arXiv subjects

Liam Burke

Publications and source records attributed to Liam Burke.

8 recordsLinked to original sources

arXiv2025

Extinction Coefficients of CdSe, CdS, and CdTe Nanoplatelets in Solution: A Practical Tool for Concentration Determination

Semiconductor nanoplatelets possess exceptional optical properties that make them promising candidates for next-generation optoelectronic applications. However, unlike quantum dots where absorption spectroscopy alone can determine both size and concentration, nanoplatelets present a significant characterization challenge: the absorption peak position reveals only thickness, providing no information about lateral dimensions or concentration. This limitation forces researchers to rely on time-consuming and costly elemental analysis techniques for complete sample characterization. Here, we present an experimentally verified theoretical framework that predicts the frequency-dependent absorption coefficient of randomly oriented CdSe, CdS, and CdTe nanoplatelets, enabling concentration determination from absorption measurements and lateral size estimates. Our model shows that the integrated absorption coefficient depends universally on nanoplatelet surface area and thickness, yielding a practical tool to extract concentrations without laborious elemental analysis. This approach bridges the characterization gap between quantum dots and nanoplatelets, offering a streamlined method for rapid sample analysis that could accelerate nanoplatelet research and applications.

cond-mat.mes-hall↗

arXiv2025

On the numerical stability of sketched GMRES

We perform a backward stability analysis of preconditioned sketched GMRES [Nakatsukasa and Tropp, SIAM J. Matrix Anal. Appl, 2024] for solving linear systems $Ax=b$, and show that the backward stability at iteration $i$ depends on the conditioning of the Krylov basis $B_{1:i}$ as long as the condition number of $A B_{1:i}$ can be bounded by $1/O(u)$, where $u$ is the unit roundoff. Under this condition, we show that sketched GMRES is backward stable as long as the condition number of $B_{1:i}$ is not too large. Under additional assumptions, we then show that the stability of a restarted implementation of sketched GMRES can be independent of the condition number of $B_{1:i}$, and restarted sketched GMRES is backward stable. We also derive sharper bounds that better capture the attainable backward error especially for cases when the basis $B_{1:i}$ is very ill-conditioned, which has been observed in the literature but not yet explained theoretically. We present numerical experiments to demonstrate the conclusions of our analysis, and also show that adaptively restarting where appropriate allows us to recover backward stability in sketched GMRES.

math.NA↗

arXiv2024

DoDo Learning: DOmain-DemOgraphic Transfer in Language Models for Detecting Abuse Targeted at Public Figures

Public figures receive a disproportionate amount of abuse on social media, impacting their active participation in public life. Automated systems can identify abuse at scale but labelling training data is expensive, complex and potentially harmful. So, it is desirable that systems are efficient and generalisable, handling both shared and specific aspects of online abuse. We explore the dynamics of cross-group text classification in order to understand how well classifiers trained on one domain or demographic can transfer to others, with a view to building more generalisable abuse classifiers. We fine-tune language models to classify tweets targeted at public figures across DOmains (sport and politics) and DemOgraphics (women and men) using our novel DODO dataset, containing 28,000 labelled entries, split equally across four domain-demographic pairs. We find that (i) small amounts of diverse data are hugely beneficial to generalisation and model adaptation; (ii) models transfer more easily across demographics but models trained on cross-domain data are more generalisable; (iii) some groups contribute more to generalisability than others; and (iv) dataset similarity is a signal of transferability.

cs.CL↗

arXiv2023

GMRES with randomized sketching and deflated restarting

We present a new Krylov subspace recycling method for solving a linear system of equations, or a sequence of slowly changing linear systems. Our approach is to reduce the computational overhead of recycling techniques while still benefiting from the acceleration afforded by such techniques. As such, this method augments an unprojected Krylov subspace. Furthermore, it combines randomized sketching and deflated restarting in a way that avoids orthogononalizing a full Krylov basis. We call this new method GMRES-SDR (sketched deflated restarting). With this new method, we provide new theory, which initially characterizes unaugmented sketched GMRES as a projection method for which the projectors involve the sketching operator. We demonstrate that sketched GMRES and its sibling method sketched FOM are an MR/OR pairing, just like GMRES and FOM. We furthermore obtain residual convergence estimates. Building on this, we characterize GMRES-SDR also in terms of sketching-based projectors. Compression of the augmented Krylov subspace for recycling is performed using a sketched version of harmonic Ritz vectors. We present results of numerical experiments demonstrating the effectiveness of GMRES-SDR over competitor methods such as GMRES-DR and GCRO-DR.

