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This paper presents an analysis of properties of two hybrid discretization methods for Gaussian derivatives, based on convolutions with either the normalized sampled Gaussian kernel or the integrated Gaussian kernel followed by central differences. The motivation for studying these discretization methods is that in situations when multiple spatial derivatives of different order are needed at the same scale level, they can be computed significantly more efficiently compared to more direct derivative approximations based on explicit convolutions with either sampled Gaussian kernels or integrated Gaussian kernels. While these computational benefits do also hold for the genuinely discrete approach for computing discrete analogues of Gaussian derivatives, based on convolution with the discrete analogue of the Gaussian kernel followed by central differences, the underlying mathematical primitives for the discrete analogue of the Gaussian kernel, in terms of modified Bessel functions of integer order, may not be available in certain frameworks for image processing, such as when performing deep learning based on scale-parameterized filters in terms of Gaussian derivatives, with learning of the scale levels. In this paper, we present a characterization of the properties of these hybrid discretization methods, in terms of quantitative performance measures concerning the amount of spatial smoothing that they imply, as well as the relative consistency of scale estimates obtained from scale-invariant feature detectors with automatic scale selection, with an emphasis on the behaviour for very small values of the scale parameter, which may differ significantly from corresponding results obtained from the fully continuous scale-space theory, as well as between different types of discretization methods.

When dealing with a large number of points was required, the traditional uniform sampling approach for approximating integrals using the Monte Carlo method becomes inefficient. In this work, we leverage the good lattice point sets from number-theoretic methods for sampling purposes and develop a deep learning framework that integrates the good lattice point sets with Physics-Informed Neural Networks. This framework is designed to address low-regularity and high-dimensional problems. Furthermore, rigorous mathematical proofs are provided for our algorithm, demonstrating its validity. Lastly, in the experimental section, we employ numerical experiments involving the Poisson equation with low regularity, the two-dimensional inverse Helmholtz equation, and high-dimensional linear and nonlinear problems to illustrate the effectiveness of our algorithm from a numerical perspective.

This work presents an abstract framework for the design, implementation, and analysis of the multiscale spectral generalized finite element method (MS-GFEM), a particular numerical multiscale method originally proposed in [I. Babuska and R. Lipton, Multiscale Model.\;\,Simul., 9 (2011), pp.~373--406]. MS-GFEM is a partition of unity method employing optimal local approximation spaces constructed from local spectral problems. We establish a general local approximation theory demonstrating exponential convergence with respect to local degrees of freedom under certain assumptions, with explicit dependence on key problem parameters. Our framework applies to a broad class of multiscale PDEs with $L^{\infty}$-coefficients in both continuous and discrete, finite element settings, including highly indefinite problems (convection-dominated diffusion, as well as the high-frequency Helmholtz, Maxwell and elastic wave equations with impedance boundary conditions), and higher-order problems. Notably, we prove a local convergence rate of $O(e^{-cn^{1/d}})$ for MS-GFEM for all these problems, improving upon the $O(e^{-cn^{1/(d+1)}})$ rate shown by Babuska and Lipton. Moreover, based on the abstract local approximation theory for MS-GFEM, we establish a unified framework for showing low-rank approximations to multiscale PDEs. This framework applies to the aforementioned problems, proving that the associated Green's functions admit an $O(|\log\epsilon|^{d})$-term separable approximation on well-separated domains with error $\epsilon>0$. Our analysis improves and generalizes the result in [M. Bebendorf and W. Hackbusch, Numerische Mathematik, 95 (2003), pp.~1-28] where an $O(|\log\epsilon|^{d+1})$-term separable approximation was proved for Poisson-type problems.

We design and investigate a variety of multigrid solvers for high-order local discontinuous Galerkin methods applied to elliptic interface and multiphase Stokes problems. Using the template of a standard multigrid V-cycle, we consider a variety of element-wise block smoothers, including Jacobi, multi-coloured Gauss-Seidel, processor-block Gauss-Seidel, and with special interest, smoothers based on sparse approximate inverse (SAI) methods. In particular, we develop SAI methods that: (i) balance the smoothing of velocity and pressure variables in Stokes problems; and (ii) robustly handles high-contrast viscosity coefficients in multiphase problems. Across a broad range of two- and three-dimensional test cases, including Poisson, elliptic interface, steady-state Stokes, and unsteady Stokes problems, we examine a multitude of multigrid smoother and solver combinations. In every case, there is at least one approach that matches the performance of classical geometric multigrid algorithms, e.g., 4 to 8 iterations to reduce the residual by 10 orders of magnitude. We also discuss their relative merits with regard to simplicity, robustness, computational cost, and parallelisation.

