Positivity Criterion for Systems of Stochastic PDEs

Author(s):  
Messoud Efendiev
2018 ◽  
Vol 174 (1-2) ◽  
pp. 177-233 ◽  
Author(s):  
Anis Matoussi ◽  
Dylan Possamaï ◽  
Wissal Sabbagh

2020 ◽  
Vol 59 (3) ◽  
pp. 1485-1493 ◽  
Author(s):  
Zeliha Korpinar ◽  
Fairouz Tchier ◽  
Mustafa Inc ◽  
Fatiha Bousbahi ◽  
Ferdous M.O. Tawfiq ◽  
...  

2013 ◽  
Vol 67 (5) ◽  
pp. 776-870 ◽  
Author(s):  
Martin Hairer ◽  
Jan Maas ◽  
Hendrik Weber
Keyword(s):  

2012 ◽  
Vol 22 (09) ◽  
pp. 1250023 ◽  
Author(s):  
JOAKIM BECK ◽  
RAUL TEMPONE ◽  
FABIO NOBILE ◽  
LORENZO TAMELLINI

In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids.


2020 ◽  
Vol 21 (Supplement_1) ◽  
Author(s):  
N Kuznetsova ◽  
C Borguezan Daros ◽  
H Zanella ◽  
Q Ciampi ◽  
L Cortigiani ◽  
...  

Abstract OnBehalf Stress Echo 2020 study group of the Italian Society of Cardiovascular Imaging Background ABCDE-stress echocardiography (SE) may explore different aspects in the heterogeneous entity of known or suspected heart failure (HF). Aim To assess ABCDE-vasodilator SE in non-ischemic HF Methods In a prospective, observational, multicenter study, we recruited 428 patients with angiographically normal coronary arteries: 148 patients with HF and preserved (>50%) ejection fraction (HFpEF, Group 1); 100 with HF and mid-range (40-50%) ejection fraction (HFmrEF, Group 2); 180 with HF and reduced (<40%) ejection fraction (HFrEF, Group 3). A group of 75 healthy asymptomatic patients (30 male, 40%, age 57 ± 14 years) referred to testing for screening was also selected (Controls). All patients underwent vasodilator SE with dipyridamole (0.84 mg/kg) in 14 accredited laboratories of 5 countries (Argentina, Brasil, Russian Federation, Serbia and Italy). The ABCDE-SE protocol was adopted: A for regional wall motion abnormalities; B for B-lines (positivity criterion: stress ≥ rest for ≥ 2 points in a 4-site simplified scan available in 181 pts and 10 controls); C for left ventricular contractile reserve (LVCR) based on force (systolic blood pressure/end-systolic volume, positivity criterion: peak/ rest ≤1.1); D for pulsed wave Doppler-based assessment of coronary flow velocity reserve (CFVR) in left anterior descending coronary artery (positivity criterion: peak/rest ≤2.0); E for EKG-based assessment of heart rate reserve (positivity criterion: peak/rest ≤1.22). Results All positivity criteria, except A, were more prevalent (p<.01) in Group 3 compared to Group 2, in Group 2 compared to Group 1, and Group 1 compared to controls: see figure. In particular, a blunted heart rate reserve was found in 4/75 controls (5%), 27/148 pts of Group 1 (18%), 28/100 of Group 2 (28%) and 98/180 of Group 3 (54%). Conclusions ABCDE-vasodilator SE can help to identify the profound pathophysiological heterogeneity underlying a similar clinical presentation in patients with known or suspected HF with angiographically normal coronary arteries. These patients rarely show stress-induced regional wall motion abnormalities (A), but may exhibit pulmonary congestion (B), reduced myocardial functional reserve (C), altered coronary microcirculation (D) and cardiac autonomic dysfunction (E). Abstract 1675 Figure. The positivity rate of ABCDE criteria


2020 ◽  
Vol 178 (3-4) ◽  
pp. 1067-1124
Author(s):  
Massimiliano Gubinelli ◽  
Nicolas Perkowski

Abstract We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for their infinitesimal generators. It was known that the domain must have trivial intersection with the usual cylinder test functions, and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions. Using the new domain, we are able to prove existence and uniqueness for the Kolmogorov backward equation and the martingale problem. We also extend the uniqueness result for “energy solutions” of the stochastic Burgers equation of Gubinelli and Perkowski (J Am Math Soc 31(2):427–471, 2018) to a wider class of equations. As applications of our approach we prove that the stochastic Burgers equation on the torus is exponentially $$L^2$$ L 2 -ergodic, and that the stochastic Burgers equation on the real line is ergodic.


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