Probabilistic Gradients for Fast Calibration of Differential Equation Models

Calibration of large-scale differential equation models to observational or experimental data is a widespread challenge throughout applied sciences and engineering. A crucial bottleneck in state-of-the art calibration methods is the calculation of local sensitivities, i.e. derivatives of the loss function with respect to the estimated parameters, which often necessitates several numerical solves of the underlying system of partial or ordinary differential equations. In this paper we present a new probabilistic approach to computing local sensitivities. The proposed method has several advantages over classical methods. Firstly, it operates within a constrained computational budget and provides a probabilistic quantification of uncertainty incurred in the sensitivities from this constraint. Secondly, information from previous sensitivity estimates can be recycled in subsequent computations, reducing the overall computational effort for iterative gradient-based calibration methods. The methodology presented is applied to two challenging test problems and compared against classical methods

A Fourier representation of kernel Stein discrepancy with application to Goodness-of-Fit tests for measures on infinite dimensional Hilbert spaces

Kernel Stein discrepancy (KSD) is a widely used kernel-based measure of discrepancy between probability measures. It is often employed in the scenario where a user has a collection of samples from a candidate probability measure and wishes to compare them against a specified target probability measure. KSD has been employed in a range of settings including goodness-of-fit testing, parametric inference, MCMC output assessment and generative modelling. However, so far the method has been restricted to finite-dimensional data. We provide the first analysis of KSD in the generality of data lying in a separable Hilbert space, for example functional data. The main result is a novel Fourier representation of KSD obtained by combining the theory of measure equations with kernel methods. This allows us to prove that KSD can separate measures and thus is valid to use in practice. Additionally, our results improve the interpretability of KSD by decoupling the effect of the kernel and Stein operator. We demonstrate the efficacy of the proposed methodology by performing goodness-of-fit tests for various Gaussian and non-Gaussian functional models in a number of synthetic data experiments.Comment: To appear in Bernoull

Statistical Inference for Generative Models with Maximum Mean Discrepancy

While likelihood-based inference and its variants provide a statistically efficient and widely applicable approach to parametric inference, their application to models involving intractable likelihoods poses challenges. In this work, we study a class of minimum distance estimators for intractable generative models, that is, statistical models for which the likelihood is intractable, but simulation is cheap. The distance considered, maximum mean discrepancy (MMD), is defined through the embedding of probability measures into a reproducing kernel Hilbert space. We study the theoretical properties of these estimators, showing that they are consistent, asymptotically normal and robust to model misspecification. A main advantage of these estimators is the flexibility offered by the choice of kernel, which can be used to trade-off statistical efficiency and robustness. On the algorithmic side, we study the geometry induced by MMD on the parameter space and use this to introduce a novel natural gradient descent-like algorithm for efficient implementation of these estimators. We illustrate the relevance of our theoretical results on several classes of models including a discrete-time latent Markov process and two multivariate stochastic differential equation models

A High-dimensional Convergence Theorem for U-statistics with Applications to Kernel-based Testing

We prove a convergence theorem for U-statistics of degree two, where the data dimension d is allowed to scale with sample size n . We find that the limiting distribution of a U-statistic undergoes a phase transition from the non-degenerate Gaussian limit to the degenerate limit, regardless of its degeneracy and depending only on a moment ratio. A surprising consequence is that a non-degenerate U-statistic in high dimensions can have a non-Gaussian limit with a larger variance and asymmetric distribution. Our bounds are valid for any finite n and d , independent of individual eigenvalues of the underlying function, and dimension-independent under a mild assumption. As an application, we apply our theory to two popular kernel-based distribution tests, MMD and KSD, whose high-dimensional performance has been challenging to study. In a simple empirical setting, our results correctly predict how the test power at a fixed threshold scales with d and the bandwidth

System Effects in Identifying Risk-Optimal Data Requirements for Digital Twins of Structures

Structural Health Monitoring (SHM) technologies offer much promise to the risk management of the built environment, and they are therefore an active area of research. However, information regarding material properties, such as toughness and strength is instead measured in destructive lab tests. Similarly, the presence of geometrical anomalies is more commonly detected and sized by inspection. Therefore, a risk-optimal combination should be sought, acknowledging that different scenarios will be associated with different data requirements. Value of Information (VoI) analysis is an established statistical framework for quantifying the expected benefit of a prospective data collection activity. In this paper the expected value of various combinations of inspection, SHM and testing are quantified, in the context of supporting risk management of a location of stress concentration in a railway bridge. The Julia code for this analysis (probabilistic models and influence diagrams) is made available. The system-level results differ from a simple linear sum of marginal VoI estimates, i.e. the expected value of collecting data from SHM and inspection together is not equal to the expected value of SHM data plus the expected value of inspection data. In summary, system-level decision making, requires system-level models

Incorporating habitat distribution in wildlife disease models: conservation implications for the threat of squirrelpox on the Isle of Arran

Emerging infectious diseases are a substantial threat to native populations. The spread of disease through naive native populations will depend on both demographic and disease parameters, as well as on habitat suitability and connectivity. Using the potential spread of squirrelpox virus (SQPV) on the Isle of Arran as a case study, we develop mathematical models to examine the impact of an emerging disease on a population in a complex landscape of different habitat types. Furthermore, by considering a range of disease parameters, we infer more generally how complex landscapes interact with disease characteristics to determine the spread and persistence of disease. Specific findings indicate that a SQPV outbreak on Arran is likely to be short lived and localized to the point of introduction allowing recovery of red squirrels to pre-infection densities; this has important consequences for the conservation of red squirrels. More generally, we find that the extent of disease spread is dependent on the rare passage of infection through poor quality corridors connecting good quality habitats. Acute, highly transmissible infectious diseases are predicted to spread rapidly causing high mortality. Nonetheless, the disease typically fades out following local epidemics and is not supported in the long term. A chronic infectious disease is predicted to spread more slowly but can remain endemic in the population. This allows the disease to spread more extensively in the long term as it increases the chance of spread between poorly connected populations. Our results highlight how a detailed understanding of landscape connectivity is crucial when considering conservation strategies to protect native species from disease threats

Nanosecond laser texturing for high friction applications

AbstractA nanosecond pulsed Nd:YAG fibre laser with wavelength of 1064nm was used to texture several different steels, including grade 304 stainless steel, grade 316 stainless steel, Cr–Mo–Al ‘nitriding’ steel and low alloy carbon steel, in order to generate surfaces with a high static friction coefficient. Such surfaces have applications, for example, in large engines to reduce the tightening forces required for a joint or to secure precision fittings easily. For the generation of high friction textures, a hexagonal arrangement of laser pulses was used with various pulse overlaps and pulse energies. Friction testing of the samples suggests that the pulse energy should be high (around 0.8mJ) and the laser pulse overlap should be higher than 50% in order to achieve a static friction coefficient of more than 0.5. It was also noted that laser processing increases the surface hardness of samples which appears to correlate with the increase in friction. Energy-Dispersive X-ray spectroscopy (EDX) measurements indicate that this hardness is caused by the formation of hard metal-oxides at the material surface