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Abstract: We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, including operators defined on networks of finite intervals, in which the coefficients of the differential operator may vary between subintervals, and arbitrary interface and boundary conditions may be imposed; differential operators with piecewise constant coefficients are thus included.

See abstract PDF file.

Abstract: I will discuss finite-time blowup for a model of the 3D Euler equation where the Helmholtz projection is replaced by a projection onto a more restrictive constraint space. The model preserves the structure of the self-advection nonlinearity entirely, and also conserves energy and helicity.

Abstract: The COVID-19 pandemic has placed an unprecedented strain on healthcare systems worldwide, making it difficult to accurately predict demand surge for acute care and allocate resources effectively. This paper proposes a novel predictive learning model that combines a differential equation model with delays and a multi-objective learning genetic algorithm to forecast hospitalizations and intensive care admissions based on predicted infections in a given population. Using data from Canada, Alberta, and Edmonton, we demonstrate the effectiveness of our model in significantly improving the ability to predict acute care capacity and manage resources during emergent infectious disease outbreaks such as COVID-19. Our approach is tailored to different population segments and considers various factors, such as variant strains of the virus and vaccination campaigns. Our model represents a significant contribution to the field, providing healthcare managers and decision-makers with a powerful tool to plan resources in the face of epidemics and major disasters. We contribute to the conversation on how we respond to public health emergencies, enabling more accurate predictions of demand for care and allocation of resources, ultimately saving lives and limiting the impact of these crises on our communities.

This is a joint work with Y. Bahri, N. Gilani, S. Ibrahim, A. Park and O. Chakroun

Abstract: One of the challenges of the accurate simulation of turbulent flows is that initial data is often incomplete, which is a significant difficulty when modeling chaotic systems whose solutions are sensitive to initial conditions. If one instead has snapshots of a system, i.e. data, one can make a better guess at the true state by incorporating the data via data assimilation. Many of the most popular data assimilation methods were developed for general physical systems, not just turbulent flows or chaotic systems. However, in the context of fluids, data assimilation works better than would be anticipated for a general physical system. In particular, turbulent fluid flows have been proven to have the property that, given enough perfect observations, one can recover the full state irrespective of the choice of initial condition. This property is unique to turbulent fluid flows, a consequence of their finite dimensionality. In this presentation, we will discuss the continuous data assimilation algorithm that was used to prove the convergence in the original, perfect data setting, present various robustness results of the continuous data assimilation algorithm, and discuss how continuous data assimilation can be used to identify and correct model error. Moreover, we will highlight the efforts of our other current research into long-standing problems in the stability of fluid flows, and how we have discovered some very interesting connections to our data assimilation research.

The aim of this talk is to present some theoretical and numerical results concerning Beltrami fields, called force-free fields in solar and fusion physics. The talk will be divided into three parts. In the first part, we will recall the physical context of the appearance of these fields, i .e. essentially the modeling of magnetic equilibria in solar physics. In the second part, we will recall some mathematical theoretical results concerning the existence and properties of fields in any three-dimensional domain. In the third and last part, we will expose some methods to compute these 3D fields and the associated physical equilibria, as well as some numerical results obtained with our codes.