An embedding technique for the solution of reaction-fiffusion equations on algebraic surfaces with isolated singularities

In this paper we construct a parametrization-free embedding technique for numerically evolving reaction-diffusion PDEs defined on algebraic curves that possess an isolated singularity. In our approach, we first desingularize the curve by appealing to techniques from algebraic geometry.\ud We create a family of smooth curves in higher dimensional space that correspond to the original curve by projection. Following this, we pose the analogous reaction-diffusion PDE on each member of this family and show that the solutions (their projection onto the original domain) approximate the solution of the original problem. Finally, we compute these approximants numerically by applying the Closest Point Method which is an embedding technique for solving PDEs on smooth surfaces of arbitrary dimension or codimension, and is thus suitable for our situation. In addition, we discuss the potential to generalize the techniques presented for higher-dimensional surfaces with multiple singularities

A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces

The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. Recently, a closest point method with explicit time-stepping was proposed that uses finite differences derived from radial basis functions (RBF-FD). Here, we propose a least-squares implicit formulation of the closest point method to impose the constant-along-normal extension of the solution on the surface into the embedding space. Our proposed method is particularly flexible with respect to the choice of the computational grid in the embedding space. In particular, we may compute over a computational tube that contains problematic nodes. This fact enables us to combine the proposed method with the grid based particle method (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]) to obtain a numerical method for approximating PDEs on moving surfaces. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for advection-diffusion equations and Cahn-Hilliard equations that are strongly coupled to the velocity of the surface are also presented

Spatially partitioned embedded Runge-Kutta Methods

We study spatially partitioned embedded Runge–Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory

Safety and Feasibility of Thoracic Malignancy Surgery During the COVID-19 Pandemic

Background: The coronavirus disease 2019 (COVID-19) pandemic has decreased surgical activity, particularly in the field of oncology, because of the suspicion of a higher risk of COVID-19–related severe events. This study aimed to investigate the feasibility and safety of thoracic cancer surgery in the most severely affected European and Canadian regions during the COVID-19 pandemic. Methods: The study investigators prospectively collected data on surgical procedures for malignant thoracic diseases from January 1 to April 30, 2020. The study included patients from 6 high-volume thoracic surgery departments: Nancy and Strasbourg (France), Freiburg (Germany), Milan and Turin (Italy), and Montreal (Canada). The centers involved in this research are all located in the most severely affected regions of those countries. An assessment of COVID-19–related symptoms, polymerase chain reaction (PCR)–confirmed COVID-19 infection, rates of hospital and intensive care unit admissions, and death was performed for each patient. Every deceased patient was tested for COVID-19 by PCR. Results: In the study period, 731 patients who underwent 734 surgical procedures were included. In the whole cohort, 9 cases (1.2%) of COVID-19 were confirmed by PCR, including 5 in-hospital contaminants. Four patients (0.5%) needed readmission for oxygen requirements. In this subgroup, 2 patients (0.3%) needed intensive care unit and mechanical ventilatory support. The total number of deaths in the whole cohort was 22 (3%). A single death was related to COVID-19 (0.14%). Conclusions: Maintaining surgical oncologic activity in the era of the COVID-19 pandemic seems safe and feasible, with very low postoperative morbidity or mortality. To continue to offer the best care to patients who do not have COVID-19, reports on other diseases are urgently needed

Differential evolution for the offline and online optimization of fed-batch fermentation processes

The optimization of input variables (typically feeding trajectories over time) in fed-batch fermentations has gained special attention, given the economic impact and the complexity of the problem. Evolutionary Computation (EC) has been a source of algorithms that have shown good performance in this task. In this chapter, Differential Evolution (DE) is proposed to tackle this problem and quite promising results are shown. DE is tested in several real world case studies and compared with other EC algorihtms, such as Evolutionary Algorithms and Particle Swarms. Furthermore, DE is also proposed as an alternative to perform online optimization, where the input variables are adjusted while the real fermentation process is ongoing. In this case, a changing landscape is optimized, therefore making the task of the algorithms more difficult. However, that fact does not impair the performance of the DE and confirms its good behaviour.(undefined

High-order spectral/hp element discretisation for reaction-diffusion problems on surfaces: application to cardiac electrophysiology

We present a numerical discretisation of an embedded two-dimensional manifold using high-order continuous Galerkin spectral/hp elements, which provide exponential convergence of the solution with increasing polynomial order, while retaining geometric flexibility in the representation of the domain. Our work is motivated by applications in cardiac electrophysiology where sharp gradients in the solution benefit from the high-order discretisation, while the compu- tational cost of anatomically-realistic models can be reduced through the surface representation. We describe and validate our discretisation and provide a demonstration of its application to modeling electrochemical propagation across a human left atrium