Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences最新文献

The aim of this review is to discuss some applications of orthogonal polynomials in quantum information processing. The hope is to keep the paper self-contained so that someone wanting a brief introduction to the theory of orthogonal polynomials and continuous time quantum walks on graphs may find it in one place. In particular, we focus on the associated Jacobi operators and discuss how these can be used to detect perfect state transfer (PST). We also discuss how orthogonal polynomials have been used to give results which are analogous to those given by Karlin and McGregor when studying classical birth and death processes. Finally, we show how these ideas have been extended to quantum walks with more than nearest-neighbour interactions using exceptional orthogonal polynomials (XOPs). We also provide a (non-exhaustive) list of related open questions.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.

The chiral quantum walk is an emerging tool for state transfer as it helps circumvent the barrier of no-go theorems in quantum transport. Yet, it remains largely unexplored. We prove a conjecture that universal monogamy violations for perfect quantum transfer in large graphs require couplings beyond the simple imaginary [Formula: see text]. This motivates our constructions of new monogamy violations on sparse graph products. Then, we show a novel violation of periodicity and the first known examples of one-way perfect quantum transfer via transcendental couplings.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.

Recent progress in quantum computing shows the need to incorporate many branches of mathematics (graph theory, matrix theory, optimization, theory of orthogonal polynomials and more) into physics, computer science and chemistry. At the 2024 SIAM Quantum Intersections Convening, Bert de Jong (Lawrence Berkeley National Laboratory) gave a talk entitled 'Quantum Science Needs Mathematicians' (Report of the SIAM Quantum Intersections Convening. Integrating Mathematical Scientists into Quantum Research, 7-9 October 2024, Tysons, Virginia (doi:10.11337/25M1741017)), since despite the growing demand for research in these domains, the mathematical sciences community has remained largely disengaged from quantum research, with only a few isolated areas of active involvement. This issue brings together researchers from different areas of mathematics to show the relation between spectral graph theory, the theory of orthogonal polynomials and numerical analysis. This interconnectedness highlights the versatility and importance of these areas of mathematics in the context of quantum computing.This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.

This paper details the development of a rapid inverse approach to determine the yield and location of an explosion through trilateration of empirical laws for blast wave arrival time. A rigorous sensitivity analysis of measurement uncertainty is first performed. From this, a probabilistic framework is proposed that utilizes Monte Carlo sampling of datasets to mitigate the effects of the variability and uncertainties typically present in blast events. Subsequently, the trilateration method is successfully applied to two existing datasets. Analysing well-controlled small-scale laboratory experiments, charge mass is predicted within 6.3% of the true yield, and position within 3.65 charge radii of the true centre. Social media footage of the 2020 Beirut explosion is then used to assess performance against data collected under in-field conditions. The predicted yield of 0.52 kt_{[Formula: see text]} shows good agreement with the literature, and charge position is predicted to within the radius of the crater. Trilateration is shown to be able to rapidly and reliably determine explosive yield and centre, despite large levels of sensor noise. The sub-second computation time of this approach offers the possibility to better model and predict the damage and injury patterns immediately after an explosion, facilitating more effective disaster response planning.This article is part of the theme issue 'Frontiers of applied inverse problems in science and engineering'.

In this paper, we leverage the structured foundation of deep image prior to delve into the complexities of positron emission tomography (PET) image reconstruction. We aim to underscore the potential of deep learning in overcoming inherent challenges associated with PET imaging. Acknowledging the limitations of conventional supervised learning in this domain, we propose an innovative unsupervised approach employing deep neural networks to enhance PET reconstruction. A central focus of our study revolves around the spectral bias issue that arises during PET image reconstruction. To tackle this challenge, we introduce a comprehensive framework that incorporates Gaussian Fourier features and Uniform Positional encoding. Our approaches undergo rigorous testing on both Brainweb data and naive rat data, revealing a noticeable improvement in image reconstruction performance. This underscores the efficacy of our framework in advancing PET imaging methodologies.This article is part of the theme issue 'Frontiers of applied inverse problems in science and engineering'.

