arXiv subjects

Tristan Pryer

Publications and source records attributed to Tristan Pryer.

At least 37 records · Page 2Linked to original sources

arXiv2025

Wireless Network Topology Inference: A Markov Chains Approach

We address the problem of inferring the topology of a wireless network using limited observational data. Specifically, we assume that we can detect when a node is transmitting, but no further information regarding the transmission is available. We propose a novel network estimation procedure grounded in the following abstract problem: estimating the parameters of a finite discrete-time Markov chain by observing, at each time step, which states are visited by multiple ``anonymous'' copies of the chain. We develop a consistent estimator that approximates the transition matrix of the chain in the operator norm, with the number of required samples scaling roughly linearly with the size of the state space. Applying this estimation procedure to wireless networks, our numerical experiments demonstrate that the proposed method accurately infers network topology across a wide range of parameters, consistently outperforming transfer entropy, particularly under conditions of high network congestion.

cs.NI↗

arXiv2025

A Positivity-Preserving Finite Element Framework for Accurate Dose Computation in Proton Therapy

We present a stabilised finite element method for modelling proton transport in tissue, incorporating both inelastic energy loss and elastic angular scattering. A key innovation is a positivity-preserving formulation that guarantees non-negative fluence and dose, even on coarse meshes. This enables reliable computation of clinically relevant quantities for treatment planning. We derive a priori error estimates demonstrating optimal convergence rates and validate the method through numerical benchmarks. The proposed framework provides a robust, accurate and efficient tool for advancing proton beam therapy.

math.NA↗

arXiv2025

A nodally bound-preserving discontinuous Galerkin method for the drift-diffusion equation

In this work, we introduce and analyse discontinuous Galerkin (dG) methods for the drift-diffusion model. We explore two dG formulations: a classical interior penalty approach and a nodally bound-preserving method. Whilst the interior penalty method demonstrates well-posedness and convergence, it fails to guarantee non-negativity of the solution. To address this deficit, which is often important to ensure in applications, we employ a positivity-preserving method based on a convex subset formulation, ensuring the non-negativity of the solution at the Lagrange nodes. We validate our findings by summarising extensive numerical experiments, highlighting the novelty and effectiveness of our approach in handling the complexities of charge carrier transport.

math.NA↗

arXiv2025

Adaptive Regularisation for PDE-Constrained Optimal Control

PDE-constrained optimal control problems require regularisation to ensure well-posedness, introducing small perturbations that make the solutions challenging to approximate accurately. We propose a finite element approach that couples both regularisation and discretisation adaptivity, varying both the regularisation parameter and mesh-size locally based on rigorous a posteriori error estimates aiming to dynamically balance induced regularisation and discretisation errors, offering a robust and efficient method for solving these problems. We demonstrate the efficacy of our analysis with several numerical experiments.

math.NA↗

arXiv2025

A nodally bound-preserving finite element method for time-dependent convection-diffusion equations

This paper presents a new method to approximate the time-dependent convection-diffusion equations using conforming finite element methods, ensuring that the discrete solution respects the physical bounds imposed by the differential equation. The method is built by defining, at each time step, a convex set of admissible finite element functions (that is, the ones that satisfy the global bounds at their degrees of freedom) and seeks for a discrete solution in this admissible set. A family of $θ$-schemes is used as time integrators, and well-posedness of the discrete schemes is proven for the whole family, but stability and optimal-order error estimates are proven for the implicit Euler scheme. Nevertheless, our numerical experiments show that the method also provides stable and optimally-convergent solutions when the Crank-Nicolson method is used.

math.NA↗

arXiv2025

A nodally bound-preserving finite element method for hyperbolic convection-reaction problems

In this article, we present a numerical approach to ensure the preservation of physical bounds on the solutions to linear and nonlinear hyperbolic convection-reaction problems at the discrete level. We provide a rigorous framework for error analysis, formulating the discrete problem as a variational inequality and demonstrate optimal convergence rates in a natural norm. We summarise extensive numerical experiments validating the effectiveness of the proposed methods in preserving physical bounds and preventing unphysical oscillations, even in challenging scenarios involving highly nonlinear reaction terms.

