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Randomised linear algebra for scientific computing

Our group specializes in integrating Randomised Numerical Linear Algebra (RNLA) with the Finite Element Method (FEM) to solve large-scale parametric elliptic PDEs in real time. In our foundational work, Lung et al. (2019), we proposed a sketched FEM framework that projects solutions onto a low-dimensional subspace and applies randomised statistical leverage-score sampling. This projection not only delivers multi-order speedups but naturally regularises the reduced system against sketching errors.

To expand these scaling capabilities, Wu and Polydorides (2020) introduced a Multilevel Monte Carlo (MLMC) estimator for matrix multiplication by treating sampling index sizes analogously to grid refinements to drastically decrease the variance of high-dimensional products. Most recently, in Polydorides et al. (2026), we designed a low-variance approach tailored for digital twins and multi-query systems. By pairing parameter-oblivious leverage-score sampling with a control variates scheme and fusing forward and inverse sketches into a regularised estimator, this architecture significantly dampens statistical error bounds while preserving the structural stability of the underlying FEM formulation.

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Atmospheric dispersion imaging with off-beam optical backscatter

We are developing a method for fast detection and tomographic imaging of chemicals dispersed in the atmosphere, enabling near real-time 3D tracking of gas plumes that remains impossible under weak-signal constraints in current setups. As outlined in the SIAM work by Lung and Polydorides (2024), our approach utilises wider fields-of-view in the detector to explicitly leverage high-order multiple scattering responses from standard pulsed laser sources. To manage the added complexity of unknown photon trajectories and radiative transfer modeling, we exploit properties of turbulent dynamics within atmospheric plumes which naturally regularise and reduce the dimensionality of the inverse problem. By collecting light from oblique directions that penetrates deeper into the plume where interactions are strongest, we achieve a highly efficient acquisition process, especially when dealing with hard-to-detect chemical species.

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Chemical species tomography for combustion diagnostics

Building upon our previous contributions to chemical species tomography for aviation gas turbine and jet exhausts, we develop computational frameworks to reconstruct gas profiles under severely limited data constraints. Our earlier work established highly efficient approaches for mapping chemical species in high-temperature, high-velocity flows, including the industrial multi-channel tomography setups validated in Polydorides et al. (2018) and the sparse-data inversion techniques presented in Polydorides et al. (2016). Extending these principles, our frameworks have been successfully validated with real experimental data, as demonstrated in Upadhyay et al. (2022), leveraging underlying structural regularities to capture cross-sectional concentration profiles. Most recently, by fusing Large Eddy Simulation (LES) ground truths with a continuous Polar Fourier Transform (PFT) formulation and the Fourier Slice Theorem, we have developed a truncated series framework capable of reconstructing entire profiles from as few as four projection angles. This allows for the precise, real-time mapping of complex turbulent structures within combustor environments via low-dimensional angular coefficients, entirely bypassing dense, pixel-by-pixel grid inversions.

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Data analytics for Laser Powder Bed Fusion (Additive Manufacturing)

Our research in this domain accelerates the numerical analysis and monitoring of Laser Powder Bed Fusion (LPBF) to make high-fidelity process insights practical for online manufacturing control. In a pair of core journal publications, including our work on Gaussian process emulation in Li and Polydorides (2022) published in Additive Manufacturing, and our model order reduction framework in Li and Polydorides (2023) published in Computer Methods in Applied Mechanics and Engineering, we developed fast surrogate models designed to expedite the computationally intensive non-linear heat transfer simulations that govern powder-to-solid phase transitions. Exploiting the smooth properties of the heat kernel, these frameworks enable rapid estimation of transient melt pool geometry and local thermal anisotropy. Bringing these capabilities into manufacturing settings, our group is currently collaborating with the National Manufacturing Institute Scotland (NMIS) under a High Value Manufacturing Catapult Researcher in Residence project. This current trajectory focuses on Scientific Machine Learning (SciML) paradigms, blending physics-informed constraints with data-driven modeling to achieve in-situ real-time anomaly detection and quality assurance from Optical Emission Spectroscopy (OES) data.

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Grants and funded projects