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Many stochastic processes in the physical and biological sciences can be modelled using Brownian dynamics with multiplicative noise. However, numerical integrators for these processes can lose accuracy or even fail to converge when the diffusion term is configuration-dependent. One remedy is to construct a transform to a constant-diffusion process and sample the transformed process instead. In this work, we explain how coordinate-based and time-rescaling-based transforms can be used either individually or in combination to map a general class of variable-diffusion Brownian motion processes into constant-diffusion ones. The transforms are invertible, thus allowing recovery of the original dynamics. We motivate our methodology using examples in one dimension before then considering multivariate diffusion processes. We illustrate the benefits of the transforms through numerical simulations, demonstrating how the right combination of integrator and transform can improve computational efficiency and the order of convergence to the invariant distribution. Notably, the transforms that we derive are applicable to a class of multibody, anisotropic Stokes-Einstein diffusion that has applications in biophysical modelling.

Many optimization problems in electrical engineering consider a large number of design parameters. A sensitivity analysis identifies the design parameters with the strongest influence on the problem of interest. This paper introduces the adjoint variable method as an efficient approach to study sensitivities of nonlinear electroquasistatic problems in time domain. In contrast to the more common direct sensitivity method, the adjoint variable method has a computational cost nearly independent of the number of parameters. The method is applied to study the sensitivity of the field grading material parameters on the performance of a 320 kV cable joint specimen, which is modeled as a Finite Element nonlinear transient electroquasistatic problem. Special attention is paid to the treatment of quantities of interest, which are evaluated at specific points in time or space. It is shown that shown that the method is a valuable tool to study this strongly nonlinear and highly transient technical example.

The increasing availability of high-dimensional, longitudinal measures of genetic expression can facilitate analysis of the biological mechanisms of disease and prediction of future trajectories, as required for precision medicine. Biological knowledge suggests that it may be best to describe complex diseases at the level of underlying pathways, which may interact with one another. We propose a Bayesian approach that allows for characterising such correlation among different pathways through Dependent Gaussian Processes (DGP) and mapping the observed high-dimensional gene expression trajectories into unobserved low-dimensional pathway expression trajectories via Bayesian Sparse Factor Analysis. Compared to previous approaches that model each pathway expression trajectory independently, our model demonstrates better performance in recovering the shape of pathway expression trajectories, revealing the relationships between genes and pathways, and predicting gene expressions (closer point estimates and narrower predictive intervals), as demonstrated in the simulation study and real data analysis. To fit the model, we propose a Monte Carlo Expectation Maximization (MCEM) scheme that can be implemented conveniently by combining a standard Markov Chain Monte Carlo sampler and an R package GPFDA (Konzen and others, 2021), which returns the maximum likelihood estimates of DGP parameters. The modular structure of MCEM makes it generalizable to other complex models involving the DGP model component. An R package has been developed that implements the proposed approach.

This paper presents a novel approach for optical flow control of Micro Air Vehicles (MAVs). The task is challenging due to the nonlinearity of optical flow observables. Our proposed Incremental Nonlinear Dynamic Inversion (INDI) control scheme incorporates an efficient data-driven method to address the nonlinearity. It directly estimates the inverse of the time-varying control effectiveness in real-time, eliminating the need for the constant assumption and avoiding high computation in traditional INDI. This approach effectively handles fast-changing system dynamics commonly encountered in optical flow control, particularly height-dependent changes. We demonstrate the robustness and efficiency of the proposed control scheme in numerical simulations and also real-world flight tests: multiple landings of an MAV on a static and flat surface with various tracking setpoints, hovering and landings on moving and undulating surfaces. Despite being challenged with the presence of noisy optical flow estimates and the lateral and vertical movement of the landing surfaces, the MAV is able to successfully track or land on the surface with an exponential decay of both height and vertical velocity at almost the same time, as desired.

We present compact semi-implicit finite difference schemes on structured grids for numerical solutions of the advection by an external velocity and by a speed in normal direction that are applicable in level set methods. The most involved numerical scheme is third order accurate for the linear advection with a space dependent velocity and unconditionally stable in the sense of von Neumann stability analysis. We also present a simple high-resolution scheme that gives a TVD (Total Variation Diminishing) approximation of the spatial derivative for the advected level set function. In the case of nonlinear advection, the semi-implicit discretization is proposed to linearize the problem. The compact form of implicit stencil in numerical schemes containing unknowns only in the upwind direction allows applications of efficient algebraic solvers like fast sweeping methods. Numerical tests to evolve a smooth and non-smooth interface and an example with a large variation of velocity confirm the good accuracy of the methods and fast convergence of the algebraic solver even in the case of very large Courant numbers.