math.NA↗

arXiv2024

Optimal control of a kinetic equation

This work addresses an optimal control problem constrained by a degenerate kinetic equation of parabolic-hyperbolic type. Using a hypocoercivity framework we establish the well-posedness of the problem and demonstrate that the optimal solutions exhibit a hypocoercive decay property, ensuring stability and robustness. Building on this framework, we develop a finite element discretisation that preserves the stability properties of the continuous system. The effectiveness and accuracy of the proposed method are validated through a series of numerical experiments, showcasing its ability to handle challenging PDE-constrained optimal control problems.

math.NA↗

arXiv2024

Cycle-Free Polytopal Mesh Sweeping for Boltzmann Transport

We introduce a novel property of bounded Voronoi tessellations that enables cycle-free mesh sweeping algorithms. We prove that a topological sort of the dual graph of any Voronoi tessellation is feasible in any flow direction and dimension, allowing straightforward application to discontinuous Galerkin (DG) discretisations of first-order hyperbolic partial differential equations and the Boltzmann Transport Equation (BTE) without requiring flux-cycle corrections. We also present an efficient algorithm to perform the topological sort on the dual mesh nodes, ensuring a valid sweep ordering. This result expands the applicability of DG methods for transport problems on polytopal meshes by providing a robust framework for scalable, parallelised solutions. To illustrate its effectiveness, we conduct a series of computational experiments showcasing a DG scheme for BTE, demonstrating both computational efficiency and adaptability to complex geometries.

math.NA↗

arXiv2024

Efficient Proton Transport Modelling for Proton Beam Therapy and Biological Quantification

In this work, we present a fundamental mathematical model for proton transport, tailored to capture the key physical processes underpinning Proton Beam Therapy (PBT). The model provides a robust and computationally efficient framework for exploring various aspects of PBT, including dose delivery, linear energy transfer, treatment planning and the evaluation of relative biological effectiveness. Our findings highlight the potential of this model as a complementary tool to more complex and computationally intensive simulation techniques currently used in clinical practice.

physics.med-ph↗

arXiv2024

A Deep Uzawa-Lagrange Multiplier Approach for Boundary Conditions in PINNs and Deep Ritz Methods

We introduce a deep learning-based framework for weakly enforcing boundary conditions in the numerical approximation of partial differential equations. Building on existing physics-informed neural network and deep Ritz methods, we propose the Deep Uzawa algorithm, which incorporates Lagrange multipliers to handle boundary conditions effectively. This modification requires only a minor computational adjustment but ensures enhanced convergence properties and provably accurate enforcement of boundary conditions, even for singularly perturbed problems. We provide a comprehensive mathematical analysis demonstrating the convergence of the scheme and validate the effectiveness of the Deep Uzawa algorithm through numerical experiments, including high-dimensional, singularly perturbed problems and those posed over non-convex domains.

math.NA↗

arXiv2024

Deep Uzawa for PDE constrained optimisation

In this work, we present a numerical solver for optimal control problems constrained by linear and semi-linear second-order elliptic PDEs. The approach is based on recasting the problem and includes an extension of Uzawa's algorithm to build approximating sequences for these constrained optimal control problems. We prove strong convergence of the iterative scheme in their respective norms, and this convergence is generalised to a class of restricted function spaces. We showcase the algorithm by demonstrating its use numerically with neural network methods that we coin Deep Uzawa Algorithms and show they perform favourably compared with some existing Deep Neural Network approaches.

math.NA↗

arXiv2024

Duality based error control for the Signorini problem

In this paper we study the a posteriori bounds for a conforming piecewise linear finite element approximation of the Signorini problem. We prove new rigorous a posteriori estimates of residual type in $L^{p}$, for $p \in (4,\infty)$ in two spatial dimensions. This new analysis treats the positive and negative parts of the discretisation error separately, requiring a novel sign- and bound-preserving interpolant, which is shown to have optimal approximation properties. The estimates rely on the sharp dual stability results on the problem in $W^{2,(4 - \varepsilon)/3}$ for any $\varepsilon \ll 1$. We summarise extensive numerical experiments aimed at testing the robustness of the estimator to validate the theory.

