Solving SODEs with large noise by balanced integration methods

Isaak E. Solving SODEs with large noise by balanced integration methods. Bielefeld: Universität Bielefeld; 2018

Stochastic C-Stability and B-Consistency of Explicit and Implicit Euler-Type Schemes

Beyn W-J, Isaak E, Kruse R. Stochastic C-Stability and B-Consistency of Explicit and Implicit Euler-Type Schemes. Journal of Scientific Computing. 2016;67(3):955-987.This paper is concerned with the numerical approximation of stochastic ordinary differential equations, which satisfy a global monotonicity condition. This condition includes several equations with super-linearly growing drift and diffusion coefficient functions such as the stochastic Ginzburg-Landau equation and the 3/2-volatility model from mathematical finance. Our analysis of the mean-square error of convergence is based on a suitable generalization of the notions of C-stability and B-consistency known from deterministic numerical analysis for stiff ordinary differential equations. An important feature of our stability concept is that it does not rely on the availability of higher moment bounds of the numerical one-step scheme. While the convergence theorem is derived in a somewhat more abstract framework, this paper also contains two more concrete examples of stochastically C-stable numerical one-step schemes: the split-step backward Euler method from Higham et al. (SIAM J Numer Anal 40(3):1041-1063, 2002) and a newly proposed explicit variant of the Euler-Maruyama scheme, the so called projected Euler-Maruyama method. For both methods the optimal rate of strong convergence is proven theoretically and verified in a series of numerical experiments

Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes

Beyn W-J, Isaak E, Kruse R. Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes. Journal of Scientific Computing. 2017;70(3):1042-1077.This paper focuses on two variants of the Milstein scheme, namely the split-step backward Milstein method and a newly proposed projected Milstein scheme, applied to stochastic differential equations which satisfy a global monotonicity condition. In particular, our assumptions include equations with super-linearly growing drift and diffusion coefficient functions and we show that both schemes are mean-square convergent of order 1. Our analysis of the error of convergence with respect to the mean-square norm relies on the notion of stochastic C-stability and B-consistency, which was set up and applied to Euler-type schemes in Beyn et al. (J Sci Comput 67(3):955-987, 2016. doi:10.1007/s10915-015-0114-4). As a direct consequence we also obtain strong order 1 convergence results for the split-step backward Euler method and the projected Euler-Maruyama scheme in the case of stochastic differential equations with additive noise. Our theoretical results are illustrated in a series of numerical experiments

Correlation between Landforms and Ground Deformation at Nisyros Volcano (Greece)

Relief represents a major element in the characterization of a landscape that can be affected even by slight modifications on its shape, often due to ground deformation. These terrain characteristics are both linked to the topography of a given area, through a Digital Elevation Model (DEM). Topographical attributes, such as slope, aspect, elevation gradients, and others, can be provided by DEM analysis and used further as input to classification methods for defining terrain units. In this study, we explore the relationship between ground deformation and landforms in Nisyros volcano, as ground deformation seems to follow existing geomorphological patterns. Possible correlations between morphological information, collected by classifying landforms on the basis of the ~30 m spatial resolution ASTER Global Digital Elevation Model, and deformation observations resulting from Differential Synthetic Aperture Radar Interferometry (DInSAR) are performed, taking also into consideration the geostructural setting of the study area. Nisyros volcano, located at the eastern part of the active Hellenic Volcanic Arc, was investigated for the time period 2002-2010, during a dormant phase where the topography is mostly described by the geological/tectonic structure. In terms of DInSAR technique, the entire archive of ENVISAT images was used, in both ascending and descending mode, to derive the deformation rates. For the geomorphic consideration Tandem Analysis was implemented through a mixed classification procedure following a principal component analysis applied to local elevation gradients, extracted considering each pixel of the DEM and its nearest neighbours. The projection of the DInSAR results on the obtained factor spaces allows evaluating the homogeneity of the deformation in the pixels belonging to the same classes

Global impact of the first coronavirus disease 2019 (COVID-19) pandemic wave on vascular services

This online structured survey has demonstrated the global impact of the COVID-19 pandemic on vascular services. The majority of centres have documented marked reductions in operating and services provided to vascular patients. In the months during recovery from the resource restrictions imposed during the pandemic peaks, there will be a significant vascular disease burden awaiting surgeons. One of the most affected specialtie

Documenting the Recovery of Vascular Services in European Centres Following the Initial COVID-19 Pandemic Peak: Results from a Multicentre Collaborative Study

Objective: To document the recovery of vascular services in Europe following the first COVID-19 pandemic peak. Methods: An online structured vascular service survey with repeated data entry between 23 March and 9 August 2020 was carried out. Unit level data were collected using repeated questionnaires addressing modifications to vascular services during the first peak (March - May 2020, "period 1"), and then again between May and June ("period 2") and June and July 2020 ("period 3"). The duration of each period was similar. From 2 June, as reductions in cases began to be reported, centres were first asked if they were in a region still affected by rising cases, or if they had passed the peak of the first wave. These centres were asked additional questions about adaptations made to their standard pathways to permit elective surgery to resume. Results: The impact of the pandemic continued to be felt well after countries' first peak was thought to have passed in 2020. Aneurysm screening had not returned to normal in 21.7% of centres. Carotid surgery was still offered on a case by case basis in 33.8% of centres, and only 52.9% of centres had returned to their normal aneurysm threshold for surgery. Half of centres (49.4%) believed their management of lower limb ischaemia continued to be negatively affected by the pandemic. Reduced operating theatre capacity continued in 45.5% of centres. Twenty per cent of responding centres documented a backlog of at least 20 aortic repairs. At least one negative swab and 14 days of isolation were the most common strategies used for permitting safe elective surgery to recommence. Conclusion: Centres reported a broad return of services approaching pre-pandemic "normal" by July 2020. Many introduced protocols to manage peri-operative COVID-19 risk. Backlogs in cases were reported for all major vascular surgeries