The paradigm of self-supervision focuses on representation learning from raw data without the need of labor-consuming annotations, which is the main bottleneck of current data-driven methods. Self-supervision tasks are often used to pre-train a neural network with a large amount of unlabeled data and extract generic features of the dataset. The learned model is likely to contain useful information which can be transferred to the downstream main task and improve performance compared to random parameter initialization. In this paper, we propose a new self-supervision task called source identification (SI), which is inspired by the classic blind source separation problem. Synthetic images are generated by fusing multiple source images and the network's task is to reconstruct the original images, given the fused images. A proper understanding of the image content is required to successfully solve the task. We validate our method on two medical image segmentation tasks: brain tumor segmentation and white matter hyperintensities segmentation. The results show that the proposed SI task outperforms traditional self-supervision tasks for dense predictions including inpainting, pixel shuffling, intensity shift, and super-resolution. Among variations of the SI task fusing images of different types, fusing images from different patients performs best.

The paper establishes the strong convergence rates of a spatio-temporal full discretization of the stochastic wave equation with nonlinear damping in dimension one and two. We discretize the SPDE by applying a spectral Galerkin method in space and a modified implicit exponential Euler scheme in time. The presence of the super-linearly growing damping in the underlying model brings challenges into the error analysis. To address these difficulties, we first achieve upper mean-square error bounds, and then obtain mean-square convergence rates of the considered numerical solution. This is done without requiring the moment bounds of the full approximations. The main result shows that, in dimension one, the scheme admits a convergence rate of order $\tfrac12$ in space and order $1$ in time. In dimension two, the error analysis is more subtle and can be done at the expense of an order reduction due to an infinitesimal factor. Numerical experiments are performed and confirm our theoretical findings.

This paper proposes a novel approach to predict epidemiological parameters by integrating new real-time signals from various sources of information, such as novel social media-based population density maps and Air Quality data. We implement an ensemble of Convolutional Neural Networks (CNN) models using various data sources and fusion methodology to build robust predictions and simulate several dynamic parameters that could improve the decision-making process for policymakers. Additionally, we used data assimilation to estimate the state of our system from fused CNN predictions. The combination of meteorological signals and social media-based population density maps improved the performance and flexibility of our prediction of the COVID-19 outbreak in London. While the proposed approach outperforms standard models, such as compartmental models traditionally used in disease forecasting (SEIR), generating robust and consistent predictions allows us to increase the stability of our model while increasing its accuracy.

A series of experiments in stationary and moving passenger rail cars were conducted to measure removal rates of particles in the size ranges of SARS-CoV-2 viral aerosols, and the air changes per hour provided by existing and modified air handling systems. Such methods for exposure assessments are customarily based on mechanistic models derived from physical laws of particle movement that are deterministic and do not account for measurement errors inherent in data collection. The resulting analysis compromises on reliably learning about mechanistic factors such as ventilation rates, aerosol generation rates and filtration efficiencies from field measurements. This manuscript develops a Bayesian state space modeling framework that synthesizes information from the mechanistic system as well as the field data. We derive a stochastic model from finite difference approximations of differential equations explaining particle concentrations. Our inferential framework trains the mechanistic system using the field measurements from the chamber experiments and delivers reliable estimates of the underlying physical process with fully model-based uncertainty quantification. Our application falls within the realm of Bayesian ``melding'' of mechanistic and statistical models and is of significant relevance to environmental hygienists and public health researchers working on assessing performance of aerosol removal rates for rail car fleets.

The volume function V(t) of a compact set S\in R^d is just the Lebesgue measure of the set of points within a distance to S not larger than t. According to some classical results in geometric measure theory, the volume function turns out to be a polynomial, at least in a finite interval, under a quite intuitive, easy to interpret, sufficient condition (called ``positive reach'') which can be seen as an extension of the notion of convexity. However, many other simple sets, not fulfilling the positive reach condition, have also a polynomial volume function. To our knowledge, there is no general, simple geometric description of such sets. Still, the polynomial character of $V(t)$ has some relevant consequences since the polynomial coefficients carry some useful geometric information. In particular, the constant term is the volume of S and the first order coefficient is the boundary measure (in Minkowski's sense). This paper is focused on sets whose volume function is polynomial on some interval starting at zero, whose length (that we call ``polynomial reach'') might be unknown. Our main goal is to approximate such polynomial reach by statistical means, using only a large enough random sample of points inside S. The practical motivation is simple: when the value of the polynomial reach , or rather a lower bound for it, is approximately known, the polynomial coefficients can be estimated from the sample points by using standard methods in polynomial approximation. As a result, we get a quite general method to estimate the volume and boundary measure of the set, relying only on an inner sample of points and not requiring the use any smoothing parameter. This paper explores the theoretical and practical aspects of this idea.