Tripwire Detection for Landmines

Pre-processed Radon transform-based detection methods were identified as the most viable technique. There are some issues that arose when testing the various methods. A) The 'log' edge detection in Matlab seemed better than the default 'sobel' method, but is still worth considering other edge detection algorithms that might deal more effectively with noisy or grainy image. B) The algorithm works best when the wire is horizontal and spans the length of the image. If constrained to thin-strip type images then a simple solution is to run the algorithm twice, once with the image and a second time with the same image rotated through 45 degrees

National Air Traffic Services

National Air Traffic Services (NATS) are concerned with ensuring low probabilities of errors in determining aircraft positions. In general, error probabilities depend on the tails of some probability distributions for which there has been no theoretical model. Analysis of radar performance is regularly undertaken by NATS to ensure radar performance is within safety limits, with the maximum range being dependent on the declared separation between aircraft. NATS brought two questions to the Study Group, involving the horizontal (azimuthal) errors in radar data and the vertical errors in altimetry system data. In both cases, NATS asked the Study Group to analyse the data and assess whether the probability distributions that are currently used are good models for the errors

Arc Phenomena in low-voltage current limiting circuit breakers

Circuit breakers are an important safety feature in most electrical circuits, and they act to prevent excessive currents caused by short circuits, for example. Low-voltage current limiting circuit breakers are activated by a trip solenoid when a critical current is exceeded. The solenoid moves two contacts apart to break the circuit. However, as soon as the contacts are separated an electric arc forms between them, ionising the air in the gap, increasing the electrical conductivity of air to that of the hot plasma that forms, and current continues to flow. The currents involved may be as large as 80,000 amperes. Critical to the success of the circuit breaker is that it is designed to cause the arc to move away from the contacts, into a widening wedge-shaped region. This lengthens the arc, and then moves it onto a series of separator plates called an arc divider or splitter. The arc divider raises the voltage required to sustain the arcs across it, above the voltage that is provided across the breaker, so that the circuit is broken and the arcing dies away. This entire process occurs in milliseconds, and is usually associated with a sound like an explosion and a bright ash from the arc. Parts of the contacts and the arc divider may melt and/or vapourise. The question to be addressed by the Study Group was to mathematically model the arc motion and extinction, with the overall aim of an improved understanding that would help the design of a better circuit breaker. Further discussion indicated that two key mechanisms are believed to contribute to the movement of the arc away from the contacts, one being self-magnetism (where the magnetic field associated with the arc and surrounding circuitry acts to push it towards the arc divider), and the other being air flow (where expansion of air combined with the design of the chamber enclosing the arc causes gas flow towards the arc divider). Further discussion also indicated that a key aspect of circuit breaker design was that it is desirable to have as fast a quenching of the arc as possible, that is, the faster the circuit breaker can act to stop current flow, the better. The relative importance of magnetic and air pressure effects on quenching speed is of central interest to circuit design

Moving Mesh Methods for Problems with Blow-Up

This is the published version, also available here: http://dx.doi.org/10.1137/S1064827594272025.In this paper we consider the numerical solution of PDEs with blow-up for which scaling invariance plays a natural role in describing the underlying solution structures. It is a challenging numerical problem to capture the qualitative behaviour in the blow-up region, and the use of nonuniform meshes is essential. We consider moving mesh methods for which the mesh is determined using so-called moving mesh partial differential equations (MMPDEs).Specifically, the underlying PDE and the MMPDE are solved for the blow-up solution and the computational mesh simultaneously. Motivated by the desire for the MMPDE to preserve the scaling invariance of the underlying problem, we study the effect of different choices of MMPDEs and monitor functions. It is shown that for suitable ones the MMPDE solution evolves towards a. (moving) mesh which close to the blow-up point automatically places the mesh points in such a manner that the ignition kernel, which is well known to be a natural coordinate in describing the behaviour of blow-up, approaches a constant as $t \to T$ (the blow-up time). Several numerical examples are given to verify the theory for these MMPDE methods and to illustrate their efficacy

An interpolation tool for aircraft surface pressure data

Airbus UK are concerned with designing efficient wings for aircraft. In the design process, the aerodynamic load on the wing is calculated for various configurations including different Mach numbers and angles of attack. The aerodynamic load is calculated from the pressure profile around the wing. Airbus use a number of different methods to calculate the pressure, primarily CFD calculations and wind tunnel experiments. However, experiments and calculations cannot be performed for all configurations. Airbus asked the Study Group to investigate interpolation methods which incorporate wind tunnel and CFD data to calculate the aerodynamic load for many different configurations

Closing the ODE-SDE gap in score-based diffusion models through the Fokker-Planck equation

Score-based diffusion models have emerged as one of the most promising frameworks for deep generative modelling, due to their state-of-the art performance in many generation tasks while relying on mathematical foundations such as stochastic differential equations (SDEs) and ordinary differential equations (ODEs). Empirically, it has been reported that ODE based samples are inferior to SDE based samples. In this paper we rigorously describe the range of dynamics and approximations that arise when training score-based diffusion models, including the true SDE dynamics, the neural approximations, the various approximate particle dynamics that result, as well as their associated Fokker--Planck equations and the neural network approximations of these Fokker--Planck equations. We systematically analyse the difference between the ODE and SDE dynamics of score-based diffusion models, and link it to an associated Fokker--Planck equation. We derive a theoretical upper bound on the Wasserstein 2-distance between the ODE- and SDE-induced distributions in terms of a Fokker--Planck residual. We also show numerically that conventional score-based diffusion models can exhibit significant differences between ODE- and SDE-induced distributions which we demonstrate using explicit comparisons. Moreover, we show numerically that reducing the Fokker--Planck residual by adding it as an additional regularisation term leads to closing the gap between ODE- and SDE-induced distributions. Our experiments suggest that this regularisation can improve the distribution generated by the ODE, however that this can come at the cost of degraded SDE sample quality