arXiv subjects

Nick Polydorides

Publications and source records attributed to Nick Polydorides.

7 recordsLinked to original sources

arXiv2023

Imaging of atmospheric dispersion processes with Differential Absorption Lidar

We consider the inverse problem of fitting atmospheric dispersion parameters based on time-resolved back-scattered differential absorption Lidar (DIAL) measurements. The obvious advantage of light-based remote sensing modalities is their extended spatial range which makes them less sensitive to strictly local perturbations/modelling errors or the distance to the plume source. In contrast to other state-of-the-art DIAL methods, we do not make a single scattering assumption but rather propose a new type modality which includes the collection of multiply scattered photons from wider/multiple fields-of-view and argue that this data, paired with a time dependent radiative transfer model, is beneficial for the reconstruction of certain image features. The resulting inverse problem is solved by means of a semi-parametric approach in which the image is reduced to a small number of dispersion related parameters and high-dimensional but computationally convenient nuisance component. This not only allows us to effectively avoid a high-dimensional inverse problem but simultaneously provides a natural regularisation mechanism along with parameters which are directly related to the dispersion model. These can be associated with meaningful physical units while spatial concentration profiles can be obtained by means of forward evaluation of the dispersion process.

math.AP↗

arXiv2020

Learned Spectral Computed Tomography

Spectral Photon-Counting Computed Tomography (SPCCT) is a promising technology that has shown a number of advantages over conventional X-ray Computed Tomography (CT) in the form of material separation, artefact removal and enhanced image quality. However, due to the increased complexity and non-linearity of the SPCCT governing equations, model-based reconstruction algorithms typically require handcrafted regularisation terms and meticulous tuning of hyperparameters making them impractical to calibrate in variable conditions. Additionally, they typically incur high computational costs and in cases of limited-angle data, their imaging capability deteriorates significantly. Recently, Deep Learning has proven to provide state-of-the-art reconstruction performance in medical imaging applications while circumventing most of these challenges. Inspired by these advances, we propose a Deep Learning imaging method for SPCCT that exploits the expressive power of Neural Networks while also incorporating model knowledge. The method takes the form of a two-step learned primal-dual algorithm that is trained using case-specific data. The proposed approach is characterised by fast reconstruction capability and high imaging performance, even in limited-data cases, while avoiding the hand-tuning that is required by other optimisation approaches. We demonstrate the performance of the method in terms of reconstructed images and quality metrics via numerical examples inspired by the application of cardiovascular imaging.

eess.IV↗

arXiv2019

Application of Randomized Quadrature Formulas to the Finite Element Method for Elliptic Equations

The implementation of the finite element method for linear elliptic equations requires to assemble the stiffness matrix and the load vector. In general, the entries of this matrix-vector system are not known explicitly but need to be approximated by quadrature rules. If the coefficient functions of the differential operator or the forcing term are irregular, then standard quadrature formulas, such as the barycentric quadrature rule, may not be reliable. In this paper we investigate the application of two randomized quadrature formulas to the finite element method for such elliptic boundary value problems with irregular coefficient functions. We give a detailed error analysis of these methods, discuss their implementation, and demonstrate their capabilities in several numerical experiments.

math.NA↗

arXiv2019

A Sketched Finite Element Method for Elliptic Models

We consider a sketched implementation of the finite element method for elliptic partial differential equations on high-dimensional models. Motivated by applications in real-time simulation and prediction we propose an algorithm that involves projecting the finite element solution onto a low-dimensional subspace and sketching the reduced equations using randomised sampling. We show that a sampling distribution based on the leverage scores of a tall matrix associated with the discrete Laplacian operator, can achieve nearly optimal performance and a significant speedup. We derive an expression of the complexity of the algorithm in terms of the number of samples that are necessary to meet an error tolerance specification with high probability, and an upper bound for the distance between the sketched and the high-dimensional solutions. Our analysis shows that the projection not only reduces the dimension of the problem but also regularises the reduced system against sketching error. Our numerical simulations suggest speed improvements of two orders of magnitude in exchange for a small loss in the accuracy of the prediction.

math.NA↗

arXiv2019

A Multilevel Monte Carlo Estimator for Matrix Multiplication

Inspired by the latest developments in multilevel Monte Carlo (MLMC) methods and randomised sketching for linear algebra problems we propose a MLMC estimator for real-time processing of matrix structured random data. Our algorithm is particularly effective in handling high-dimensional inner products and matrix multiplication, in applications of image analysis and large-scale supervised learning.

math.NA↗

arXiv2019

A row-sampling-based randomised finite element method for elliptic partial differential equations

We consider a randomised implementation of the finite element method (FEM) for elliptic partial differential equations on high-dimensional models. This is motivated by applications where model predictions are essential for real-time process diagnostics. In these circumstances it is imperative to expedite prediction without a significant compromise in the model's fidelity, which in turn relies on the rapid assembly and solution of the associated system of equations typically at the many-query context. Our approach involves converting the solution of the linear, symmetric positive definite FEM system into an over-determined least squares problem, whose solution is then projected onto a low-dimensional subspace. The resulting low-dimensional system can be effectively sketched as a product of two high-dimensional matrices using a parameter-dependent non-uniform sampling distribution, utilising only a small subset of the model's parameters. Although different to the optimal sampling distributions based on the statistical leverage-scores of the rows of the matrices, we show that the distance between them shrinks for an appropriate choice of the projection subspace. For the approximate solution we bound the incurring errors due to the projection, subspace approximation and sketching and show that the overall error is dominated by the condition number of the projected stiffness matrix. Our approach is tested on simulations on the Dirichlet and Neumann problems for the steady-state diffusion equation. The results show that our approach has on average a tenfold improvement on the computational times compared to the classical deterministic framework at the expense of a moderately small approximation error.

math.NA↗

arXiv2012

High-order regularized regression in Electrical Impedance Tomography

We present a novel approach for the inverse problem in electrical impedance tomography based on regularized quadratic regression. Our contribution introduces a new formulation for the forward model in the form of a nonlinear integral transform, that maps changes in the electrical properties of a domain to their respective variations in boundary data. Using perturbation theory the transform is approximated to yield a high-order misfit unction which is then used to derive a regularized inverse problem. In particular, we consider the nonlinear problem to second-order accuracy, hence our approximation method improves upon the local linearization of the forward mapping. The inverse problem is approached using Newton's iterative algorithm and results from simulated experiments are presented. With a moderate increase in computational complexity, the method yields superior results compared to those of regularized linear regression and can be implemented to address the nonlinear inverse problem.

physics.geo-ph↗