math.NA↗

arXiv2023

Krylov Subspace Recycling With Randomized Sketching For Matrix Functions

A Krylov subspace recycling method for the efficient evaluation of a sequence of matrix functions acting on a set of vectors is developed. The method improves over the recycling methods presented in [Burke et al., arXiv:2209.14163, 2022] in that it uses a closed-form expression for the augmented FOM approximants and hence circumvents the use of numerical quadrature. We further extend our method to use randomized sketching in order to avoid the arithmetic cost of orthogonalizing a full Krylov basis, offering an attractive solution to the fact that recycling algorithms built from shifted augmented FOM cannot easily be restarted. The efficacy of the proposed algorithms is demonstrated with numerical experiments.

math.NA↗

arXiv2023

Augmented unprojected Krylov subspace methods

Augmented Krylov subspace methods aid in accelerating the convergence of a standard Krylov subspace method by including additional vectors in the search space. A residual projection framework based on residual (Petrov-) Galerkin constraints was presented in [Gaul et al. SIAM J. Matrix Anal. Appl 2013], and later generalised in a recent survey on subspace recycling iterative methods [Soodhalter et al. GAMM-Mitt. 2020]. The framework describes augmented Krylov subspace methods in terms of applying a standard Krylov subspace method to an appropriately projected problem. In this work we show that the projected problem has an equivalent unprojected formulation, and that viewing the framework in this way provides a similar description for the class of unprojected augmented Krylov subspace methods. We derive the first unprojected augmented Full Orthogonalization Method (FOM), and demonstrate its effectiveness as a recycling method. We then show how the R$^{3}$GMRES algorithm fits within the framework. We show that unprojected augmented short recurrence methods fit within the framework, but can only be implemented in practice under certain conditions on the augmentation subspace. We demonstrate this using the Augmented Conjugate Gradient (AugCG) algorithm as an example.

math.NA↗

arXiv2022

Krylov Subspace Recycling For Matrix Functions

We derive an augmented Krylov subspace method with subspace recycling for computing a sequence of matrix function applications on a set of vectors. The matrix is either fixed or changes as the sequence progresses. We assume consecutive matrices are closely related, but make no assumptions on the relationship between the vectors. We present three versions of the method with different practical implementations. We demonstrate the effectiveness of the method using a range of numerical experiments with a selection of functions and matrices. We primarily focus our attention on the sign function arising in the overlap formalism of lattice QCD.

math.NA↗

arXiv2022

Unprojected recycled block Krylov subspace methods for shifted systems

The use of block Krylov subspace methods for computing the solution to a sequence of shifted linear systems using subspace recycling was first proposed in [Soodhalter, SISC 2016], where a recycled shifted block GMRES algorithm (rsbGMRES) was proposed. Such methods use the equivalence of the shifted system to a Sylvester equation and exploit the shift invariance of the block Krylov subspace generated from the Sylvester operator. This avoids the need for initial residuals to span the same subspace and allows for a viable restarted Krylov subspace method with recycling for solving sequences of shifted systems. In this paper we propose to develop these types of methods using unprojected Krylov subspaces. In doing so we show how one can overcome the difficulties associated with developing methods based on projected Krylov subspaces such as rsbGMRES, while also allowing for practical methods to fit within a well known residual projection framework. In addition, unprojected methods are known to be advantageous when the projector is expensive to apply, making them of significant interest for High-Performance Computing applications. We develop an unprojected rsbFOM and unprojected rsbGMRES. We also develop a procedure for extracting shift dependent harmonic Ritz vectors over an augmented block Krylov subspace for shifted systems yielding an approach for selecting a new recycling subspace after each cycle of the algorithm. Numerical experiments demonstrate the effectiveness of our methods.

math.NA↗