconditioners and the NFFT in significatively reducing computation times and hardware requirements for solving nonlocal problems. Phase separation studies the evolutive formation of two or more distinct layers from a homogeneous mixture. Several PDE models have been proposed to capture this behaviour, effectively describing the dynamics of multi-component systems where different patterns spontaneously form and evolve. One of these models is the Cahn-Hilliard equation, a quasilinear PDE describing a separation and coarsening stage. Characterised by fourth-order derivative terms, the equation relies on high regularity assumptions for the solution. A less restrictive extension is the nonlocal Cahn-Hilliard equation, allowing for more flexibility in the involved nonlinear terms and less regularity for the solution. Determining the mobility term is a question that can be addressed using optimal control theory. When discretising the system, we first encounter the problem of adequately approximating the action of a singular kernel. We propose a pseudospectral-element method for obtaining detailed and accurate solutions of DEs in the presence of singular kernels. It is based on the MultiShape extension of the 2DChebClass library that allows us to isolate the singularity of the kernel on an arbitrarily small neighbourhood, effectively approximating the action of the singular kernel to machine precision. We present an extensive set of tests to showcase the robustness of our method and provide computationally cost-effective solutions to the nonlocal system.