Microtexture regions (MTRs) are collections of grains with similar crystallographic orientation. When present in aerospace components, they can potentially limit component life. As such, a non-destructive evaluation (NDE) method to detect and characterize MTR is desired. One potential solution is to use an electromagnetic NDE method known as eddy current testing (ECT), which is sensitive to local conductivity variations associated with MTR. Recent work has shown that MTR boundaries and orientation can be determined from ECT data using a variant of matching component analysis (MCA) combined with a regularization method originally developed for image deblurring. However, this method has only been demonstrated on simulated ECT data. In this work, we apply the previously developed method to experimental ECT data of a large grain titanium specimen. We show that we are able to determine grain boundaries and orientation from experimental ECT data, serving as a first step to full MTR characterization.This article is part of the theme issue 'Frontiers of applied inverse problems in science and engineering'.

We propose a Coefficient-to-Basis Network (C2BNet), a novel framework for solving inverse problems within the operator learning paradigm. C2BNet efficiently adapts to different discretizations through fine-tuning, using a pre-trained model to significantly reduce computational cost while maintaining high accuracy. Unlike traditional approaches that require retraining from scratch for new discretizations, our method enables seamless adaptation without sacrificing predictive performance. Furthermore, we establish theoretical approximation and generalization error bounds for C2BNet by exploiting low-dimensional structures in the underlying datasets. Our analysis demonstrates that C2BNet adapts to low-dimensional structures without relying on explicit encoding mechanisms, highlighting its robustness and efficiency. To validate our theoretical findings, we conducted extensive numerical experiments that showcase the superior performance of C2BNet on several inverse problems. The results confirm that C2BNet effectively balances computational efficiency and accuracy, making it a promising tool to solve inverse problems in scientific computing and engineering applications.This article is part of the theme issue 'Frontiers of applied inverse problems in science and engineering'.

This paper examines the current challenges in computed tomography (CT), with a critical exploration of existing methodologies from a mathematical perspective. Specifically, it aims to identify research directions to enhance image quality in low-dose, cost-effective cone beam CT (CBCT) systems, which have recently gained widespread use in general dental clinics. Dental CBCT offers a substantial cost advantage over standard medical CT, making it affordable for local dental practices; however, this affordability brings significant challenges related to image quality degradation, further complicated by the presence of metallic implants, which are particularly common in older patients. This paper investigates metal-induced artefacts stemming from mismatches in the forward model used in conventional reconstruction methods and explains an alternative approach that bypasses the traditional Radon transform model. Additionally, it examines both the potential and limitations of deep learning-based methods in tackling these challenges, offering insights into their effectiveness in improving image quality in low-dose dental CBCT.This article is part of the theme issue 'Frontiers of applied inverse problems in science and engineering'.

The Volterra series has been used in nonlinear system identification (NLSI) for decades; its frequency-domain counterpart allows a generalization of 'resonance curves' for nonlinear systems-so-called higher-order frequency-response functions (HFRFs). Estimating the terms in the series has often proved to be a challenge; however, the (comparatively) recent uptake of machine-learning technology into engineering dynamics has led to advances in the identification of the series-both for the Volterra kernels themselves and for the HFRFs. The current paper provides an overview of a number of approaches based on neural networks, Gaussian processes (GPs) and reproducing kernel Hilbert spaces (RKHSs), and presents new results for multi-input multi-output (MIMO) systems based on neural networks.This article is part of the theme issue 'Frontiers of applied inverse problems in science and engineering'.

It has been hypothesized that during a motion task the central nervous system controls the skeletal muscles partitioning them into synergetic groups, hence effectively reducing the dimensionality of the control problem. The identification of muscle groups that are co-activated remains an open problem: its solution could have important implications in the design of training or rehabilitation protocols. In this article, we combine Bayesian inverse problem techniques and data science algorithms to identify muscle synergies in human motion from the motion tracker time series of positions of fiducial markers on the body during the task. The inverse problem of estimating the muscle activation patterns from the motion tracking data is cast in the Bayesian framework, and the posterior distribution of muscle activations is explored using Myobolica, a Gibbs-sampler-based Markov chain Monte Carlo sampler. A low-rank approximation of the muscle activation patterns is then obtained via a sparsity promoting Bayesian non-negative matrix factorization of the sample mean, where the sparse coefficient vectors correspond to groups of muscles that show co-activation over the sample.This article is part of the theme issue 'Frontiers of applied inverse problems in science and engineering'.