Recent surge in Large Language Model (LLM) availability has opened exciting avenues for research. However, efficiently interacting with these models presents a significant hurdle since LLMs often reside on proprietary or self-hosted API endpoints, each requiring custom code for interaction. Conducting comparative studies between different models can therefore be time-consuming and necessitate significant engineering effort, hindering research efficiency and reproducibility. To address these challenges, we present prompto, an open source Python library which facilitates asynchronous querying of LLM endpoints enabling researchers to interact with multiple LLMs concurrently, while maximising efficiency and utilising individual rate limits. Our library empowers researchers and developers to interact with LLMs more effectively and allowing faster experimentation, data generation and evaluation. prompto is released with an introductory video (//youtu.be/lWN9hXBOLyQ) under MIT License and is available via GitHub (//github.com/alan-turing-institute/prompto).

Detection of abrupt spatial changes in physical properties representing unique geometric features such as buried objects, cavities, and fractures is an important problem in geophysics and many engineering disciplines. In this context, simultaneous spatial field and geometry estimation methods that explicitly parameterize the background spatial field and the geometry of the embedded anomalies are of great interest. This paper introduces an advanced inversion procedure for simultaneous estimation using the domain independence property of the Karhunen-Lo\`eve (K-L) expansion. Previous methods pursuing this strategy face significant computational challenges. The associated integral eigenvalue problem (IEVP) needs to be solved repeatedly on evolving domains, and the shape derivatives in gradient-based algorithms require costly computations of the Moore-Penrose inverse. Leveraging the domain independence property of the K-L expansion, the proposed method avoids both of these bottlenecks, and the IEVP is solved only once on a fixed bounding domain. Comparative studies demonstrate that our approach yields two orders of magnitude improvement in K-L expansion gradient computation time. Inversion studies on one-dimensional and two-dimensional seepage flow problems highlight the benefits of incorporating geometry parameters along with spatial field parameters. The proposed method captures abrupt changes in hydraulic conductivity with a lower number of parameters and provides accurate estimates of boundary and spatial-field uncertainties, outperforming spatial-field-only estimation methods.

In Coevolving Latent Space Networks with Attractors (CLSNA) models, nodes in a latent space represent social actors, and edges indicate their dynamic interactions. Attractors are added at the latent level to capture the notion of attractive and repulsive forces between nodes, borrowing from dynamical systems theory. However, CLSNA reliance on MCMC estimation makes scaling difficult, and the requirement for nodes to be present throughout the study period limit practical applications. We address these issues by (i) introducing a Stochastic gradient descent (SGD) parameter estimation method, (ii) developing a novel approach for uncertainty quantification using SGD, and (iii) extending the model to allow nodes to join and leave over time. Simulation results show that our extensions result in little loss of accuracy compared to MCMC, but can scale to much larger networks. We apply our approach to the longitudinal social networks of members of US Congress on the social media platform X. Accounting for node dynamics overcomes selection bias in the network and uncovers uniquely and increasingly repulsive forces within the Republican Party.

To enhance the accuracy of power load forecasting in wind farms, this study introduces an advanced combined forecasting method that integrates Variational Mode Decomposition (VMD) with an Improved Particle Swarm Optimization (IPSO) algorithm to optimize the Extreme Learning Machine (ELM). Initially, the VMD algorithm is employed to perform high-precision modal decomposition of the original power load data, which is then categorized into high-frequency and low-frequency sequences based on mutual information entropy theory. Subsequently, this research profoundly modifies the traditional multiverse optimizer by incorporating Tent chaos mapping, exponential travel distance rate, and an elite reverse learning mechanism, developing the IPSO-ELM prediction model. This model independently predicts the high and low-frequency sequences and reconstructs the data to achieve the final forecasting results. Simulation results indicate that the proposed method significantly improves prediction accuracy and convergence speed compared to traditional ELM, PSO-ELM, and PSO-ELM methods.

In this work we develop an a posteriori error estimator for mixed finite element methods of Darcy flow problems with Robin-type jump interface conditions. We construct an energy-norm type a posteriori error estimator using the Stenberg post-processing. The reliability of the estimator is proved using an interface-adapted Helmholtz-type decomposition and an interface-adapted Scott--Zhang type interpolation operator. A local efficiency and the reliability of post-processed pressure are also proved. Numerical results illustrating adaptivity algorithms using our estimator are included.

When dealing with a large number of points was required, the traditional uniform sampling approach for approximating integrals using the Monte Carlo method becomes inefficient. In this work, we leverage the good lattice point sets from number-theoretic methods for sampling purposes and develop a deep learning framework that integrates the good lattice point sets with Physics-Informed Neural Networks. This framework is designed to address low-regularity and high-dimensional problems. Furthermore, rigorous mathematical proofs are provided for our algorithm, demonstrating its validity. Lastly, in the experimental section, we employ numerical experiments involving the Poisson equation with low regularity, the two-dimensional inverse Helmholtz equation, and high-dimensional linear and nonlinear problems to illustrate the effectiveness of our algorithm from a numerical perspective.