math.NA↗

arXiv2024

Breakup of oil slicks on wavy water surfaces

We hypothesize that the spread of oil slicks on the water's surface during oil spills is significantly influenced by water wave motion at the initial or intermediate spreading stages, well before emulsification processes have a substantial impact on the oil film's state. We demonstrate that the spreading dynamics of an oil slick on the water surface are facilitated by water waves, employing the thin film approximation. It is shown that water wave motion can rapidly deplete any oil slick, reducing the oil layer's thickness to nearly zero. This mechanism may act as a precursor to emulsification processes, leading to the accelerated depletion of oil spills into a distribution of droplets that form an emulsion.

physics.flu-dyn↗

arXiv2024

Discretisation of an Oldroyd-B viscoelastic fluid flow using a Lie derivative formulation

In this article we present a numerical method for the Stokes flow of an Oldroyd-B fluid. The viscoelastic stress evolves according to a constitutive law formulated in terms of the upper convected time derivative. A finite difference method is used to discretise along fluid trajectories to approximate the advection and deformation terms of the upper convected derivative in a simple, cheap and cohesive manner, as well as ensuring that the discrete conformation tensor is positive definite. A full implementation with coupling to the fluid flow is presented, along with detailed discussion of the issues that arise with such schemes. We demonstrate the performance of this method with detailed numerical experiments in a lid-driven cavity setup. Numerical results are benchmarked against published data, and the method is shown to perform well in this challenging case.

math.NA↗

arXiv2023

A nodally bound-preserving finite element method for reaction-convection-diffusion equations

This paper introduces a novel approach to approximate a broad range of reaction-convection-diffusion equations using conforming finite element methods while providing a discrete solution respecting the physical bounds given by the underlying differential equation. The main result of this work demonstrates that the numerical solution achieves accuracy of $O(h^k)$ in the energy norm, where $k$ represents the underlying polynomial degree. To validate the approach, a series of numerical experiments is conducted for various problem instances. Comparisons with the linear continuous interior penalty stabilised method, and the algebraic flux-correction scheme (for the piecewise linear finite element case) have been carried out, where we can observe the favourable performance of the current approach.

math.NA↗

arXiv2023

A Bayesian Inverse Approach to Proton Therapy Dose Delivery Verification

This study presents a proof-of-concept for a novel Bayesian inverse method in a one-dimensional setting, aimed at proton beam therapy treatment verification. Our methodology is predicated on a hypothetical scenario wherein strategically positioned sensors detect prompt-γ's emitted from a proton beam when it interacts with defined layers of tissue. Using this data, we employ a Bayesian framework to estimate the proton beam's energy deposition profile. We validate our Bayesian inverse estimations against a closed-form approximation of the Bragg Peak in a uniform medium and a layered lung tumour.

stat.AP↗

arXiv2023

A Nodally Bound-Preserving Finite Element Method

This work proposes a nonlinear finite element method whose nodal values preserve bounds known for the exact solution. The discrete problem involves a nonlinear projection operator mapping arbitrary nodal values into bound-preserving ones and seeks the numerical solution in the range of this projection. As the projection is not injective, a stabilisation based upon the complementary projection is added in order to restore well-posedness. Within the framework of elliptic problems, the discrete problem may be viewed as a reformulation of a discrete obstacle problem, incorporating the inequality constraints through Lipschitz projections. The derivation of the proposed method is exemplified for linear and nonlinear reaction-diffusion problems. Near-best approximation results in suitable norms are established. In particular, we prove that, in the linear case, the numerical solution is the best approximation in the energy norm among all nodally bound-preserving finite element functions. A series of numerical experiments for such problems showcase the good behaviour of the proposed bound-preserving finite element method.

math.NA↗

arXiv2021

Conservative Galerkin methods for dispersive Hamiltonian problems

An energy conservative discontinuous Galerkin scheme for a generalised third order KdV type equation is designed. Based on the conservation principle, we propose techniques that allow for the derivation of optimal a priori bounds for the linear KdV equation and a posteriori bounds for the linear and modified KdV equation. Extensive numerical experiments showcasing the good long time behaviour of the scheme are summarised which are in agreement with the analysis proposed.

math.NA↗