The Burgers' equation with stochastic transport: shock formation, local and global existence of smooth solutions

In this work, we examine the solution properties of the Burgers' equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine-Hugoniot condition that the shocks satisfy. Next, we establish the local existence and uniqueness of smooth solutions in the inviscid case and construct a blow-up criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions

Transport noise restores uniqueness and prevents blow-up in geometric transport equations

In this work, we demonstrate well-posedness and regularisation by noise results for a class of geometric transport equations that contains, among others, the linear transport and continuity equations. This class is known as linear advection of $k$-forms. In particular, we prove global existence and uniqueness of $L^p$-solutions to the stochastic equation, driven by a spatially $\alpha$-H\"older drift $b$, uniformly bounded in time, with an integrability condition on the distributional derivative of $b$, and sufficiently regular diffusion vector fields. Furthermore, we prove that all our solutions are continuous if the initial datum is continuous. Finally, we show that our class of equations without noise admits infinitely many $L^p$-solutions and is hence ill-posed. Moreover, the deterministic solutions can be discontinuous in both time and space independently of the regularity of the initial datum. We also demonstrate that for certain initial data of class $C^\infty_{0},$ the deterministic $L^p$-solutions blow up instantaneously in the space $L^{\infty}_{loc}$. In order to establish our results, we employ characteristics-based techniques that exploit the geometric structure of our equations

Iterative State Estimation in Non-linear Dynamical Systems Using Approximate Expectation Propagation

Bayesian inference in non-linear dynamical systems seeks to find good posterior approximations of a latent state given a sequence of observations. Gaussian filters and smoothers, including the (extended/unscented) Kalman filter/smoother, which are commonly used in engineering applications, yield Gaussian posteriors on the latent state. While they are computationally efficient, they are often criticised for their crude approximation of the posterior state distribution. In this paper, we address this criticism by proposing a message passing scheme for iterative state estimation in non-linear dynamical systems, which yields more informative (Gaussian) posteriors on the latent states. Our message passing scheme is based on expectation propagation (EP). We prove that classical Rauch--Tung--Striebel (RTS) smoothers, such as the extended Kalman smoother (EKS) or the unscented Kalman smoother (UKS), are special cases of our message passing scheme. Running the message passing scheme more than once can lead to significant improvements of the classical RTS smoothers, so that more informative state estimates can be obtained. We address potential convergence issues of EP by generalising our state estimation framework to damped updates and the consideration of general alpha-divergences

Gaussian Processes on Cellular Complexes

In recent years, there has been considerable interest in developing machine learning models on graphs in order to account for topological inductive biases. In particular, recent attention was given to Gaussian processes on such structures since they can additionally account for uncertainty. However, graphs are limited to modelling relations between two vertices. In this paper, we go beyond this dyadic setting and consider polyadic relations that include interactions between vertices, edges and one of their generalisations, known as cells. Specifically, we propose Gaussian processes on cellular complexes, a generalisation of graphs that captures interactions between these higher-order cells. One of our key contributions is the derivation of two novel kernels, one that generalises the graph Mat\'ern kernel and one that additionally mixes information of different cell types

Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels

Gaussian processes are machine learning models capable of learning unknown functions in a way that represents uncertainty, thereby facilitating construction of optimal decision-making systems. Motivated by a desire to deploy Gaussian processes in novel areas of science, a rapidly-growing line of research has focused on constructively extending these models to handle non-Euclidean domains, including Riemannian manifolds, such as spheres and tori. We propose techniques that generalize this class to model vector fields on Riemannian manifolds, which are important in a number of application areas in the physical sciences. To do so, we present a general recipe for constructing gauge independent kernels, which induce Gaussian vector fields, i.e. vector-valued Gaussian processes coherent withgeometry, from scalar-valued Riemannian kernels. We extend standard Gaussian process training methods, such as variational inference, to this setting. This enables vector-valued Gaussian processes on Riemannian manifolds to be trained using standard methods and makes them accessible to machine learning practitioners

Actually Sparse Variational Gaussian Processes

Gaussian processes (GPs) are typically criticised for their unfavourable scaling in both computational and memory requirements. For large datasets, sparse GPs reduce these demands by conditioning on a small set of inducing variables designed to summarise the data. In practice however, for large datasets requiring many inducing variables, such as low-lengthscale spatial data, even sparse GPs can become computationally expensive, limited by the number of inducing variables one can use. In this work, we propose a new class of inter-domain variational GP, constructed by projecting a GP onto a set of compactly supported B-spline basis functions. The key benefit of our approach is that the compact support of the B-spline basis functions admits the use of sparse linear algebra to significantly speed up matrix operations and drastically reduce the memory footprint. This allows us to very efficiently model fast-varying spatial phenomena with tens of thousands of inducing variables, where previous approaches failed.Comment: 14 pages, 5 figures, published in AISTATS 202

Comparison of Various Methods of Assaying the Cytotoxic Effects of Ethanol on Human Hepatoblastomaells (HUH-6 Line)

The sensitivity of five kinds of cytotoxicity assays using ethanol on human hepatoblastoma cells (HUH-6 line), which were cultured as monolayers or spheroids, was compared. Ethanol was chosen as a test because it acts on cell membranes directly without being metabolized and exerts its cytotoxicity. The assay methods used were as follows: 3- (4, 5-dimethylthiazol-2-yl)-2, 5-diphenyltetrazolium bromide (MTT), lactate dehydrogenase (LDH), colony formation, cell growth and DNA assays. The sensitivity of the assays was: LDH &#60; DNA &#60; cell growth &#60; MTT &#60; colony formation. LDH assay had the advantage that the same culture could be used for multiple assays, but when a small number of cells were assayed, no significant increase in the release of LDH was detected in the assay cultures compared with the control cultures. Although the DNA and cell growth assays were more sensitive than the LDH assay, the extent of cell damage may be underestimated because the damaged cells and DNA present in the cultures are included in the assay samples. On the other hand, both MTT and colony formation assays showed a high sensitivity. The MTT assay was done within 24 h after ethanol was added to the cultures and was applicable to both monolayer and spheroid cultures, while the colony formation assay required 1-2 weeks and it was applicable only to monolayer cultures. Taken together, the MTT assay was the most suitable method to evaluate the cytotoxic effects of ethanol on HUH-6 cells cultured as either monolayers or spheroids.</p

Spheroid Cultures of Human Hepatoblastoma Cells (HuH-6 Line) and Their Application for Cytotoxicity Assay of Alcohols

&#60;P&#62;Spheroid cultures of human hepatoblastoma cells (HuH-6 line) were established by rotating 3 x 10(6) cells/3 ml culture medium in 25-ml Erlenmeyer flasks on a gyratory shaker. The size of the spheroids rapidly increased until 4 days of culture, and thereafter their size gradually increased until 8 days of culture. A considerable amount of lactate dehydrogenase (LDH) was detected in the culture medium at 24h after seeding because of cell damage by subculturing, but thereafter the amount released was small, indicating that the spheroids were in healthy condition. Albumin production, one of the differentiated functions of hepatocytes, was higher in spheroid cultures than in monolayer cultures. Using this spheroid culture model, the cytotoxic effects of alcohols on HuH-6 cells were studied by measuring the activity of LDH released in the medium from damaged cells. The results indicate that the increasing order of toxicity of the alcohols was as follows: methanol &#60; ethanol &#60; propanol.&#60;/P&#62;</p